These students don’t realize how lucky they are to have such an effective and passionate teacher. Really makes all the difference in the world to keep the students engaged whilst educating them.
Absolutely. I watch a lot of maths related videos on UA-cam because I find the subject fascinating, but watching this is the first time I've actually thought about getting back into learning it formally. One bad teacher can destroy a student's interest for years - as unfortunately happened to me - but one good teacher can turn it around and this guy is great. If he'd been my teacher, I'd have a maths degree right now.
Very few teachers will have the kind of enthusiasm to show students all the great tips and techniques in any subject. Most probably are just biting their fingers waiting for their paycheck to come every month 🤔
I love how he literally stopped his whole class for the sake of the one kid who had stopped paying attention, and the kid who told him to get off his phone sounded legitimately angry. I feel like this guy has succeeded as a teacher because of the fact that he has his students so engaged and he cares enough about the individual to stop the class to allow them to regain their focus. These are the teachers we need more of. Its always refreshing to see a teacher who genuinely cares and also tries to make learning fun, while at the same time trying to make the content he gives to his students more easily understood. He simplifies things as much as possible in order to benefit everyone in his class, and I think that deserves tons of respect.
Hi Dimitri, interested in math competitions? If so, take a look at ua-cam.com/video/rkzxdMFEEtw/v-deo.html and others in the Olympiad playlist. You will see a lot of tricky and somewhat complex problems to try. Hope to see you there.
Some of my teachers were like this too, some of them...not so much. If you pay teachers well and give them good working conditions you'll probably get better teachers. I have no idea how it is in Australia though (where this guy is from).
I love how he stopped the class for that one unfocused kid. Teachers who are passionate, kind, caring and engaging sometimes can be disrespected by students because he is, well, too nice. That's just how teenagers function sometimes, no judgements. Stopping the class and politely ask for attention can serve as a hint or "warning" to the students that while the teacher is kind and caring, he has standards and is to be respected. He then smoothly moves on without making the student too uncomfortable and impacting the flow of the class. This makes me think not only is he great at teaching, but also understands how students think and respects that.
I am 59 years old. I've had 100's of teachers in my life. This guy is about the best I've ever seen. Passionate, polite, patient & motivating! I hope his students in future classrooms appreciate, respect & award this guy! He deserves it!!!
@@Lazerboomtv I'm 31, I can say I had about 100 teachers. I dont really remember how many subjects I had in middle school, it was a lot, the number 9 pops into my head. I remember I had pretty much the same teachers from grades 1 through 4, then a whole new roster from 5 to 8 (my country's ed system has 8 years of middle school, then 3 years high school, then college, btw) then I went to a new school for grade 1 high school, then a new school for grades 2-3... Just that already goes over 100. Then technical school (sort of middle ground between high school and college, idk what it's called in NA) I had about 12 different teachers, then college where I had way over 20 teachers... I wouldn't be surprised if I had over 200 teachers in my lifetime, so 100+ is not unbelievable. I, too, like this teacher. Very nice way to explain limits and the students seemed to get interested when the calculator part rolled in, that's a sign of a good teacher. My calculus teacher in college was great, too. He explained limits in a very different, more complicated way since he went deeper in, explaining why the increase starts between .4 and .3 (euler's number) and some other rules that I honestly dont rememebr cause I dont really use limits in my field, but he had a great way of putting things that made it easy for you to visualize stuff, and he was great with little songs/mnemonics to remember stuff for tests that I still remember to this day.
“Now that’s a good question, and i don’t have an answer to that at least i mean i dont have one right now probably because i haven’t really thought about it all that much to be honest. There’s always reasons for everything. So maybe we could find out where they go.” Things done by just these sentence: -Appreciate student for asking means appreciating them for using their logic and also appreciates their courage to voice their logic. -Being honest to what that is not known by the teacher, rather than making up answers or saying “that is just how it works”. Admitting incapabilities or mistakes in certain area will not decrease your integrity as a teacher, rather it build trust with your student and increase your integrity as a teacher. -Rather than discouraging students and teach them not to overthink things too much, he took the other way around and told “there’s always a reason for everything” then leaves the sentence ends with encouragement that we could find out about it. Good job ed
@@landsgevaer Huh. You know, it never before clicked in my head that e is the tipping point when adding to the exponent scales faster than adding to the base.
x^x reaches a minimum when ln(x^x) reaches a minimum. This simplifies to d/dx x*ln(x) = 0, therefore x*(1/x) + 1*ln(x) = 0, so ln(x) = -1, so x = 1/e. #
@@tawfiqclick2481 / Yeah, no. And, even if they were, they're usually not that good at explaining why things are the way they are. After an entire year with two really bad professors, I was lucky to get a good one, who actually made me like math; I can assure you not everybody have the same lucky :\
Yeah I've had a bad math teacher only one year but that was one heck of a year. Literally got an amazing racer ext year and was s oring fulls. Also I'm not saying current teachers are bad it's just that he teaches a lot better than others
Aarsh Agrawal Absolutely correct, Aarsh. I was comparing Professor Woo with various teachers from my youth just as I saw your comment. He’s smart, engaging and appears to genuinely love mathematics. Those prime elements of teaching coalesce only rarely in real-world educators.
@@prototropo yeah and the best part was he didn't punish the student for not listening. That would completely ruin the spirit of learning. It's often saddening to see that someone who either doesn't understand or loses attention is punished thus making them act like they are listening just to not have to deal with the punishment and thus end up hating the subject and the teacher. Do this with multiple teachers and ultimately they're hating school.
Wow, I wish I had you as a teacher in my Math Class when I was young. Thank you Eddie for sharing this. I'm 66 now and still learning thanks to guys like you.
@@thunderboltcloud3675 if y = x^x then dy/dx = (lnx+1)x^x www.quora.com/What-is-d-dx-x-x-1 setting dy/dx = 0 yields lnx = -1 or x = 1/e. the function x^x has minimum value (1/e)^(1/e) = 0.692200627... at x = (1/e) = 0.367879441...
@Sol iloquy if y = x^x then dy/dx = (lnx+1)x^x www.quora.com/What-is-d-dx-x-x-1 setting dy/dx = 0 yields lnx = -1 or x = 1/e. the function x^x has minimum value (1/e)^(1/e) = 0.692200627... at x = (1/e) = 0.367879441...
I’m not saying anything you don’t already know, but this method of teaching is superb. Anyone can tell students that zero raised to the power of zero is equal to one. But you let the students discover it. They will never forget. When I was in grade school many(oh so many) years ago, my teacher told us to go home and measure the circumference of tin cans and the diameter and bring the results back to class. None of us knew why and none of us understood the meaning of Pi. We did the next day! Thanks so much for these lessons you post.
im a college graduate who already learned this throughout multiple classes and he still kept me engaged enough to watch a 14 minute youtube video in its entirety
He could have just said: "it's 1 remember it, it'll be on the exam" , and continued, but he's done more: he gave them meaning, passion, curiosity, knowledge. The fact people hate Maths, or anything for that matter, it's because they learn it from the wrong people. It's because some doctors, or business men, could have been good teachers and they chose to be something else to make the money they think they deserve. I hope this teacher continues to be the same and contaminate his students with his passion.
Dihcar bahari bhai,i am convinced with what u said, In this world there is a problem of money everywhere no one does anything with passion for it they just perfect something for earning and because of obvious pressure from society ,generally they are called middle class but it can be anyone post script i know u r talking about mathematics here but what i believe about the world i said that
Don’t entirely agree, alot of doctors chose to study medicine for the soul purpose on wanting to help people. Not saying that there aren’t doctors who do it for the money, but alot of doctors, as kids, probably dreamt about helping people for a career. Regardless of the money, majority of doctors are there because they want to be.
Your totally right, I'm an engineer so naturally I have had a lot of math teachers. Some of them fantastic, some of them so so and some of them made me hate life. At one point I thought I was terrible at math but it took one good teacher to show me I actually have a good mind for it. A good teacher can be the difference between understanding and frustration. A good teacher can teach anyone regardless whether they have the mind for it or not and a bad teacher can make someone who does have a mind for it hate the subject.
kind of a teacher i had when i was student. great teacher. The trick to understand this is to rewrite the equation : x^x = (e(ln(x))^x = e(x*ln(x)), then you can derivative etc to study the function (minimum, limit etc)
My favorite subjects in high school was the higher abstract math like algebra and geometry because I had interesting and engaging teachers who were passionate and made sure everyone understood the concepts before going on to the next exercise. Also gave real-world examples of practical uses for formulas, like principle to interest ratios and amortization schedules.
13:56 For anyone wondering why this number starts increasing again specifically between .4 and .3, it's because the minimum value of x^x (x to the x) happens at x = 1/e, aka. the inverse of Euler's number, aka. ~.36788, which, as you would notice, is between .4 and .3. You're welcome.
@@joemarshall4226 Follow the spirt of the lesson and plot it for yourself ! Normal exponential behaviour i.e rapidly rising curve above x=1, then a v shape between x=0 and x=1. For x < -1, a flat curve that gradually rises from -1 towards 0 as x gets increasingly negative. The bit between x=0 and x = -1 is left as an exercise for the reader.
Yes, you're absolutely right. I noticed this before I read any of these comments. As I started watching this video, I generated a list of decreasing numbers by using MS Excel. In column A, I made a list of the positive integers in increasing order. I made each entry in column B to be the reciprocal of the corresponding entry in column A, and each entry in column C to be the value in column B (of the same row) raised to the power of itself. Seeing that the value in column C was indeed the same for the 2nd row and the 4th row, i.e., (1/2)^(1/2) = (1/4)^(1/4) = 1/sqrt(2) = half of the square root of 2 (which I already knew it would be), I created 9 rows between 2 and 3, and put in column A 2.1 through 2.9, and carried the equations for columns B and C into the corresponding rows; then, seeing the results in column C, I inserted 9 rows between 2.7 and 2.8 and placed 2.71 through 2.79 in column A, and repeated the process for the new rows in columns B and C; then, seeing the results in column C, I inserted 9 rows between 2.71 and 2.72, and placed (in column A) 2.711 through 2.719, and finished that step with the corresponding entries for columns B and C. It was here that I got the idea that the number in column A that produced the smallest value in column C might be Euler's Constant (e). I continued this process until I had the smallest value in column C when the value in column A was 2.7182818. But then I computed the first derivative of x^x = (x^x)(1 + lnx), and setting this equal to zero, 1 + lnx = 0 and lnx = -1 and so x = e^(-1). So, yes, the minimum value for the function f(x) = x^x is found when x = (1/e). And yes, the limit of x^x as x approaches zero is 1.
Yeah man, most people don't acknowledge they don't know everything and being a teacher doesn't mean you have to know everything, and he knows that and embraces that.
0^0 is an interesting one in that math textbooks and math teachers will usually insist that it's undefined (because if you take x^y, what you get in the limit x,y -> 0 depends on what path you take through the x-y space), but professionals in most contexts just take it to be 1. If you think about it, every time you write a polynomial or power series using a summation notation over powers, you're implicitly assuming that x^0 = 1 for all x; otherwise you'd need to include a special case for x=0 every time you did this.
Great comment! I also want to add a bit of a historical note. In the 1700s, pretty much every mathematician accepted 0^0 = 1 as true. So how did we go from 0^0 = 1 is true to so many people shouting that 0^0 is undefined? It's because of the switch from infinitesimals to limits. In the late 1700s and throughout the 1800s, mathematicians were becoming wary of the way infinitesimals were used in Calculus. In the early 1800s, Augustin-Louis Cauchy introduced the notion of the limit of a function as the input approaches a certain value. So instead of plugging in infinitely small values, they could now take the limit as the input tends to 0. Cauchy also included a list of indeterminate forms. This list of indeterminate forms included 0^0, and Cauchy was right to include it. 0^0 really is an indeterminate limiting form. This caused the mathematical community to have a bit of a panic concerning the arithmetical value of 0^0 because 0^0 was an indeterminate limiting form, so they un-defined 0^0. No longer was it considered 1; it was now undefined. But this panicked move was not based on sound reasoning. 0^0 being an indeterminate limiting form means that if f(x) approaches 0 and g(x) approaches 0, then knowing this information is insufficient to tell what f(x)^g(x) approaches. The limit of f(x)^g(x) could, conceivably, be any nonnegative real number, positive infinity, or not exist. But the important thing about limits is that *_limits do not have to agree with the function value._* The limit of f(x) as x approaches c could be different from f(c) itself. So just because 0^0 is an indeterminate limiting form does not imply that 0^0 must be undefined as an arithmetical operation. With more development into discrete mathematics, abstract algebra, universal algebra, set theory, and category theory, we know now that 0^0 = 1 will *always be correct* in the context of discrete exponentiation. What I mean by this is the following: if you ever have a formula involving exponents where the exponents represent repeated multiplication, if you can get 0^0 from that formula, the formula will give the right answer (for what the formula is supposed to represent) if and only if 0^0 is evaluated as 1. It's not an accident. It's not a happy coincidence. There's a perfectly reasonable mathematical idea which shows us why this will always work (the empty product). But unfortunately, the mistaken reasoning from the 1800s is still perpetuated to this day.
@@MuffinsAPlenty _"if you ever have a formula involving exponents where the exponents represent repeated multiplication"_ Isn't that equivalent to saying "the exponent is a non-negative integer"? Well sure, if we're only going to deal with non-negative values of x, then we only need consider the limit as x tends to 0 of x^x coming from above, and that's well-behaved and equal to one. But what about the cases where the exponents _aren't_ all positive integers? Then the value of 0^0 doesn't exist, so perhaps the reasoning from the 1800s was sensible after all?
@@RexxSchneider Why would you say that 0^0 in the context of discrete exponentiation should match up with lim(x→0+) x^x? Why not match it up with lim(x→0+) x^0? Why not match it up with lim(x→0+) 0^x? Why not match it up with lim(x→0+) x^(-1/ln(x))? The issue in the 19th century reasoning is pretty much a categorical error. It's saying that f(c) can't be 1 because lim(x→c) f(x) isn't 1. It's a fundamental misunderstanding of what limits mean and what their purpose is. The definition of "lim(x→c) f(x)" intentionally ignores f(c). Arithmetic is different from limits. It would be, in no way, a contradiction for 0^0 = 1 to be true while lim(x→0+) 0^x = 0 is also true. All this means is that 0^x is not continuous from the right at x = 0. Here's another example for you. Let ⌊x⌋ denote the greatest integer less than or equal to x, when x is a real number. By definition, ⌊0⌋ is the greatest integer less than or equal to 0, so ⌊0⌋ = 0. On the other hand, you could consider things like lim(x→0) ⌊x⌋, which doesn't exist, lim(x→0) ⌊x^2⌋ = 0, or lim(x→0) ⌊-x^2⌋ = -1 From this perspective, we could say that ⌊0⌋ is an indeterminate form. If you have a limit, and evaluating the function at the point in question gives ⌊0⌋, this is not enough information to determine the value of the limit. No one in their right mind would use this justification to say that ⌊0⌋ must be undefined. But that's the reasoning that 19th century mathematicians used to un-define 0^0. Just because not every limit gives 1 when you approach 0^0 doesn't mean that 0^0 itself must be undefined. Now, you could get into discussions about how when we move from rational number exponents to real number exponents, we need to use some analytic tools in order to define exponentiation. And sure, depending on what your definition of exponentiation is, 0^0 is probably undefined. (But also, depending on what your definition of exponentiation is, your definition of 0^2 might also be undefined. For example, z^w = exp(w*log(z)) leaves all powers of 0, including things as simple as 0^2, undefined.) Additionally, in pretty much every situation, analytic exponentiation is an extension of discrete exponentiation. For example, if you go the exp(w*log(z)) route, you typically would define exp(x) as a power series, which involves discrete exponentiation of x. So you are building up from the definition where 0^0 = 1 is the only way to go. To make things clear, if you want to study continuous functions, un-define 0^0 in that context. It's a good choice in that one specific context. But it shouldn't be seen as the correct thing to do generally. We shouldn't view 0^0 as actually undefined, except in any practical application, where 0^0 is defined as 1 for convenience; rather, we should see 0^0 = 1 except in the one instance where it's convenient to un-define 0^0. I would also argue that the 19th century reasoning causes problems with students understanding indeterminate forms in general. Particularly things like 1^∞. A lot of students will be very confused about 1^∞ being an indeterminate form. They will view 1^∞ as 1*1*1*..., and they can't see how 1*1*1*... can possibly have any value other than 1. And there's something to their confusion. In any context where 1^∞ can be viewed as actual arithmetic, either as 1*1*1*... or as 1^α where α is a transfinite number (e.g., cardinal number, ordinal number, hyperreal number, surreal number), the value is always 1. *_But_* if f(x)→1 and g(x)→∞, this isn't enough to tell us what f(x)^g(x) approaches. And that's because limits are not the same thing as arithmetic. I think a nice way to see my point is to try defining "indeterminate form" without simply writing a list of indeterminate forms.
@@MuffinsAPlenty Thanks for the wall of text. Did you read what I wrote? I'm pretty certain that I stated quite clearly "if we're only going to deal with non-negative values of x, then we only need consider the limit as x tends to 0 of x^x coming from above, and that's well-behaved and equal to one." Then you proceed to spend reams of text arguing that if we only deal with non-negative values of x, we can define the value of 0^0 to be 1. *I already said that was the case.* Anyway, the reason we don't take f(x)=x^0 or f(x)=0^x is that those give differing values as x tends to zero, so they are not the function we're dealing with, which is f(x) = x^x. You tell me that "Just because not every limit gives 1 when you approach 0^0 doesn't mean that 0^0 itself must be undefined." But that's exactly what an indeterminate form is. The problem is not that we don't define 0^0, it's that the process of taking a limit isn't leading us to an unambiguous value. Surely you understand the difference? 1/0 is undefined, but 0/0 is indeterminate. The 19th century reasoning is comes about because they actually understood limits, and defined a limit as applying to functions that are continuous in the region containing the limit, which x^x is not in the neighbourhood of x=0. The whole point of having a limit to stand in place of f(c) when f(c) is arithmetically indeterminate is that it preserves the continuity of f(x) when x passes through the region containing c. When x is discontinuous in that region, then you can't use a limit. Yes, of course, you can choose to assign the value 1 to 0^0 when x is a non-negative integer, because x^x is a continuous function for x>0, but that's just a choice made for convenience; there's nothing fundamental that requires 0^0 to take on that value. I understand you take the opposite viewpoint, and assert that 0^0 has the value 1, except when we choose to call it undefined. I do view 0^0 as undefined, except for those cases where it is convenient to treat it as 1, and I maintain that the latter is an artificial choice.
@@RexxSchneider I think you _might_ under the impression that the definition of discrete exponentiation doesn't assign a value to 0^0. And if you have that impression, I understand why you would view 0^0 = 1 as artificial. After all, we have the original definition, then we extend using algebra, and then we extend using analysis. Now, if you believe that 0^0 isn't defined, then using algebra to extend the definition of exponentiation won't pin down a unique value of 0^0 (this is probably what you mean by "arithmetically indeterminate"). And using analysis to extend the definition of exponentiation won't pin down a unique value of 0^0 either (because 0^0 is an indeterminate limiting form). And if it were the case that the original definition didn't assign a value to 0^0, I would agree that this is compelling enough reason to declare 0^0 not defined and all places where 0^0 is replaced with 1 to be artificial context-dependent patches. But the original definition of exponentiation as repeated multiplication _does_ assign a value to 0^0. For a nonnegative integer n, x^n is the product with n factors where each factor is x. In the case that n = 0, we have the product with no factors (each factor being x is vacuously true). This is known as the empty product, and it has a value of 1. Now, the empty product is a choice, and you may consider this to be artificial. However, what I consider to be important here is the reason that we assign the value of 1 to the empty product. And that's because, if we expect the empty product to be consistent with the generalized associative property of multiplication, it _must_ have a value of 1. If we assigned the empty product any value other than 1, the generalized associative property of multiplication would be false when empty products were involved. Knowing this, 0^0 is the empty product, and thus, has a value of 1, right from the definition of exponentiation as repeated multiplication. Now, the generalized associative property of multiplication is extremely important for exponentiation. It's the entire reason why exponentiation is even well-defined in the first place [is x^4 equal to ((x∙x)∙x)∙x or (x∙(x∙x))∙x or x∙((x∙x)∙x) or x∙(x∙(x∙x)) or (x∙x)∙(x∙x)? It doesn't matter because they're all the same thanks to the generalized associative property of multiplication]. The generalized associative property of multiplication is how we prove the most important property of exponentiation: x^n∙x^m = x^(n+m), which we use in pretty much every extension of the definition of exponentiation. The most important property of exponentiation is used in pretty much every formula involving discrete exponents. This is why I can have confidence to say 0^0 = 1 will always work in any context where we have discrete exponents. The reasoning that gives us 0^0 = 1 is the same reasoning that makes exponentiation well-defined in the first place and gives us the workhorse of exponentiation, which we use to extend the definition and use to develop every formula using exponentiation. From this perspective, 0^0 = 1 has the same amount of justification as 0! = 1. When we consider the empty product, 0! = 1 pops right out of the original definition of factorials. Factorials are only well-defined because of the generalized associative property of multiplication, the most important property of factorials is that (n+1)∙n! = (n+1)!, and this is proven using the generalized associative property of multiplication. Every formula involving factorials is built up with this most important property. Extending the factorial function to the Gamma or Pi function uses the most important property of factorials (since 0! = 1 can be justified using (n+1)∙n! = (n+1)! alone, the analyticity and log convexity of the Gamma/Pi functions have nothing to do with 0! = 1). And every argument used to "justify" that 0! = 1 should be true uses the generalized associative property of multiplication. I doubt you're going to be telling me that 0! is actually undefined, but 0! = 1 is an artificial patch which always works in any meaningful situation. But hey! You might tell me that. So that's the story. 0^0 = 1 falls out from the definition of exponentiation at its most basic level. When we extend the definition of exponentiation using algebra, keeping 0^0 = 1 doesn't produce any contradictions. So while algebra wouldn't be able to pin down a unique value for 0^0 if 0^0 weren't already defined, there are no issues with keeping the definition what it originally was. The same thing is also true for extending the definition of exponentiation using analysis, with one small caveat. The fact that 0^0 is an indeterminate limiting form does not contradict 0^0 being defined as an arithmetic expression (as my previous post argues). The caveat is that, depending on your definition of exponentiation using analysis, it's possible that all powers of 0 are undefined. In such a case, 0^0 would, indeed, be undefined using that definition, but so too would things like 0^1 and 0^2. This is why I say that analysts un-define 0^0. And sure, it's not a bad decision if you're dealing with continuous functions (unless you're only dealing with analytic functions, in which case you want 0^0 = 1 again). And this is perhaps why it felt like I ignored your previous post when you said "if we're only going to deal with non-negative values of x, then we only need consider the limit as x tends to 0 of x^x coming from above, and that's well-behaved and equal to one." It's because this suggestion doesn't make any sense when we're talking about discrete exponentiation. Limits don't make sense when talking about discrete exponents. And maybe you believe there isn't a discrete way to make sense of 0^0, so taking the limit of a particular function is the only way to justify a value of 0^0. But that's not correct.
The best part is when he answers a student's question with "I don't have the answer right now, but we could find out." So important to show that it's okay to not know things.
@@richardchurch9709 He knows what's going to happen and why, It isn't just coincidence he drew up the table with 7 boxes on the left side to hold the decreasing values then increasing on the right. It is important for a good teacher to teach students that no one knows everything but you can find out. and as Bedër Butka, pointed out the answer is quite simple and he would already know it and why it is so.
Yes, it is important to be honest. However, a trained mathematician should know the properties of such a simple function, it is inappropriate to say "I don't know" here. People who are not good professionals should not even teach children.
I'm so happy that the student asked why it turned around, because upon seeing what you were doing, and after plugging the problem into desmos, I worked out that because anything to the power of a fraction is basically just taking the root of that number, and the square root of a number less than one will get larger, therefore eventually there is a threshold where the number gains more from the root than it loses from the base.
But not too often... And they better go and check it up to have an answer by the next period ! But I sure agree that claiming omniscience is a disservice to your student.
Same here. Some of my best math teachers were the ones that were actually excited about the subject and the concepts they were teaching at that time. Wish there were more like him.
a possible cause for maths teachers losing enthusiasm is that after, probably, doing higher level, more advanced math, going back to teach relatively basic concepts doesn’t bring any joy. it’s like secondary school students revisiting 2x3 (the general logic behind it, not the calculation specifically).
12:19 Student asked a great question. Eddie only calculated lim_x->0+ of x^x = 1. The lim_x->0- of x^x = 0-j1. I think it's best if you treat this as z = x^y. Depending on the direction of the limit you get different answers. So wouldn't that make it undefined? (similarly to 1/0)
I really like his teaching style. He's very engaging and honest enough to say "I don't know, but I can find out." My understanding of maths is too basic to really appreciate the lesson though.
You're probably not as far away from understanding as you think. The equations on the board at the start are just giving you the notation, The top line is telling you that 'this is short form for that'. a^m = a times a times a times a, m number of times. Next line I will just give an example if a = 2 and m = 3 and n = 2, that means (a^m) x (a^n) = (2x2x2) x (2x2) which equals 2^5 = a^(m+n). I don't know if that helps, maybe, maybe that wasn't even something you were stuck on haha. But don't give up on understanding if you want to get better at math, I always found the "That looks too complicated" feeling to be my worst enemy when learning it.
@zerere I moved around a lot as a kid, roughly every two years. I never got my head around basic maths, and because my level was poor, I got put into a class with all the dropouts. It is interesting, though, and he is a good teacher.
Actually zero to the power of zero can't be defined. However, if you take the limit of x to the power of x with x approaching zero you end up with 1. Thats not to say that zero to the power of zero is exactly equal to one. The correct answer is that it can't be defined.
@@ziyanoffl you, a boy/girl of today's era don't know how was the life when this phone u and me are holding was not there, these curiosity questions arousing in a child can not be answered by google,these are basically learnt from surroundings, and teachers come in these surroundings you don't know,it was an altogether different feeling to ask question from teacher when teacher gave answer it gave satisfaction to a student so please if u don't know the feel of the situation just stay away from the situation and don't comment
I'm experiencing being math teacher for the first time. I did just 6 months yet but it was enough to realize my own weaknesses and strengths. This guy not just only saying the right words. He uses extremely perfect tempo and intonation. He makes you believe you understand and since you believe that you already do understand. Amazing! His communication and way of explanation and confidence is perfect.
I've been a teacher for many years and I've trained and mentored many teachers. There are many many ways to be a good teacher. But here's my tips. Understand your own persona and play to your strengths. Have a clear goal and a well thought out route to that goal. Keep up the pace. Be respectful and demand respect in return. Be kind but be the boss. Enjoy leading your students down the path you've chosen. But you're absolutely right...let the students see the answer just before you "show" it to them.
His tempo is actually very fast. It is good for the lecture, but not good for the actual learning. Believing in understanding something isn't the same as actual understanding and reproducing results. And of course you need to consider the age of pupils. Speed of understanding the material isn't the same at 13yo, 15yo, 17yo or 20yo students...
One thing I like about his videos is the way he can recover from making a mistake or a poor notational choice without getting flustered or losing the trail of the argument. That is real skill.
I love how the students in this class actually answer the teacher's questions....my class is usually dead silent with a few people murmuring the answer. I guess that's what you get for constantly telling kids to shut their mouths since early primary school.
I feel the same way dude. I'm not a professor, but I am a student (and at school 12th grade is last year), and I understand. When I was in 4th-10th grade/12, I was in a school were classes were really distracted off the subject of learning due to bad student behaviour, but I was always siding with the teacher while listening intently to the lectures. When I entered 11th grade at another school, the classes seemed to be too silent, and I was the only person shouting the math, Bio, chemistry... answers aloud(due to my passions), which was kind of disrupting at first, but because I'm clever (answer most questions correctly), I didn't receive much warning(not more then 2) last year 11th year. And now same as last year, I'm in grade 12/12th grade, and I'm still being the only speaker in class which is still weird, but I will hang on to being active in class. Proud to see professors somewhat advocating for more of this.
@@solcarzemog5232 Correct. In this case it is a switch between the predominance of the the value towards its factor. Then you wonder why you deposit zero dollar to the power of zero in your bank, and the stupid clerk does not give you dollar. Could take time to convince him.
as a math major I'm not satisfied with this answer. There are many things where the limit does not equal the value. It could have been the answer that the limit was 1 but the true value was undefined or possibly 0 but maybe it can't be that I forget. After looking it up, it says that this is a very rare case where it depends on the context. For us viewers, it would have been nice it they went over that because this gives us the false impression that the value always equals the limit.
I noticed that too. I interpreted this as a slightly self-conscious gesture, the act of a man who is perhaps a little shy by nature but very much into delivering a lesson - an attempt at nonchalance. While I'm commenting I should note also that I like the video a lot. I myself am a retired secondary level math teacher, for whatever that might be worth.
@@grinpick You: giving an insightful little psychoanalysis based on that one action Us: yo he frickin yeeted that thing! Your career experience is definitely worth something!
You're an amazing teacher! The moment at 10:38 when the guy in the class room says "I've done the next one" without you even asking for it which is proof that you have gotten them interested and invested in finding the answer together with you and started to take their own initiatives. Great work!
The biggest problem with math and the way it's taught in schools is that they tell you so much "this will be useful, you will need to know this, here are some obscure word problems to show you that you need to know this, blah blah blah..." I promise you, you will probably never need to know how to graph a parabola in your lifetime, assuming you go into any sort of normal career (obviously STEM is different). And if you do know, you can google it. So why learn it? Well, this teacher clearly has the answer. Math teachers *should* teach math for the same reasons why your English teachers have you read classic literature - it's good for you. Math is the study of logical truth, of taking a set of rules and carrying them out to the letter and seeing what you get, then finding out what that answer means. English teachers don't have to tell their kids: "You will need to know what the major themes of the Great Gatsby", because they definitely won't, they just have to tell the kids that it's good for them to learn these things. Because it is good for them, and a good English teacher will make the kids love English, which will be good for their minds and for their souls. Kids are taught to hate math because they're taught it like it's something they'll need to know, something that's boring but hey, you gotta learn it. Because it's math and you just should know it. That's a horrible reason to learn math; no wonder kids hate it! Math is good because it sharpens the mind, not because it will help you count your groceries. Math is good because it is the study of truth, and logic, and how numbers are in the world around us interacting with a set of rules that are constant, but maluable. More than that, it's just plain *fascinating*. People need to learn what math really is, not what common core is teaching them.
The fundamental issue I, personally, have with teaching stuff that "sharpens the mind" isn't the "sharpening of the mind"-part. It's how these, technically, "useless" subjects end up determining your future prospects. Depending on where you live, your average will determine whether you're even allowed to apply for that dream-education you've always fantasized about - and convoluted subjects, that require you to memorize pointless info, just make that journey more difficult and more frustrating, for any aspiring student. Math tends to be one of those roadblocks - as it's a heavy subject by nature, requires extensive dedication and time investment to grasp, is demanded by virtually all STEM fields, yet is only really necessary for a few of said fields, and usually, their mathematical demands are highly specific anyway, to the point where most of the math you've been taught never had any relevance to your field of choice in the first place. Having said that, it goes both ways - if you're an aspiring mathematician, and you love the subject, why should your knowledge of history, or your ability to analyze the "major themes of the Great Gatsby" have an impact on your future? It's very demotivating, to HAVE to grind away at a subject you know won't matter to you; you know, you don't care about; you know you're forced to sit through, not because of necessity, but just so the educational system can "weed out" those it deems unworthy. Kids hate math, not just because of how it's presented, but because of how it seems like a massive man-made hurdle, that they are told they HAVE to overcome if they want the "good jobs" - they don't ask, "Eh, do I really need to know this? Does the hill I have to climb really have to be this steep...?" just because their teacher is boring, but because they genuinely don't care to know or need to know, how to graph a parabola.
Teachers that don't speak with energy and excitement often hate what they do, which is why you know Eddie will teach you something, it's because he wants to not because he has to.
I taught maths for 33years and this guy engages me. He knows his stuff, is relaxed and thorough in his presentation taking time to explain every detail. He is a rare species in todays classrooms.
When I was in 9th grade my first advanced math class was an Algebra class taught by a lady who had no patience for questions and she would chalk equations on the board so quickly and impatiently that debris and dust from the chalk flew everywhere. I foolishly thought that I hated math. If I had a math teacher like this man then the trajectory of my life would have been very different.
Rama G I had the same experience, Rama, and dread thinking that same possibility about having fewer life choices as a result. I believe you are right. Nevertheless, even as a curious adult, algebra engages my brain like a piano falling from space, whereas geometry has from 5th grade been utterly logical and lovable, and still is. Since I never needed algebra in my life, but have had regular cause to calculate things geometrically, maybe things worked out. Serendipitously. But in an ideal world, everyone would have great teachers for every academic discipline.
Reee Flex Well, one thing is for sure-it wouldn’t help anyone learn something new to tell them they’re moronic, with a low IQ, and “that can’t be fixed.”
Despite being a 3rd year maths undergrad and already knowing everything said in the video, I still watched the whole thing. This is a really fun way to introduce limits to the students. Maths is such a beautiful subject and I really think if more teachers were like you a greater proportion of students would take an interest in it. Hats off to you
I still don't know how don't they know the limits already tho, it seems like they're in college I first was introduced to it at highschool that's weird
Still don't understand how 0^(a number that has infinite 0s followed by a 1) still = 0, but then decreasing that by an infinitieth changes it entirely 🤔
Im 47 now and I have completed my high school 32 years back. I'm now working on a completely different field. But Why am I listening to his lectures again and again?
As someone who teaches college and grad school, just let me say that "Oooo" and "What?" @7:25 is an absolutely beautiful sound. Those are the moments for which we teach.
He's an amazing teacher, there's no doubt about that; but I think some of those sounds might have been sarcasm by some disrespectful student, or not; we may never now for sure.
I have been teaching for 40 years and I know a good teacher when I see one. You can have all the academic theories of pedagogy you like but if the teacher a) is obviously personally interested in what he/she is teaching b) is constantly asking doable questions from the class c) has a bit of humour d) shows interest in the class as people then you will not go far wrong. I try to teach like this and succeed sometimes. The hard part is the fact that doing this for four or five hours a a day for five days is incredibly tiring. Most other jobs have down time during the actual doing of the job - teaching does not. This man is good but he will be utterly exhausted at the end of the week.
Frank Gibson - agreed. And the most exhausting classes are the non-Honors classes that don't want to learn. I am practically yelling the lesson just to talk over them.
They are not shitty and you are teaching people not chemistry. Take time from the chemistry to get to know them. Aim that each lesson should be an individual dialogue with each student and make eye contact with every student in the room at least once every few minutes - that is what Eddie Woo is doing. Talk about chemistry in their lives, tell them about experiments that went wrong, discuss pollution with them. I call it teaching by stealth - or teaching in the cracks. It may sound like taking time away from teaching but in fact you are building the bridges that will allow communication in the future.
OK learning should be cooperative. The reason I have stuck with teaching for 40 years is because I like to learn and understand stuff. If as a student you are genuine about wanting to learn and you are sure that your teacher has a good understanding of the topic then get the ones together who want to learn and form a cooperating group with the teacher. Ignore the dickheads or even better tell them full up front to fuck off and stop wasting everybody's time. They will either come on side or leave. This is a win win for everybody concerned.
@@ewaldatok611 Not every teacher connects with every student and that’s a giant flaw in a lot of education systems. Asking for a better teacher isn’t the same as asking them to do the work for you. You can be really hard working but never understand a certain subject in school until a certain teacher finally lets you understand it. Just because you understand a certain subject well doesn’t mean every other student on the planet does and some need more help in certain areas than others. I hope you understand that the end goal of school is to make sure that every kid understands what they are learning and not just grading assignments.
Imagine being that girl who got embarrassed Infront of millions of people and having one of the most happiest and intricate teachers, who actually loves what he does.
Johnson Pootisman you get a break twice a day and two days a week. It’s school after all, not a fuckin nursery. Don’t go to school if you’re that mentally tired, it’s not the teachers fault you only slept three hours because you were playing a game or some shit.
Mathematicallly 0 power 0 is undefined, maybe for convenience we can keep it as 1. If you apply the logic of factorial which same teacher explained, 0 power0 becomes 0. It can lead to different results and subjective.
I have taken math from arithmetic to calculus, and I have had a good many instructors. They mostly just regurgitated mathematical facts, as if some kind of mantra. This guy goes to the trouble reaching beyond just the facts, and explained the concept behind the facts. This promotes understanding, instead of just memorization (like memorizing a phone number), which puts the information into the logic portion of the brain. Even after you forget the equations, you can regenerate them by applying the logic of the concept. If all teachers used this method, or a similar one, people would not hate math, which is so common!
This was so mysterious to me I couldn't help continuing with my calculator myself after watching. I needed to locate the exact place the function started growing again... and I did, and it's at... *drum roll* ... 0.3678794412 !!! Oh well, I thought... So I thought, naaah, there's got to be something about this, it can't be just another accidental weird decimal number... so I tried to do 1/e just to see if ... and there it is!!! In other words the function has its lowest value at (1/e)^(1/e). Math is such a beautiful mystery :-)
Thank you all for your interest :-) e, or Euler's number, is a mathematical constant, a so called irrational and transcendent number - in class with the number pi. e = 2,7182818284590452353602 (approximately) en.m.wikipedia.org/wiki/E_(mathematical_constant) Peace and light to you
@@list9016 unfortunately, I realized this after graduating high school and my major has no relations with mathematics whatsoever , but I can still enjoy his videos so maybe, it's not so bad after all
@@bao6287 dont worry. studying is a continuous process. it wont stop until your brain cells starts to decline in numbers (as you start losing memory, motor functions, cognition aka grow old) its only a matter of whether you wanna accept new information or not. you can say studying has an exponential growth so past a certain level of information you will start to see things a lot more clearly and you will pick up new informations more quickly and that often results in information overload which is why many geniuses out there have very peculiar traits or hobbies to distract them from information overload.
I note that he openly accepts comments from his students. In the US, at least high school, does not want you interrupting the teacher ever. Most teachers there will make you raise your hand or won't even acknowledge you if you don't. Love his teaching style.
The fact he has his class so engaged, and the fact he is happy to find out a students questions that he doesn't know, really gives me hope for the future of those kids in his class.
Spaghettificated Before you do that, just approach it from the left side (negative side), you get -1 so 0^0 does not = 1. Also, if you take 0.000000000000000...1i^0.000000000000000...1i, you get 1/infinity or if you take -0.000000000000000...1i^-0.000000000000000...1i = -1/infinity.
Regarding the question @13:50, the "turnaround point" (i.e. what's the number in the first column when the second column starts to increase again) is 1/e or 0.3678....
There's a really lovely result for the question asked by the student at the end (i.e. "why does it change at around 0.4?")... If you differentiate y=x^x you get dy/dx=x^x(ln(x)+1) (Don't worry if this isn't obvious - this is not an 'in your head' result... it's a bit of a slog to reach it). The inflection point is of course the zero of the derivative. So where does dy/dx = 0? The zero happens to occur at around 0.4 - just as predicted from the graph. But it's not just vaguely near 0.4 ... the zero actually occurs at exactly 1/e (or approximately 0.367879441). 1/e huh. Isn't maths just one mindblow after another? Thanks Mr Woo.
Same, although he didn't know it was the gate to the deeper hell when he asked. I think the teacher resisted the urge so hard to drag him into the neutral log (ln) and neutral number and derivative stuff, as he just told them the idea of limits which is the foundation of differentiation and integration.
ahh. i differentiated it too. but came out with f'(X) = X^1 * X^(X-1) = X^X. (i havent learnt log stuff yet) however i will do soon. is there a short explanation as to why mine is wrong?
I don't think you can do it without taking logs of both sides first (ok - maybe you can, but I can't :) y=x^x ------(1) Take logs of both sides you get ln(y) = ln (x^x) = x ln(x) So then if you differentiate both sides wrt x you get 1/y dy/dx = ln(x) + 1 [on the left hand side you use the chain rule to get d/dx ln(y) = 1/y dy/dx] [on the right hand side you use the product rule... u (dv/dx) +v (du/dx) ... to get d/dx x ln(x) = ln(x)+1] Then if you rearrange 1/y dy/dx = ln(x) + 1 you get dy/dx=(y).(ln(x)+1) -----(2) ... but from our initial point (1) we know y = x^x So you substitute (1) back into (2) you get dy/dx=(x^x)(ln(x)+1). Then you find where that is equal to zero and you're done :)
Short version: because when you do differentiation, you need to do the process on every single variable and individuals too. for example: X^2+x+1 become 2X+1+0 , therefore you cannot not treat the power X like a constant.
2:08 "0 to the power of are you paying attention?" Teacher: "... ... ... I'm waiting" Student: "What? Other Student: "He's waiting for you! ... Phone away" Teacher: "Cool, thank you."
@@milesgaming5075 “She” being the one who wasn’t paying attention. And he’s right. The way she said, _’Whuuttttt’,_ either she’s a total bitch or complete stoner, imo.
I watched this many times, but it never fails to amaze me. My teacher would just say “it’s 1, remember that, and write it on your notebook so you don’t forget” and if you ask her why its equal to 1, she would just say it’s a law. I’m glad you are doing this mr Eddie!
That Eddie should be fired for spreading false information among his students. And your teacher would be right and save you from that. Because it is define like that and not proved. You can just look at wiki and easily see that. I can't believe so many people in the comments are so ignorant.
@@Phymacss that you can calculate, deduce or prove that 0^0=1 through some function analysis (it's actually the opposite) or at all. As I said you can just check at least wiki for more thorough explanations.
love the fact that the other students were annoyed at the girl with the phone, like they actually want to be there to learn. I wish my maths teacher was as good as this!
@@mr.fufucudlypoops8207 I'm almost certain that 99% of people who "wish he was my math teacher" never clicked around and tried to learn any math from the guy. They wish they could surf UA-cam and get course credits. Sure, it's clever media use or whatever, but these are mostly the same people who only remember three things from math and flip out when Common Core teaches their kids a 4th way to think about something.
This guy is a great teacher. Learning the 'mechanics' behind limit functions. And showing that some laws are not absolute, but just the best theory we have right now. And that is what academic students need to grab.
He just made them type in numbers in their calculator to get an unexpected result. Of what use is this other than a bad bar trick if you don't understand why it works like it does? It seems to me just like a magic trick, an unexpected result that you don't understand, learning nothing unless the trick is explained. Although it is not exactly the same since a magic trick doesn't necessarily trick people into thinking they've learned anything useful(unless you can tell me that there is useful information in this and what it is, apart from the fact that it spikes curiosity.).
@@joellasoe3218 he was demonstrating the method of limit chasing to provide proofs for competing laws. He set up the example of two laws everyone knows but conflict when they hit eachother so that he could show them an example of how to work toward a limit so that he could explain the proper notation for limits. He was literally doing exactly what you were complaining he wasn't. Perhaps the flaw is not with the lesson but with the observer. The applications for this method are wide as basically every major mathematical proof still waiting to be found at some point will involve the need to address limits and discuss the exponentially declining approach toward some limit. Just because you don't understand it or it's applications does not mean it is a magic trick lacking in practical application.
this was so engaging that i brought out my own calculator and did the calculations as he went along, such a good teacher also one interesting thing that i noticed was when i had 20 zeroes (0.000000000000000000001^0.000000000000000000001), the calculator just said 1. and it's not like my calculator is limited to the display, i have seen it produce recurring decimals that take 30 seconds to scroll to the end of. so i tried adding in five more zeroes, and the answer was still 1. so could that mean that there is a very specific, infinitesimally small number after which all x^x = 1?
Nghi Tran Are you a teacher yourself? If not, then you should probably stay silent. Also, you're in fact wrong. 0^0 is not undefined, but rather indeterminate: it is a form that takes one every single possible number simultaneously.
You probably already know these channels but Veritasium (Derek Muller) and Stand-Up Maths (Matt Powell) are two pretty good maths educators, also Australian.
Maybe because that thing’s like 120 euros (atleast mine is, TI 84). Throwing it on the table with paper and books on it really won’t damage it. But say it falls on the ground, you’d be fucked.
1. Engaging everyone. 2. Learning for all. 3. Making everyone curious. 4.Accepting when he don't know the answer. I wish I had someone like him to teach me Maths.
What a nice teacher! And for people who wondered, the reason why the number started to increase at around 0.4 and 0.3 can be explained by differentiating the function y=x^x, which gives dy/dx = x^x(lnx + 1). When the dy/dx is smaller than 0, the function will decrease; and if it's larger than 0, the function increases. lnx + 10) x
@@0brine No problem! Differentiating functions can help you find the range of x where the function is increasing (f'(x) > 0)/decreasing (f'(x) < 0). I live in HK, and we learnt how to draw functions with the calculus lessons in my secondary school. This is a crucial step we had to learn back then.
Very true, but with a slight quibble. Differentiation tells you exactly what is happening and precisely at what point in the function finds its local minima, but it doesn't actually explain why the original function that is being differentiated behaved that way, in and of itself.
I can see in the way he speaks and moves his arms that he is really excited to teach that. I dare to say that this guy loves teaching math. And it's just beautiful.
GREAT GUY! Very clear, concise and insightful showing meaningful context for the examples leading to limits/asymptotes. Not simply saying stuff by rote or dictum. Facile understanding of the true meaning of teaching vs "reviewing material"! I saw comments noting he's Chinese meaning I suppose genetically gifted at math and science! Too funny!!!
to answer the kids question of why the turning point is there (good question): let y=x^x to find a local minima we can use first differential of y dy/dx=x^x(lnx+1) we let dy/dx=0, solve the equation for x and you get 1/e, which is 0.367879
From a calculus perspective , the limit as X approaches 0 for X^X actually does not exist. just to clarify, the reason why he says that it "breaks down" when you use negative numbers is because when you set x to very tiny negative numbers you instead approach -1 instead of 1. In calculus, if the limit from the left and the right aren't equal, then the limit simply does not exist. if you approach X^X from the left you get -1, and if you approach X^X From the right you get 1, therefore 0^0 actually doesn't have any real value.
14:03 One student asked very important question “why does the function value start to increase at x=0.4, to answer that you have to know that it will inflect at exactly x=1/e=0.3678 this x value is when the first derivative of the function x^x equal to zero , f’(x)= (x^x)(1+ln(x)) =0
I mean, the first derivative helps to identify the minimum/maximum point of the curve the function is describing; so this answers at which value the curves reaches this point, but still doesn't explain why it happens at 1/e. Having "e" involved is a really nice clue though.
This video addresses a problem I've been dealing with for some time. Although it could be demonstrated using the Taylor series of e^x , this method is amazing. Thank you for sharing it. I greatly appreciate it.
Mr. Woo, I just came across this and want you to know how much I appreciate your teaching style. Math has fascinated me my entire life (which began in 1951) and it's teachers like you that made it fascinating. Bravo! Best of everything, Mike
THIS IS THE BEST WAY to teach Maths. What make Maths difficult for some is that it has to follow detailed rules and a lot of sets of logic that human invented while observing patterns in numbers and in nature. When you get to associate the reason of these rules; like what Eddie Woo is doing, then Maths become easier to understand. At younger age it is quite challenging to understand Maths because few teacher get to explain that every operation must follow its own set of rules. Most teacher don't bother to explain them because it demands time, effort and most importantly a fun way to get students attention. If we explain to children that traffic lights exist to enforce the law, then they will never appreciate their role. If you convince children that these traffic lights make traffic organized, give everyone his share of the road and save lives, then the children will definitively appreciate them. This is exactly what Eddie Woo is doing and I hope the other Maths teachers and professors will adopt his enjoyable way of teaching.
If I may make a comment that has nothing to do with the video, but supports your argument: Gerald Wilde published an e-book on traffic, and his conclusion (one of them) was that traffic rules are not about rules (or for that matter safety), but rather to allow more mobility, i.e. easy of movement.
That lesson gave the kids a chance to express their understanding of maths and air their conceptions...many of which were partly wrong. You can't drag people up to a level of understanding...they have to think it out for themselves. That lesson was good for them.
I had a math teacher at the university who was so passionate about his teachings. He always made time for his students. we could knock on his door and ask for help and he'd sit with you one on one for a while and really help you and explain things. Thanks to him I managed to go from a C in the first year to an A in the last year. We need more people like this :)
Yeah I'm also confused, because the students sound much older than I am and they have never been introduced to limits somehow. I suppose they have more knowledge on other mathematical subjects than I do?
Im literally in uni doing engineering but no one's ever really delved into stuff like these as deeply as Eddie Woo, so whoever his students are must be so lucky. Also what class is this? like MAN, no one ever really teaches cool math theories like this in high school or uni unless you're taking a course in pure mathematics.
I don't know, for me it's not real maths, it's just using your calculator and get the intuition of what's happening. There's no proof nor any understanding of the concepts behind exponentiation, just an observation that anyone can make using a calculator. A limit has a rigorous definition, it has to be more than "it looks like it's 1". I feel like this way of approaching the problem teaches nothing, it only gives you the answer with no explanation of why that is true. The real way of getting to this result is to calculate the limit of e^(x ln x) when x approaches 0. Since the exponential function is continuous everywhere, this is the same as the exponential of the limit of x ln x when x approaches 0. And for that there's Hospital's rule, which, to be honest, I just found out about. I just remember when I was doing math we knew a bunch of limits and didn't have to re-demonstrate them (x ln x when x -> 0 being one of them). Anyway, that would be a more mathematical approach than using a calculator.
@@dicidicee Its important to highlight that this is a high school class, whilst it does not dwell into the advanced mathematics, Eddie did a great job on teaching student to think. To consider why 0 to the power of 0 is different than what they are expecting. He's teaching them how to think flexibly which in my opinion is a very important traits.
@@dicidicee the point is to first introduce the the concept in a easily digestible way(less things to consider like e and natural log), then expand on it without drawing up a table of values. It would be an issue if there's no continuation after this, but I'm pretty sure that's what he's doing off camera. idk, it would be like using exponents to teach kids multiplication. Quite a bit more extreme but I hope you get the point.
Zero to the power of ain't paying attention, all with a smile on his face. Made class interact, let them see logic and start to understand without being "told". Amazing teacher :)
These students don’t realize how lucky they are to have such an effective and passionate teacher. Really makes all the difference in the world to keep the students engaged whilst educating them.
2:10 That really made me sad. The way that student said "what".
Well obviously the student wasn't engaged.
Absolutely. I watch a lot of maths related videos on UA-cam because I find the subject fascinating, but watching this is the first time I've actually thought about getting back into learning it formally. One bad teacher can destroy a student's interest for years - as unfortunately happened to me - but one good teacher can turn it around and this guy is great. If he'd been my teacher, I'd have a maths degree right now.
Why?
Very few teachers will have the kind of enthusiasm to show students all the great tips and techniques in any subject. Most probably are just biting their fingers waiting for their paycheck to come every month 🤔
"0 to the power of are you paying attention?" mate that was brilliant
Rudy Julian it would be approaching one since he was giving 0 attention.
I love it when this guy taps his chin and goes "hmm... what should I call this?"
and then you hear..........umm... what
LOL
"wut"
Timestamp pls.
I love how he literally stopped his whole class for the sake of the one kid who had stopped paying attention, and the kid who told him to get off his phone sounded legitimately angry. I feel like this guy has succeeded as a teacher because of the fact that he has his students so engaged and he cares enough about the individual to stop the class to allow them to regain their focus. These are the teachers we need more of. Its always refreshing to see a teacher who genuinely cares and also tries to make learning fun, while at the same time trying to make the content he gives to his students more easily understood. He simplifies things as much as possible in order to benefit everyone in his class, and I think that deserves tons of respect.
Hi Dimitri, interested in math competitions? If so, take a look at ua-cam.com/video/rkzxdMFEEtw/v-deo.html and others in the Olympiad playlist. You will see a lot of tricky and somewhat complex problems to try. Hope to see you there.
If you want excellent teachers, you need to completely revamp the system. The current system makes teachers overworked, underpaid, and burnt out.
Some of my teachers were like this too, some of them...not so much.
If you pay teachers well and give them good working conditions you'll probably get better teachers. I have no idea how it is in Australia though (where this guy is from).
I love how he stopped the class for that one unfocused kid. Teachers who are passionate, kind, caring and engaging sometimes can be disrespected by students because he is, well, too nice. That's just how teenagers function sometimes, no judgements. Stopping the class and politely ask for attention can serve as a hint or "warning" to the students that while the teacher is kind and caring, he has standards and is to be respected. He then smoothly moves on without making the student too uncomfortable and impacting the flow of the class. This makes me think not only is he great at teaching, but also understands how students think and respects that.
That sounded like a girl to me.
As a Maths teacher as well, I must confess that this guy is really good. Choosing the right words and keeping the students engaged.
I am 59 years old. I've had 100's of teachers in my life. This guy is about the best I've ever seen. Passionate, polite, patient & motivating! I hope his students in future classrooms appreciate, respect & award this guy! He deserves it!!!
How do you have hundreds of teachers did u move schools alot?
@@Lazerboomtv going to the end of grad school is almost 200 teachers already, then you add other types of teachers like at jobs and hobby classes
@@Lazerboomtv I'm 31, I can say I had about 100 teachers. I dont really remember how many subjects I had in middle school, it was a lot, the number 9 pops into my head. I remember I had pretty much the same teachers from grades 1 through 4, then a whole new roster from 5 to 8 (my country's ed system has 8 years of middle school, then 3 years high school, then college, btw) then I went to a new school for grade 1 high school, then a new school for grades 2-3... Just that already goes over 100. Then technical school (sort of middle ground between high school and college, idk what it's called in NA) I had about 12 different teachers, then college where I had way over 20 teachers... I wouldn't be surprised if I had over 200 teachers in my lifetime, so 100+ is not unbelievable.
I, too, like this teacher. Very nice way to explain limits and the students seemed to get interested when the calculator part rolled in, that's a sign of a good teacher.
My calculus teacher in college was great, too. He explained limits in a very different, more complicated way since he went deeper in, explaining why the increase starts between .4 and .3 (euler's number) and some other rules that I honestly dont rememebr cause I dont really use limits in my field, but he had a great way of putting things that made it easy for you to visualize stuff, and he was great with little songs/mnemonics to remember stuff for tests that I still remember to this day.
@@Ins4n1ty_ don't need ur biography
@@reenaoswal634 He was answering the person who questioned how he could have 100’s of teachers. Maybe read the other comments to keep up?
“Now that’s a good question, and i don’t have an answer to that at least i mean i dont have one right now probably because i haven’t really thought about it all that much to be honest. There’s always reasons for everything. So maybe we could find out where they go.”
Things done by just these sentence:
-Appreciate student for asking means appreciating them for using their logic and also appreciates their courage to voice their logic.
-Being honest to what that is not known by the teacher, rather than making up answers or saying “that is just how it works”. Admitting incapabilities or mistakes in certain area will not decrease your integrity as a teacher, rather it build trust with your student and increase your integrity as a teacher.
-Rather than discouraging students and teach them not to overthink things too much, he took the other way around and told “there’s always a reason for everything” then leaves the sentence ends with encouragement that we could find out about it.
Good job ed
Yes I was looking for this comment! The answer I would have received was " that's just how it is ..."
It turns around at 1/e, obviously. ;-)
@@landsgevaer Huh. You know, it never before clicked in my head that e is the tipping point when adding to the exponent scales faster than adding to the base.
x^x reaches a minimum when ln(x^x) reaches a minimum. This simplifies to d/dx x*ln(x) = 0, therefore x*(1/x) + 1*ln(x) = 0, so ln(x) = -1, so x = 1/e. #
Great analysis. BTW before I read your comment I was impressed with those lines too
If everybody had such great teachers the world would be different...
We have, usually math teachers are kind, and funny
@@tawfiqclick2481 / Yeah, no. And, even if they were, they're usually not that good at explaining why things are the way they are. After an entire year with two really bad professors, I was lucky to get a good one, who actually made me like math; I can assure you not everybody have the same lucky :\
Yeah I've had a bad math teacher only one year but that was one heck of a year. Literally got an amazing racer ext year and was s oring fulls. Also I'm not saying current teachers are bad it's just that he teaches a lot better than others
Aarsh Agrawal Absolutely correct, Aarsh. I was comparing Professor Woo with various teachers from my youth just as I saw your comment. He’s smart, engaging and appears to genuinely love mathematics. Those prime elements of teaching coalesce only rarely in real-world educators.
@@prototropo yeah and the best part was he didn't punish the student for not listening. That would completely ruin the spirit of learning. It's often saddening to see that someone who either doesn't understand or loses attention is punished thus making them act like they are listening just to not have to deal with the punishment and thus end up hating the subject and the teacher. Do this with multiple teachers and ultimately they're hating school.
Wow, I wish I had you as a teacher in my Math Class when I was young. Thank you Eddie for sharing this. I'm 66 now and still learning thanks to guys like you.
Math gets so great when you start thinking about infinity. It's not taught that way in high school. ;/
Teachers that love their jobs are so much better... This guy is awesome.
If you are in that level of math, and you dont pay attention, you deserve to be left behind
@@insideoutsideupsidedown2218 do i smell nerd in your words?
@@insideoutsideupsidedown2218 jesse what the fuck are you talking about
@@dio8429 holy shit 😆
@@dio8429 ah, i see you were one of the left behind….
Me: finally it’s Saturday
Also me: watches math videos
patthemightymat W A T C H E S ***
You saw nothing
😂😂😂😂
It’s Friday for me! :p
Wait, it's saturday today. And your comment is from 4 days ago. What the fuck?
Me: *is about to sleep*
Me: *sees this in my recommended*
Me: sleep is for the weak, I NEED ANSWERS
Im reading this in this beautiful Australian accent lol
@Sol iloquy you can find it using differentiation.... you just need to find the first derivative and put zero as its value and solve for x.
Hello 😂😂
@@thunderboltcloud3675
if y = x^x
then dy/dx = (lnx+1)x^x
www.quora.com/What-is-d-dx-x-x-1
setting dy/dx = 0
yields lnx = -1
or x = 1/e.
the function x^x has minimum value (1/e)^(1/e) = 0.692200627...
at x = (1/e) = 0.367879441...
@Sol iloquy
if y = x^x
then dy/dx = (lnx+1)x^x
www.quora.com/What-is-d-dx-x-x-1
setting dy/dx = 0
yields lnx = -1
or x = 1/e.
the function x^x has minimum value (1/e)^(1/e) = 0.692200627...
at x = (1/e) = 0.367879441...
I’m not saying anything you don’t already know, but this method of teaching is superb. Anyone can tell students that zero raised to the power of zero is equal to one. But you let the students discover it. They will never forget. When I was in grade school many(oh so many) years ago, my teacher told us to go home and measure the circumference of tin cans and the diameter and bring the results back to class. None of us knew why and none of us understood the meaning of Pi. We did the next day! Thanks so much for these lessons you post.
i would just like to say i’ve met him and he’s the only teacher ever that made me excited for math
I would say maybe you should meet 3blue1brown but they’re not really a teacher just an online educator but still amazing.
I had a math teacher in my intermediate years who was possibly as good. He was much respected by us students.
woaahh you lucky lucky
I also suggest Brian Mclogan. He is such a good math teacher and I always watch him when I have any math problem
im a college graduate who already learned this throughout multiple classes and he still kept me engaged enough to watch a 14 minute youtube video in its entirety
He could have just said: "it's 1 remember it, it'll be on the exam" , and continued, but he's done more: he gave them meaning, passion, curiosity, knowledge. The fact people hate Maths, or anything for that matter, it's because they learn it from the wrong people. It's because some doctors, or business men, could have been good teachers and they chose to be something else to make the money they think they deserve. I hope this teacher continues to be the same and contaminate his students with his passion.
Dihcar bahari bhai,i am convinced with what u said, In this world there is a problem of money everywhere no one does anything with passion for it they just perfect something for earning and because of obvious pressure from society ,generally they are called middle class but it can be anyone post script i know u r talking about mathematics here but what i believe about the world i said that
In this world the main problem is teachers themselves don't have full knowledge....What would they feed to their students
Don’t entirely agree, alot of doctors chose to study medicine for the soul purpose on wanting to help people. Not saying that there aren’t doctors who do it for the money, but alot of doctors, as kids, probably dreamt about helping people for a career. Regardless of the money, majority of doctors are there because they want to be.
Your totally right, I'm an engineer so naturally I have had a lot of math teachers. Some of them fantastic, some of them so so and some of them made me hate life. At one point I thought I was terrible at math but it took one good teacher to show me I actually have a good mind for it. A good teacher can be the difference between understanding and frustration. A good teacher can teach anyone regardless whether they have the mind for it or not and a bad teacher can make someone who does have a mind for it hate the subject.
He can’t because the right answer is undefined
“0 to the power of paying attention?!”
-sun tzu the art of math
Kid has been forever immortalized on YT for not paying attention
Just the way he said that without missing a beat. That was brilliant.
perfection cubed
kind of a teacher i had when i was student. great teacher. The trick to understand this is to rewrite the equation : x^x = (e(ln(x))^x = e(x*ln(x)), then you can derivative etc to study the function (minimum, limit etc)
I'm a retired math teacher and I love it when a teacher shows such love and enthusiasm for his subject. Great vid and great teacher.
Spot on. He loves what he is doing.
I read retarded instead of retired and I was amazed for a second
@@francisbacon4363 thank you for putting that amazing and hilarious thought in my head
My favorite subjects in high school was the higher abstract math like algebra and geometry because I had interesting and engaging teachers who were passionate and made sure everyone understood the concepts before going on to the next exercise. Also gave real-world examples of practical uses for formulas, like principle to interest ratios and amortization schedules.
You guys are such nerds. Its wonderful.
13:56 For anyone wondering why this number starts increasing again specifically between .4 and .3, it's because the minimum value of x^x (x to the x) happens at x = 1/e, aka. the inverse of Euler's number, aka. ~.36788, which, as you would notice, is between .4 and .3. You're welcome.
Gratias, Amigo. I'd like to see a graph of the relationship of x and x raised to the power of x
It happens because of witch craft!
@@joemarshall4226 Follow the spirt of the lesson and plot it for yourself ! Normal exponential behaviour i.e rapidly rising curve above x=1, then a v shape between x=0 and x=1. For x < -1, a flat curve that gradually rises from -1 towards 0 as x gets increasingly negative. The bit between x=0 and x = -1 is left as an exercise for the reader.
@@jonathansmith2734 hi the devil sent me.... he loves your work
Yes, you're absolutely right. I noticed this before I read any of these comments. As I started watching this video, I generated a list of decreasing numbers by using MS Excel. In column A, I made a list of the positive integers in increasing order. I made each entry in column B to be the reciprocal of the corresponding entry in column A, and each entry in column C to be the value in column B (of the same row) raised to the power of itself. Seeing that the value in column C was indeed the same for the 2nd row and the 4th row, i.e., (1/2)^(1/2) = (1/4)^(1/4) = 1/sqrt(2) = half of the square root of 2 (which I already knew it would be), I created 9 rows between 2 and 3, and put in column A 2.1 through 2.9, and carried the equations for columns B and C into the corresponding rows; then, seeing the results in column C, I inserted 9 rows between 2.7 and 2.8 and placed 2.71 through 2.79 in column A, and repeated the process for the new rows in columns B and C; then, seeing the results in column C, I inserted 9 rows between 2.71 and 2.72, and placed (in column A) 2.711 through 2.719, and finished that step with the corresponding entries for columns B and C. It was here that I got the idea that the number in column A that produced the smallest value in column C might be Euler's Constant (e). I continued this process until I had the smallest value in column C when the value in column A was 2.7182818. But then I computed the first derivative of x^x = (x^x)(1 + lnx), and setting this equal to zero, 1 + lnx = 0 and lnx = -1 and so x = e^(-1). So, yes, the minimum value for the function f(x) = x^x is found when x = (1/e). And yes, the limit of x^x as x approaches zero is 1.
'I don't have an answer for that yet.' That is a sign of a good teacher. He's honest and even said he will have a look at it.
Yeah man, most people don't acknowledge they don't know everything and being a teacher doesn't mean you have to know everything, and he knows that and embraces that.
yeah and he did
Lol so 0to the power of one is 0 x 0 so all 0 to the power of 0 is would be 0. Lol not difficult equation. After all what's 1 to the power of 0 its 1.
@@Sh4dxwxz 0 to the power of 1 is 0 not 0x0
@@andr_sh 0^1 is 0×0 = 0. So you missed a step there buddy.
0^0 is an interesting one in that math textbooks and math teachers will usually insist that it's undefined (because if you take x^y, what you get in the limit x,y -> 0 depends on what path you take through the x-y space), but professionals in most contexts just take it to be 1. If you think about it, every time you write a polynomial or power series using a summation notation over powers, you're implicitly assuming that x^0 = 1 for all x; otherwise you'd need to include a special case for x=0 every time you did this.
Great comment! I also want to add a bit of a historical note. In the 1700s, pretty much every mathematician accepted 0^0 = 1 as true. So how did we go from 0^0 = 1 is true to so many people shouting that 0^0 is undefined?
It's because of the switch from infinitesimals to limits.
In the late 1700s and throughout the 1800s, mathematicians were becoming wary of the way infinitesimals were used in Calculus. In the early 1800s, Augustin-Louis Cauchy introduced the notion of the limit of a function as the input approaches a certain value. So instead of plugging in infinitely small values, they could now take the limit as the input tends to 0. Cauchy also included a list of indeterminate forms. This list of indeterminate forms included 0^0, and Cauchy was right to include it. 0^0 really is an indeterminate limiting form.
This caused the mathematical community to have a bit of a panic concerning the arithmetical value of 0^0 because 0^0 was an indeterminate limiting form, so they un-defined 0^0. No longer was it considered 1; it was now undefined.
But this panicked move was not based on sound reasoning. 0^0 being an indeterminate limiting form means that if f(x) approaches 0 and g(x) approaches 0, then knowing this information is insufficient to tell what f(x)^g(x) approaches. The limit of f(x)^g(x) could, conceivably, be any nonnegative real number, positive infinity, or not exist. But the important thing about limits is that *_limits do not have to agree with the function value._* The limit of f(x) as x approaches c could be different from f(c) itself. So just because 0^0 is an indeterminate limiting form does not imply that 0^0 must be undefined as an arithmetical operation.
With more development into discrete mathematics, abstract algebra, universal algebra, set theory, and category theory, we know now that 0^0 = 1 will *always be correct* in the context of discrete exponentiation. What I mean by this is the following: if you ever have a formula involving exponents where the exponents represent repeated multiplication, if you can get 0^0 from that formula, the formula will give the right answer (for what the formula is supposed to represent) if and only if 0^0 is evaluated as 1. It's not an accident. It's not a happy coincidence. There's a perfectly reasonable mathematical idea which shows us why this will always work (the empty product).
But unfortunately, the mistaken reasoning from the 1800s is still perpetuated to this day.
@@MuffinsAPlenty _"if you ever have a formula involving exponents where the exponents represent repeated multiplication"_ Isn't that equivalent to saying "the exponent is a non-negative integer"? Well sure, if we're only going to deal with non-negative values of x, then we only need consider the limit as x tends to 0 of x^x coming from above, and that's well-behaved and equal to one. But what about the cases where the exponents _aren't_ all positive integers? Then the value of 0^0 doesn't exist, so perhaps the reasoning from the 1800s was sensible after all?
@@RexxSchneider Why would you say that 0^0 in the context of discrete exponentiation should match up with lim(x→0+) x^x? Why not match it up with lim(x→0+) x^0? Why not match it up with lim(x→0+) 0^x? Why not match it up with lim(x→0+) x^(-1/ln(x))?
The issue in the 19th century reasoning is pretty much a categorical error. It's saying that f(c) can't be 1 because lim(x→c) f(x) isn't 1. It's a fundamental misunderstanding of what limits mean and what their purpose is. The definition of "lim(x→c) f(x)" intentionally ignores f(c). Arithmetic is different from limits. It would be, in no way, a contradiction for 0^0 = 1 to be true while lim(x→0+) 0^x = 0 is also true. All this means is that 0^x is not continuous from the right at x = 0.
Here's another example for you. Let ⌊x⌋ denote the greatest integer less than or equal to x, when x is a real number. By definition, ⌊0⌋ is the greatest integer less than or equal to 0, so ⌊0⌋ = 0. On the other hand, you could consider things like
lim(x→0) ⌊x⌋, which doesn't exist,
lim(x→0) ⌊x^2⌋ = 0, or
lim(x→0) ⌊-x^2⌋ = -1
From this perspective, we could say that ⌊0⌋ is an indeterminate form. If you have a limit, and evaluating the function at the point in question gives ⌊0⌋, this is not enough information to determine the value of the limit. No one in their right mind would use this justification to say that ⌊0⌋ must be undefined. But that's the reasoning that 19th century mathematicians used to un-define 0^0.
Just because not every limit gives 1 when you approach 0^0 doesn't mean that 0^0 itself must be undefined.
Now, you could get into discussions about how when we move from rational number exponents to real number exponents, we need to use some analytic tools in order to define exponentiation. And sure, depending on what your definition of exponentiation is, 0^0 is probably undefined. (But also, depending on what your definition of exponentiation is, your definition of 0^2 might also be undefined. For example, z^w = exp(w*log(z)) leaves all powers of 0, including things as simple as 0^2, undefined.) Additionally, in pretty much every situation, analytic exponentiation is an extension of discrete exponentiation. For example, if you go the exp(w*log(z)) route, you typically would define exp(x) as a power series, which involves discrete exponentiation of x. So you are building up from the definition where 0^0 = 1 is the only way to go.
To make things clear, if you want to study continuous functions, un-define 0^0 in that context. It's a good choice in that one specific context. But it shouldn't be seen as the correct thing to do generally. We shouldn't view 0^0 as actually undefined, except in any practical application, where 0^0 is defined as 1 for convenience; rather, we should see 0^0 = 1 except in the one instance where it's convenient to un-define 0^0.
I would also argue that the 19th century reasoning causes problems with students understanding indeterminate forms in general. Particularly things like 1^∞. A lot of students will be very confused about 1^∞ being an indeterminate form. They will view 1^∞ as 1*1*1*..., and they can't see how 1*1*1*... can possibly have any value other than 1. And there's something to their confusion. In any context where 1^∞ can be viewed as actual arithmetic, either as 1*1*1*... or as 1^α where α is a transfinite number (e.g., cardinal number, ordinal number, hyperreal number, surreal number), the value is always 1. *_But_* if f(x)→1 and g(x)→∞, this isn't enough to tell us what f(x)^g(x) approaches. And that's because limits are not the same thing as arithmetic.
I think a nice way to see my point is to try defining "indeterminate form" without simply writing a list of indeterminate forms.
@@MuffinsAPlenty Thanks for the wall of text. Did you read what I wrote? I'm pretty certain that I stated quite clearly "if we're only going to deal with non-negative values of x, then we only need consider the limit as x tends to 0 of x^x coming from above, and that's well-behaved and equal to one."
Then you proceed to spend reams of text arguing that if we only deal with non-negative values of x, we can define the value of 0^0 to be 1. *I already said that was the case.*
Anyway, the reason we don't take f(x)=x^0 or f(x)=0^x is that those give differing values as x tends to zero, so they are not the function we're dealing with, which is f(x) = x^x.
You tell me that "Just because not every limit gives 1 when you approach 0^0 doesn't mean that 0^0 itself must be undefined." But that's exactly what an indeterminate form is. The problem is not that we don't define 0^0, it's that the process of taking a limit isn't leading us to an unambiguous value. Surely you understand the difference? 1/0 is undefined, but 0/0 is indeterminate.
The 19th century reasoning is comes about because they actually understood limits, and defined a limit as applying to functions that are continuous in the region containing the limit, which x^x is not in the neighbourhood of x=0.
The whole point of having a limit to stand in place of f(c) when f(c) is arithmetically indeterminate is that it preserves the continuity of f(x) when x passes through the region containing c. When x is discontinuous in that region, then you can't use a limit.
Yes, of course, you can choose to assign the value 1 to 0^0 when x is a non-negative integer, because x^x is a continuous function for x>0, but that's just a choice made for convenience; there's nothing fundamental that requires 0^0 to take on that value.
I understand you take the opposite viewpoint, and assert that 0^0 has the value 1, except when we choose to call it undefined. I do view 0^0 as undefined, except for those cases where it is convenient to treat it as 1, and I maintain that the latter is an artificial choice.
@@RexxSchneider I think you _might_ under the impression that the definition of discrete exponentiation doesn't assign a value to 0^0. And if you have that impression, I understand why you would view 0^0 = 1 as artificial. After all, we have the original definition, then we extend using algebra, and then we extend using analysis. Now, if you believe that 0^0 isn't defined, then using algebra to extend the definition of exponentiation won't pin down a unique value of 0^0 (this is probably what you mean by "arithmetically indeterminate"). And using analysis to extend the definition of exponentiation won't pin down a unique value of 0^0 either (because 0^0 is an indeterminate limiting form). And if it were the case that the original definition didn't assign a value to 0^0, I would agree that this is compelling enough reason to declare 0^0 not defined and all places where 0^0 is replaced with 1 to be artificial context-dependent patches.
But the original definition of exponentiation as repeated multiplication _does_ assign a value to 0^0. For a nonnegative integer n, x^n is the product with n factors where each factor is x. In the case that n = 0, we have the product with no factors (each factor being x is vacuously true). This is known as the empty product, and it has a value of 1. Now, the empty product is a choice, and you may consider this to be artificial. However, what I consider to be important here is the reason that we assign the value of 1 to the empty product. And that's because, if we expect the empty product to be consistent with the generalized associative property of multiplication, it _must_ have a value of 1. If we assigned the empty product any value other than 1, the generalized associative property of multiplication would be false when empty products were involved. Knowing this, 0^0 is the empty product, and thus, has a value of 1, right from the definition of exponentiation as repeated multiplication.
Now, the generalized associative property of multiplication is extremely important for exponentiation. It's the entire reason why exponentiation is even well-defined in the first place [is x^4 equal to ((x∙x)∙x)∙x or (x∙(x∙x))∙x or x∙((x∙x)∙x) or x∙(x∙(x∙x)) or (x∙x)∙(x∙x)? It doesn't matter because they're all the same thanks to the generalized associative property of multiplication]. The generalized associative property of multiplication is how we prove the most important property of exponentiation: x^n∙x^m = x^(n+m), which we use in pretty much every extension of the definition of exponentiation. The most important property of exponentiation is used in pretty much every formula involving discrete exponents. This is why I can have confidence to say 0^0 = 1 will always work in any context where we have discrete exponents. The reasoning that gives us 0^0 = 1 is the same reasoning that makes exponentiation well-defined in the first place and gives us the workhorse of exponentiation, which we use to extend the definition and use to develop every formula using exponentiation.
From this perspective, 0^0 = 1 has the same amount of justification as 0! = 1. When we consider the empty product, 0! = 1 pops right out of the original definition of factorials. Factorials are only well-defined because of the generalized associative property of multiplication, the most important property of factorials is that (n+1)∙n! = (n+1)!, and this is proven using the generalized associative property of multiplication. Every formula involving factorials is built up with this most important property. Extending the factorial function to the Gamma or Pi function uses the most important property of factorials (since 0! = 1 can be justified using (n+1)∙n! = (n+1)! alone, the analyticity and log convexity of the Gamma/Pi functions have nothing to do with 0! = 1). And every argument used to "justify" that 0! = 1 should be true uses the generalized associative property of multiplication.
I doubt you're going to be telling me that 0! is actually undefined, but 0! = 1 is an artificial patch which always works in any meaningful situation. But hey! You might tell me that.
So that's the story. 0^0 = 1 falls out from the definition of exponentiation at its most basic level. When we extend the definition of exponentiation using algebra, keeping 0^0 = 1 doesn't produce any contradictions. So while algebra wouldn't be able to pin down a unique value for 0^0 if 0^0 weren't already defined, there are no issues with keeping the definition what it originally was. The same thing is also true for extending the definition of exponentiation using analysis, with one small caveat. The fact that 0^0 is an indeterminate limiting form does not contradict 0^0 being defined as an arithmetic expression (as my previous post argues). The caveat is that, depending on your definition of exponentiation using analysis, it's possible that all powers of 0 are undefined. In such a case, 0^0 would, indeed, be undefined using that definition, but so too would things like 0^1 and 0^2. This is why I say that analysts un-define 0^0. And sure, it's not a bad decision if you're dealing with continuous functions (unless you're only dealing with analytic functions, in which case you want 0^0 = 1 again).
And this is perhaps why it felt like I ignored your previous post when you said "if we're only going to deal with non-negative values of x, then we only need consider the limit as x tends to 0 of x^x coming from above, and that's well-behaved and equal to one." It's because this suggestion doesn't make any sense when we're talking about discrete exponentiation. Limits don't make sense when talking about discrete exponents. And maybe you believe there isn't a discrete way to make sense of 0^0, so taking the limit of a particular function is the only way to justify a value of 0^0. But that's not correct.
I like how he has this bulletproof velcro case for his calculator and then just throws it on the desk
BudderBoy Gaming probably the reason he has the bullet proof case
He tossed it gently that's not going to damage it.
Instead of bullet proof. Could it be a case that protects against surges or electromagnetic stuff?
@@claudeyaz sounds like some late night history channel conspiracy theory
Why do ya think he has it?
The best part is when he answers a student's question with "I don't have the answer right now, but we could find out." So important to show that it's okay to not know things.
I just want to know why the teacher didn't ask this question himself when he was studying for the lesson.
@@richardchurch9709 if you're wondering, the turning point is 1/e
@@richardchurch9709 He knows what's going to happen and why, It isn't just coincidence he drew up the table with 7 boxes on the left side to hold the decreasing values then increasing on the right.
It is important for a good teacher to teach students that no one knows everything but you can find out. and as
Bedër Butka, pointed out the answer is quite simple and he would already know it and why it is so.
He should have known when .5 to the .5 = the magic .707. It's so magic Boeing named an aircraft after it.
Yes, it is important to be honest. However, a trained mathematician should know the properties of such a simple function, it is inappropriate to say "I don't know" here. People who are not good professionals should not even teach children.
I can't believe this guy kept me engaged enough to sit through a math class. Props to this man.
a 14 min explanation of an easy limit? You are for a deception with a real class, like 2 hours of partial differential equations
He didn't. You were engaged for 14 minutes. A math class is 45 minutes every weekday for 10 months.
@@jonmarcki do 2 hours of maths everyday 😂it's quite fun
Im tryna sleep but im too engaged
@@pablomalaga4676😊😊q😊😊😊😊😊q😊😊
I'm so happy that the student asked why it turned around, because upon seeing what you were doing, and after plugging the problem into desmos, I worked out that because anything to the power of a fraction is basically just taking the root of that number, and the square root of a number less than one will get larger, therefore eventually there is a threshold where the number gains more from the root than it loses from the base.
I cannot believe that I am watching this for entertainment...
Me too😂BUT IT'S SUPER INTERESTING🥺
Same here bro
I know, it just appeared on the right side of my youtube channel. I really enjoyed this. I even understood where he was going with the limits thing.
LOL same
Me too
The best teachers are the ones that can answer: "i don't know" to a question from a student.
But not too often... And they better go and check it up to have an answer by the next period !
But I sure agree that claiming omniscience is a disservice to your student.
@@chaddaifouche536 I could not agree more
"Teacher, can I go to the bathroom?"
*I don't know* , can you?
@@whywhy8276 This is so true. Every single teacher is at least once like this!
A good teacher would say "I don't know, but here's how you could find out"
If only my secondary school maths teachers were as enthusiastic as Eddie. Great video.
EKGaming were*
Eddie is still young. Maybe after saying the same thing for 30 years he'll be less enthusiastic.
I'd prefer someone less enthusiastic, more concerned with finishing the course..
Same here. Some of my best math teachers were the ones that were actually excited about the subject and the concepts they were teaching at that time.
Wish there were more like him.
a possible cause for maths teachers losing enthusiasm is that after, probably, doing higher level, more advanced math, going back to teach relatively basic concepts doesn’t bring any joy. it’s like secondary school students revisiting 2x3 (the general logic behind it, not the calculation specifically).
12:19 Student asked a great question. Eddie only calculated lim_x->0+ of x^x = 1. The lim_x->0- of x^x = 0-j1.
I think it's best if you treat this as z = x^y. Depending on the direction of the limit you get different answers.
So wouldn't that make it undefined? (similarly to 1/0)
I really like his teaching style. He's very engaging and honest enough to say
"I don't know, but I can find out."
My understanding of maths is too basic to really appreciate the lesson though.
You're probably not as far away from understanding as you think.
The equations on the board at the start are just giving you the notation,
The top line is telling you that 'this is short form for that'. a^m = a times a times a times a, m number of times.
Next line I will just give an example if a = 2 and m = 3 and n = 2, that means (a^m) x (a^n) = (2x2x2) x (2x2) which equals 2^5 = a^(m+n).
I don't know if that helps, maybe, maybe that wasn't even something you were stuck on haha.
But don't give up on understanding if you want to get better at math, I always found the "That looks too complicated" feeling to be my worst enemy when learning it.
Here he’s laying the baseline for the fundamental theorem of calculus, which is where maths takes off.
@@HouseTre007 Thanks for the reply. Now I shall have to look up what 'calculus ' means because I didn't learn about that in school.
@@localbod maybe in your curriculum it was called Math analysis or pre math analysis
@zerere I moved around a lot as a kid, roughly every two years. I never got my head around basic maths, and because my level was poor, I got put into a class with all the dropouts.
It is interesting, though, and he is a good teacher.
23 years ago I asked my maths teacher this problem. Today I get my answer.
Actually zero to the power of zero can't be defined. However, if you take the limit of x to the power of x with x approaching zero you end up with 1. Thats not to say that zero to the power of zero is exactly equal to one. The correct answer is that it can't be defined.
You could've googled it 🙄
@@quantumnick i think no numbers are actually defined, they all are defined as a taylor series, a form of limits
@@ziyanoffl you, a boy/girl of today's era don't know how was the life when this phone u and me are holding was not there, these curiosity questions arousing in a child can not be answered by google,these are basically learnt from surroundings, and teachers come in these surroundings you don't know,it was an altogether different feeling to ask question from teacher when teacher gave answer it gave satisfaction to a student so please if u don't know the feel of the situation just stay away from the situation and don't comment
@@jennyispoop4 yes... because the actual value of any no. Say 'x' is x - 0.9999999999999999....
i.e 5=4.999999999999999999999999999.......
I'm experiencing being math teacher for the first time. I did just 6 months yet but it was enough to realize my own weaknesses and strengths. This guy not just only saying the right words. He uses extremely perfect tempo and intonation. He makes you believe you understand and since you believe that you already do understand. Amazing! His communication and way of explanation and confidence is perfect.
I've been a teacher for many years and I've trained and mentored many teachers.
There are many many ways to be a good teacher. But here's my tips.
Understand your own persona and play to your strengths.
Have a clear goal and a well thought out route to that goal.
Keep up the pace.
Be respectful and demand respect in return.
Be kind but be the boss.
Enjoy leading your students down the path you've chosen.
But you're absolutely right...let the students see the answer just before you "show" it to them.
His tempo is actually very fast. It is good for the lecture, but not good for the actual learning. Believing in understanding something isn't the same as actual understanding and reproducing results.
And of course you need to consider the age of pupils. Speed of understanding the material isn't the same at 13yo, 15yo, 17yo or 20yo students...
One thing I like about his videos is the way he can recover from making a mistake or a poor notational choice without getting flustered or losing the trail of the argument. That is real skill.
500th like😁
I don’t think it’s about tempo and intonation. I think it’s about passion.
why do i actually enjoy this all he says is basically "ok its like this but why?" but he just says it in a very engaging way
*has a massive protective bag for his calculator
*proceeds to smash it on the table
He’s high on math
Looks like a Casio fx-300ES Plus. Probably has the case just so it doesn't get rekt inside his bag.
@@gustavoperez3223 you have a good eye Gustavo!
Day 55 of Quarantine:
Am now watching math for entertainment. Send help pls.
same. and amen
Help is futile....you’re gone bro
As Ridley Scott would say......In quarantine, no-one can hear you scream.
Nero Bernardino, lol. This is a Perfect Comment.
@@TolriasHoly Hell, That is happening? The Romanians catching pidgeons isn't what surprised me, seagulls eating rats? That's... Complicated.
0 to the power of
*a r e y o u p a y i n g a t t e n t i o n*
*anyone paying attention*
@@skylarkenneth3784 Guess it's Annie
I read this at the exact moment he said it lmao
Keep it at 4:20
Punkenstine GG too bad
I love this energy while teaching. You convey the subject really good and keep the students interested and it really sticks.
I love how the students in this class actually answer the teacher's questions....my class is usually dead silent with a few people murmuring the answer.
I guess that's what you get for constantly telling kids to shut their mouths since early primary school.
And Im sitting here wondering why the hell everyone is talking so much.
+
@@Thee_Sinner lol
I feel the same way dude.
I'm not a professor, but I am a student (and at school 12th grade is last year), and I understand.
When I was in 4th-10th grade/12, I was in a school were classes were really distracted off the subject of learning due to bad student behaviour, but I was always siding with the teacher while listening intently to the lectures.
When I entered 11th grade at another school, the classes seemed to be too silent, and I was the only person shouting the math, Bio, chemistry... answers aloud(due to my passions), which was kind of disrupting at first, but because I'm clever (answer most questions correctly), I didn't receive much warning(not more then 2) last year 11th year.
And now same as last year, I'm in grade 12/12th grade, and I'm still being the only speaker in class which is still weird, but I will hang on to being active in class.
Proud to see professors somewhat advocating for more of this.
It’s really weird that most of the students have the voices of 14/15 yr olds and don’t even know limits or why 0^0 is undefined
"Now that's a good question and I don't have an answer to that.... So maybe we could find out...."
We need MORE teachers willing to say this.
Of course there is an answer to that question ! It's called a "local minimum" of a function
@@solcarzemog5232 Correct. In this case it is a switch between the predominance of the the value towards its factor. Then you wonder why you deposit zero dollar to the power of zero in your bank, and the stupid clerk does not give you dollar. Could take time to convince him.
as a math major I'm not satisfied with this answer. There are many things where the limit does not equal the value. It could have been the answer that the limit was 1 but the true value was undefined or possibly 0 but maybe it can't be that I forget.
After looking it up, it says that this is a very rare case where it depends on the context. For us viewers, it would have been nice it they went over that because this gives us the false impression that the value always equals the limit.
@@bztube888 what? how is it a minimum? what do you mean? I have a math degree
The x^x function's minimum at x=1/e (about 0.3679). Of course it's something e. But that's for another lecture so he gave the perfect answer.
Eddie: keeps his calculator in a protective case
Also Eddie: yeets it onto the table
That's what amazed me too😂
I noticed that too. I interpreted this as a slightly self-conscious gesture, the act of a man who is perhaps a little shy by nature but very much into delivering a lesson - an attempt at nonchalance. While I'm commenting I should note also that I like the video a lot. I myself am a retired secondary level math teacher, for whatever that might be worth.
@@ashbeelqadir7858 0009;0
@@grinpick
You: giving an insightful little psychoanalysis based on that one action
Us: yo he frickin yeeted that thing!
Your career experience is definitely worth something!
To yeet?
You're an amazing teacher! The moment at 10:38 when the guy in the class room says "I've done the next one" without you even asking for it which is proof that you have gotten them interested and invested in finding the answer together with you and started to take their own initiatives. Great work!
Monotone teacher explaining stuff, versus an enthusiastic teacher like this guy, is a night and day difference.
@Zeek Banistor Tf kind of a question is that?
@Bode He's teaching limits. That's fundamental.
Daen de Leon - also achieving the level of engagement he does means these kids can learn anything.
@@daendk Limits? what limits? he talks about Number zero! it's completely irrelevant to the real word.
@@s.muller8688 And I care what you think because ...?
The biggest problem with math and the way it's taught in schools is that they tell you so much "this will be useful, you will need to know this, here are some obscure word problems to show you that you need to know this, blah blah blah..." I promise you, you will probably never need to know how to graph a parabola in your lifetime, assuming you go into any sort of normal career (obviously STEM is different). And if you do know, you can google it. So why learn it?
Well, this teacher clearly has the answer. Math teachers *should* teach math for the same reasons why your English teachers have you read classic literature - it's good for you. Math is the study of logical truth, of taking a set of rules and carrying them out to the letter and seeing what you get, then finding out what that answer means.
English teachers don't have to tell their kids: "You will need to know what the major themes of the Great Gatsby", because they definitely won't, they just have to tell the kids that it's good for them to learn these things. Because it is good for them, and a good English teacher will make the kids love English, which will be good for their minds and for their souls. Kids are taught to hate math because they're taught it like it's something they'll need to know, something that's boring but hey, you gotta learn it. Because it's math and you just should know it. That's a horrible reason to learn math; no wonder kids hate it!
Math is good because it sharpens the mind, not because it will help you count your groceries. Math is good because it is the study of truth, and logic, and how numbers are in the world around us interacting with a set of rules that are constant, but maluable. More than that, it's just plain *fascinating*. People need to learn what math really is, not what common core is teaching them.
This is the best answer I've ever heard for this question. If only more people could explain it like this!
The fundamental issue I, personally, have with teaching stuff that "sharpens the mind" isn't the "sharpening of the mind"-part. It's how these, technically, "useless" subjects end up determining your future prospects. Depending on where you live, your average will determine whether you're even allowed to apply for that dream-education you've always fantasized about - and convoluted subjects, that require you to memorize pointless info, just make that journey more difficult and more frustrating, for any aspiring student.
Math tends to be one of those roadblocks - as it's a heavy subject by nature, requires extensive dedication and time investment to grasp, is demanded by virtually all STEM fields, yet is only really necessary for a few of said fields, and usually, their mathematical demands are highly specific anyway, to the point where most of the math you've been taught never had any relevance to your field of choice in the first place.
Having said that, it goes both ways - if you're an aspiring mathematician, and you love the subject, why should your knowledge of history, or your ability to analyze the "major themes of the Great Gatsby" have an impact on your future? It's very demotivating, to HAVE to grind away at a subject you know won't matter to you; you know, you don't care about; you know you're forced to sit through, not because of necessity, but just so the educational system can "weed out" those it deems unworthy.
Kids hate math, not just because of how it's presented, but because of how it seems like a massive man-made hurdle, that they are told they HAVE to overcome if they want the "good jobs" - they don't ask, "Eh, do I really need to know this? Does the hill I have to climb really have to be this steep...?" just because their teacher is boring, but because they genuinely don't care to know or need to know, how to graph a parabola.
This is true fact.
The art of being concise is also going away
exactly
Great teacher, challenging students. Much respect.
Wish i had a teacher like this
I find him exhausting. But I guess I one of the only ones who like teachers who speak slowly and without energy.
Teachers that don't speak with energy and excitement often hate what they do, which is why you know Eddie will teach you something, it's because he wants to not because he has to.
the "students" are quite stupid... hope they are just acting.
Twitchy so true
I taught maths for 33years and this guy engages me. He knows his stuff, is relaxed and thorough in his presentation taking time to explain every detail. He is a rare species in todays classrooms.
When I was in 9th grade my first advanced math class was an Algebra class taught by a lady who had no patience for questions and she would chalk equations on the board so quickly and impatiently that debris and dust from the chalk flew everywhere. I foolishly thought that I hated math. If I had a math teacher like this man then the trajectory of my life would have been very different.
Rama G I had the same experience, Rama, and dread thinking that same possibility about having fewer life choices as a result. I believe you are right. Nevertheless, even as a curious adult, algebra engages my brain like a piano falling from space, whereas geometry has from 5th grade been utterly logical and lovable, and still is. Since I never needed algebra in my life, but have had regular cause to calculate things geometrically, maybe things worked out. Serendipitously. But in an ideal world, everyone would have great teachers for every academic discipline.
Reee Flex Well, one thing is for sure-it wouldn’t help anyone learn something new to tell them they’re moronic, with a low IQ, and “that can’t be fixed.”
Reee Flex u have got to be trolling right, if not ur a dumbass
@@dylan230 hes legit spitting facts
U hit right
Despite being a 3rd year maths undergrad and already knowing everything said in the video, I still watched the whole thing. This is a really fun way to introduce limits to the students. Maths is such a beautiful subject and I really think if more teachers were like you a greater proportion of students would take an interest in it. Hats off to you
Damn what are you studying? Sounds very studious! (Just curious 🧐)
... here i am reading 10th
I still don't know how don't they know the limits already tho, it seems like they're in college I first was introduced to it at highschool that's weird
Still don't understand how 0^(a number that has infinite 0s followed by a 1) still = 0, but then decreasing that by an infinitieth changes it entirely 🤔
@beta male but what type of math? Like arithmetic, algebra, you know, like how science isn’t just “science.”
My teacher: What's 0^0
Me: I don't care
UA-cam: What's 0^0
Me: I NEED TO KNOW
Maybe because in your classroom there isn't silence
Same here.
Lmao
If you have a really engaging teacher - like this guy - it would be the same. He's really good.
Dued the answer is obviously time travel
Im 47 now and I have completed my high school 32 years back. I'm now working on a completely different field. But Why am I listening to his lectures again and again?
How can students not pay attention to such an awesome teacher...
Even me, he got my attention
not gonna lie if I had a math teacher like that I would have actually payed attention
*Paid
God乡 Dєѕтяσуєя payed
@@braedenmattson1829 Paiyed*
@@ceruleanmemoir paieyed*
WhiTeX (Timo:) no you wouldn’t have
Her: He's probably thinking about other girls
Him: What's 0 to the power of 0?
😂
0^0 = 1
@Che Ku Nur Aiman it's close enough to 1.
@@cidguy - The LIMIT as X^x approaches 0 is 1, but 0^0 is not 1.
@@Max_Griswald 0.9999999999…
This UA-cam video has me more interested than any of my classes.
As someone who teaches college and grad school, just let me say that "Oooo" and "What?" @7:25 is an absolutely beautiful sound. Those are the moments for which we teach.
I was watching this lying in my bed and I am not lying i sat up straight the moment i saw that part
Here, it’s a completely fake sound
He's an amazing teacher, there's no doubt about that; but I think some of those sounds might have been sarcasm by some disrespectful student, or not; we may never now for sure.
@@Macron87 tf bro
I have been teaching for 40 years and I know a good teacher when I see one. You can have all the academic theories of pedagogy you like but if the teacher a) is obviously personally interested in what he/she is teaching b) is constantly asking doable questions from the class c) has a bit of humour d) shows interest in the class as people then you will not go far wrong. I try to teach like this and succeed sometimes. The hard part is the fact that doing this for four or five hours a a day for five days is incredibly tiring. Most other jobs have down time during the actual doing of the job - teaching does not. This man is good but he will be utterly exhausted at the end of the week.
Frank Gibson - agreed. And the most exhausting classes are the non-Honors classes that don't want to learn. I am practically yelling the lesson just to talk over them.
Frank Gibson can u teach my shitty chemistry class please
They are not shitty and you are teaching people not chemistry. Take time from the chemistry to get to know them. Aim that each lesson should be an individual dialogue with each student and make eye contact with every student in the room at least once every few minutes - that is what Eddie Woo is doing. Talk about chemistry in their lives, tell them about experiments that went wrong, discuss pollution with them. I call it teaching by stealth - or teaching in the cracks. It may sound like taking time away from teaching but in fact you are building the bridges that will allow communication in the future.
OK learning should be cooperative. The reason I have stuck with teaching for 40 years is because I like to learn and understand stuff. If as a student you are genuine about wanting to learn and you are sure that your teacher has a good understanding of the topic then get the ones together who want to learn and form a cooperating group with the teacher. Ignore the dickheads or even better tell them full up front to fuck off and stop wasting everybody's time. They will either come on side or leave. This is a win win for everybody concerned.
I agree telling students to fuck off is a solution, if coming from a teacher its god like
Please, save this teacher's DNA so in the future we can start making a copy of him for every high-school math lecture in the world.
Or he could put his lectures on youtube
Or just fucking do the work on ur own and work ur way up fucking dumbass
@@ewaldatok611 Not every teacher connects with every student and that’s a giant flaw in a lot of education systems. Asking for a better teacher isn’t the same as asking them to do the work for you. You can be really hard working but never understand a certain subject in school until a certain teacher finally lets you understand it. Just because you understand a certain subject well doesn’t mean every other student on the planet does and some need more help in certain areas than others. I hope you understand that the end goal of school is to make sure that every kid understands what they are learning and not just grading assignments.
Anyone wanna let this guy know that it isn't your DNA that decides what kind of career you choose?
or just have him train teachers.
i studied limits 20 years ago.
I memorized most of the limit stuff cus nobody summarized it so brilliantly as you did.
Thank you.
I am watching math while procrastinating on math. Sometimes my genius... it scares me.
Edit: I've never received so many likes. Thanks
I’m in online math class watching this math
sameeeeeeeeeeeeeeeeeeeeeee
Literally im watching this instead of doing my calculus homework. Wait till my teacher hears this excuse lol
@@glyph__ lol
Sometimes... my genius... it generates gravity
I was scrolling through the comments and he was like "are you paying attention?!" I got scared for a second there
For once, I was paying attention.
😂😂
The way you Express is just hilarious..
😂
my PTSD triggered there for a second
Imagine being that girl who got embarrassed Infront of millions of people and having one of the most happiest and intricate teachers, who actually loves what he does.
Sky TrexZ I mean she was the one who wasn’t paying attention
Imagine being a teacher and being required to play babysitter rather than just being able to teach those mentally present and ignore those not.
blargg so by your logic, fuck the tired ones or the ones that zoned out.
@@FFeras And by your logic, that means we should instead force the mentally tired ones into learning instead of giving them a break
Johnson Pootisman you get a break twice a day and two days a week.
It’s school after all, not a fuckin nursery.
Don’t go to school if you’re that mentally tired, it’s not the teachers fault you only slept three hours because you were playing a game or some shit.
Mathematicallly 0 power 0 is undefined, maybe for convenience we can keep it as 1. If you apply the logic of factorial which same teacher explained, 0 power0 becomes 0. It can lead to different results and subjective.
Sounds like how my boss calculate my overtime wages.
F
69 likes? Nice
-F*-F
lets see here... 0 hours at 0$ per hour...
sonofashrub equals $1
I like how he takes his calculator out of his cushy protective case... and just plops it on the desk 😂
ikr my thoghts exactly XD
Shit
Why all teachers have same habits...they are very organized and care for tiny things💕
And his phone is just in his pocket.
"How did the universe came into existence from nothing?"
"0 to the power of 0".
underrated
LMFAOO
I really know any number that has a zero exponent would be 1.
It was 42
@@glennoc8585
87^0=1
2^0=1
0^0=1?
I have taken math from arithmetic to calculus, and I have had a good many instructors. They mostly just regurgitated mathematical facts, as if some kind of mantra. This guy goes to the trouble reaching beyond just the facts, and explained the concept behind the facts. This promotes understanding, instead of just memorization (like memorizing a phone number), which puts the information into the logic portion of the brain. Even after you forget the equations, you can regenerate them by applying the logic of the concept.
If all teachers used this method, or a similar one, people would not hate math, which is so common!
This was so mysterious to me I couldn't help continuing with my calculator myself after watching. I needed to locate the exact place the function started growing again... and I did, and it's at... *drum roll* ... 0.3678794412 !!!
Oh well, I thought...
So I thought, naaah, there's got to be something about this, it can't be just another accidental weird decimal number... so I tried to do 1/e just to see if ... and there it is!!! In other words the function has its lowest value at (1/e)^(1/e). Math is such a beautiful mystery :-)
Yea! But..Why there..?
I don't understand what you just said but I love your passion, so I believe you.
Thank you all for your interest :-)
e, or Euler's number, is a mathematical constant, a so called irrational and transcendent number - in class with the number pi.
e = 2,7182818284590452353602 (approximately)
en.m.wikipedia.org/wiki/E_(mathematical_constant)
Peace and light to you
@@johantknudsen Thank you!
Calculators can't be programmed to be logical. We need a machine to be really wrong.
I love how his calculator is in a military navy seal proof bag but then he takes it out and just slaps it on the desk
💀
Then doesn’t use it😂
Seal proof? Does that mean designed to fool attempts by SEAL's to open it?
@@bulwinkle for sure, he has the solution for 0^0 on the calculator, the SEAL's must not see it or else he has to kill them
That's what even I noticed
No mere words can describe how good of a teacher he is. I've always been avoiding maths, but he makes me feel that maths is actually interesting.
To say that maths are interesting is an understatement. I had to start studying math on my own to see that though.
Studying is interesting. The sooner you realise that, the better.
@@list9016 unfortunately, I realized this after graduating high school and my major has no relations with mathematics whatsoever , but I can still enjoy his videos so maybe, it's not so bad after all
@@bao6287 dont worry. studying is a continuous process. it wont stop until your brain cells starts to decline in numbers (as you start losing memory, motor functions, cognition aka grow old)
its only a matter of whether you wanna accept new information or not. you can say studying has an exponential growth so past a certain level of information you will start to see things a lot more clearly and you will pick up new informations more quickly and that often results in information overload which is why many geniuses out there have very peculiar traits or hobbies to distract them from information overload.
I note that he openly accepts comments from his students. In the US, at least high school, does not want you interrupting the teacher ever. Most teachers there will make you raise your hand or won't even acknowledge you if you don't. Love his teaching style.
The fact he has his class so engaged, and the fact he is happy to find out a students questions that he doesn't know, really gives me hope for the future of those kids in his class.
This is called REAL TEACHING
totally
i was also expecting some Indian reviews
lol ya got'em now!
Yeah, but I'm still curious what happens when you approach this problem from imaginary directrions.
Spaghettificated Before you do that, just approach it from the left side (negative side), you get -1 so 0^0 does not = 1. Also, if you take 0.000000000000000...1i^0.000000000000000...1i, you get 1/infinity or if you take -0.000000000000000...1i^-0.000000000000000...1i = -1/infinity.
Stupid students not giving attention to such an amazing teacher ❤️
You are more active in commenting rather than uploading bruh.. 👁️👄👁️
@@degavathkalavathi lmao
You Jack ass. We're the students
2:17
@Pure True Too bad 0^0 has been proven to be 1.
I can't believe the student was on their phone even in such and interesting class, that's stupid
The student got a message about her sister being admitted to hospital.
@@megalomaniacal people jumping to conclusions.
Jumping to conclusions...
because hes forced to be there and ur choosing to watch it on youtube
an*
Regarding the question @13:50, the "turnaround point" (i.e. what's the number in the first column when the second column starts to increase again) is 1/e or 0.3678....
*Is 2am*
UA-cam: "Wanna learn maths?"
Absolutely
Math*
@@AlanGresov Learn English from England first then try and correct me.
@@chyaboi11 tell me what a singular math is then
Aluminum. Color. Gasoline. Train. Restroom/bathroom. Eggplant. Bite me
There's a really lovely result for the question asked by the student at the end (i.e. "why does it change at around 0.4?")...
If you differentiate y=x^x you get dy/dx=x^x(ln(x)+1) (Don't worry if this isn't obvious - this is not an 'in your head' result... it's a bit of a slog to reach it).
The inflection point is of course the zero of the derivative.
So where does dy/dx = 0?
The zero happens to occur at around 0.4 - just as predicted from the graph.
But it's not just vaguely near 0.4 ... the zero actually occurs at exactly 1/e (or approximately 0.367879441).
1/e huh.
Isn't maths just one mindblow after another?
Thanks Mr Woo.
Same, although he didn't know it was the gate to the deeper hell when he asked.
I think the teacher resisted the urge so hard to drag him into the neutral log (ln) and neutral number and derivative stuff, as he just told them the idea of limits which is the foundation of differentiation and integration.
ahh. i differentiated it too. but came out with f'(X) = X^1 * X^(X-1) = X^X. (i havent learnt log stuff yet) however i will do soon. is there a short explanation as to why mine is wrong?
I don't think you can do it without taking logs of both sides first (ok - maybe you can, but I can't :)
y=x^x ------(1)
Take logs of both sides you get ln(y) = ln (x^x) = x ln(x)
So then if you differentiate both sides wrt x you get 1/y dy/dx = ln(x) + 1
[on the left hand side you use the chain rule to get d/dx ln(y) = 1/y dy/dx]
[on the right hand side you use the product rule... u (dv/dx) +v (du/dx) ... to get d/dx x ln(x) = ln(x)+1]
Then if you rearrange 1/y dy/dx = ln(x) + 1 you get
dy/dx=(y).(ln(x)+1) -----(2)
... but from our initial point (1) we know y = x^x
So you substitute (1) back into (2) you get dy/dx=(x^x)(ln(x)+1).
Then you find where that is equal to zero and you're done :)
Short version: because when you do differentiation, you need to do the process on every single variable and individuals too. for example: X^2+x+1 become 2X+1+0 , therefore you cannot not treat the power X like a constant.
ohh cheers
2:08 "0 to the power of are you paying attention?"
Teacher: "... ... ... I'm waiting"
Student: "What?
Other Student: "He's waiting for you! ... Phone away"
Teacher: "Cool, thank you."
Student: You’re welcome
She’s sounds like a fucken bitch
Respect
@@iYouVee shut up
@@milesgaming5075 “She” being the one who wasn’t paying attention. And he’s right. The way she said, _’Whuuttttt’,_ either she’s a total bitch or complete stoner, imo.
I watched this many times, but it never fails to amaze me. My teacher would just say “it’s 1, remember that, and write it on your notebook so you don’t forget” and if you ask her why its equal to 1, she would just say it’s a law. I’m glad you are doing this mr Eddie!
That Eddie should be fired for spreading false information among his students. And your teacher would be right and save you from that. Because it is define like that and not proved. You can just look at wiki and easily see that. I can't believe so many people in the comments are so ignorant.
@@LyConstantine what “wrong information” are you talking about?
@@Phymacss that you can calculate, deduce or prove that 0^0=1 through some function analysis (it's actually the opposite) or at all. As I said you can just check at least wiki for more thorough explanations.
“let’s go zero point zero zero one, what do you get?”
*”yes”*
From a computing standpoint, that’s a valid answer
Martin Reid lol
Puts phone away smiles* thank you :)
That would be me tho 😂🤣
0.001
love the fact that the other students were annoyed at the girl with the phone, like they actually want to be there to learn. I wish my maths teacher was as good as this!
Everyone else on UA-cam wants to skip the math and watch Star Wars battles in Microsoft Windows Desktop.
@@mtaur4113 fuck outta here with your superiority complex. Different people enjoy different things. Many people enjoy both of those things.
Trigonometry for AS and A level ua-cam.com/video/-WzZRx4vVxI/v-deo.html
Watch and share and subscribe
He actually made it interesting!
@@mr.fufucudlypoops8207 I'm almost certain that 99% of people who "wish he was my math teacher" never clicked around and tried to learn any math from the guy. They wish they could surf UA-cam and get course credits.
Sure, it's clever media use or whatever, but these are mostly the same people who only remember three things from math and flip out when Common Core teaches their kids a 4th way to think about something.
This guy is a great teacher. Learning the 'mechanics' behind limit functions. And showing that some laws are not absolute, but just the best theory we have right now. And that is what academic students need to grab.
He just made them type in numbers in their calculator to get an unexpected result. Of what use is this other than a bad bar trick if you don't understand why it works like it does? It seems to me just like a magic trick, an unexpected result that you don't understand, learning nothing unless the trick is explained.
Although it is not exactly the same since a magic trick doesn't necessarily trick people into thinking they've learned anything useful(unless you can tell me that there is useful information in this and what it is, apart from the fact that it spikes curiosity.).
@@joellasoe3218 Sometimes the math comes first and applications follow.
@@andtrrrot what math?
@@joellasoe3218 en.wikipedia.org/wiki/Abstraction_(mathematics)
@@joellasoe3218 he was demonstrating the method of limit chasing to provide proofs for competing laws. He set up the example of two laws everyone knows but conflict when they hit eachother so that he could show them an example of how to work toward a limit so that he could explain the proper notation for limits. He was literally doing exactly what you were complaining he wasn't. Perhaps the flaw is not with the lesson but with the observer.
The applications for this method are wide as basically every major mathematical proof still waiting to be found at some point will involve the need to address limits and discuss the exponentially declining approach toward some limit.
Just because you don't understand it or it's applications does not mean it is a magic trick lacking in practical application.
this was so engaging that i brought out my own calculator and did the calculations as he went along, such a good teacher
also one interesting thing that i noticed was when i had 20 zeroes (0.000000000000000000001^0.000000000000000000001), the calculator just said 1. and it's not like my calculator is limited to the display, i have seen it produce recurring decimals that take 30 seconds to scroll to the end of. so i tried adding in five more zeroes, and the answer was still 1. so could that mean that there is a very specific, infinitesimally small number after which all x^x = 1?
Omg, I would love to have you as a math teacher! You have so much enthusiasm and you actually get around to somewhat trivial questions like this.
+Eddie Woo Where do you teach math ?
0^0 = undefined (any number to the 0 power equal 1, EXCEPT 0)
+Nghi Tran moron! he is not saying 0^0 definitely 1, but it tends to 1.
Nghi Tran Are you a teacher yourself? If not, then you should probably stay silent.
Also, you're in fact wrong. 0^0 is not undefined, but rather indeterminate: it is a form that takes one every single possible number simultaneously.
Dean Ghosh rhl to be specific....
Mark Jamieson 0/0 is not undefined, but rather indeterminate.
If Netflix make a show with this guy explaining all of the Math , i would sit there and binge it all. 😆😆😆😆
😂😂😂😂😂😂😂😂😂😂😂😂😂👏👏👏👏👏👏👏👍👍👍👍👍👍👍
If netflix made a show of it they'd use cheesy actors and you'd be back watching his UA-cam videos before you finish 1 episode.
5 years of episodes watched nonstop in 2 weeks, yeah! xD
You probably already know these channels but Veritasium (Derek Muller) and Stand-Up Maths (Matt Powell) are two pretty good maths educators, also Australian.
@@Casey093😢😂🎉🎉🎉🎉🎉🎉🎉😢😢😢😮😮
*Has pouch for calculator*
Me: He must love that calculator, He has a bed for it and eve-
*Throws it on the table without any care*
Me: wait wha ...
Exactly what I thought 😂😂
Great minds think alike 😂
The pouch is not for the calculator but for its cover.
He's not being respectful to the calculator.
Maybe because that thing’s like 120 euros (atleast mine is, TI 84). Throwing it on the table with paper and books on it really won’t damage it. But say it falls on the ground, you’d be fucked.
1. Engaging everyone.
2. Learning for all.
3. Making everyone curious.
4.Accepting when he don't know the answer.
I wish I had someone like him to teach me Maths.
What a nice teacher!
And for people who wondered, the reason why the number started to increase at around 0.4 and 0.3 can be explained by differentiating the function y=x^x, which gives dy/dx = x^x(lnx + 1).
When the dy/dx is smaller than 0, the function will decrease; and if it's larger than 0, the function increases.
lnx + 10)
x
Thank you, i was wandering what the lowest number was, i didnt think of Differentiating the function makes a Lot of sence.
@@0brine No problem!
Differentiating functions can help you find the range of x where the function is increasing (f'(x) > 0)/decreasing (f'(x) < 0). I live in HK, and we learnt how to draw functions with the calculus lessons in my secondary school. This is a crucial step we had to learn back then.
@@0brine yep, differentiating and solving the result when equal to zero is one way to find local minima or maxima of a graph.
Very true, but with a slight quibble. Differentiation tells you exactly what is happening and precisely at what point in the function finds its local minima, but it doesn't actually explain why the original function that is being differentiated behaved that way, in and of itself.
@@disecke "why" in mathematics is sometimes elusive. To me, 1/e seems like an eminently satisfying place for that reversal to happen.
I can see in the way he speaks and moves his arms that he is really excited to teach that. I dare to say that this guy loves teaching math. And it's just beautiful.
@Rusty doesn't matter
Dude, His accent is 100% British (unless we follow the theme of the session, then his accent is 99.99999977% British)
GREAT GUY! Very clear, concise and insightful showing meaningful context for the examples leading to limits/asymptotes. Not simply saying stuff by rote or dictum. Facile understanding of the true meaning of teaching vs "reviewing material"! I saw comments noting he's Chinese meaning I suppose genetically gifted at math and science! Too funny!!!
@@senorrones Australian, not British
@@spectywecty i mean, isnt that a compliment?
to answer the kids question of why the turning point is there (good question):
let y=x^x
to find a local minima we can use first differential of y
dy/dx=x^x(lnx+1)
we let dy/dx=0, solve the equation for x and you get 1/e, which is 0.367879
was looking for this even tho i dont get that.. thanks man
Natural logrithms blow my mind.
good ol natural log
@@anderivative always showing up where you least expect it
on u toob as per blackpenredpen 0^0 is undefined v=12Nae7qYxs4. prof dave explains see calculus v=rBVi_9qAKTU
From a calculus perspective , the limit as X approaches 0 for X^X actually does not exist.
just to clarify, the reason why he says that it "breaks down" when you use negative numbers is because when you set x to very tiny negative numbers you instead approach -1 instead of 1.
In calculus, if the limit from the left and the right aren't equal, then the limit simply does not exist.
if you approach X^X from the left you get -1, and if you approach X^X From the right you get 1, therefore 0^0 actually doesn't have any real value.
It is just a 'perfect' step signal from an engineer perspective.
14:03 One student asked very important question “why does the function value start to increase at x=0.4, to answer that you have to know that it will inflect at exactly x=1/e=0.3678 this x value is when the first derivative of the function x^x equal to zero , f’(x)= (x^x)(1+ln(x)) =0
I mean, the first derivative helps to identify the minimum/maximum point of the curve the function is describing; so this answers at which value the curves reaches this point, but still doesn't explain why it happens at 1/e. Having "e" involved is a really nice clue though.
NeoDaedalusCz the number e is everywhere, by the way the function x^x can be represented also as : e^(x*ln(x))
Thanks for answering that, I had to search the whole comment section for the answer.
1+lnx=0 lnx=-1 e^lnx=e^-1 x=1/e
Me, a Business and English student kinda guy in his mid 20s
"sorry i don't speak martian"
Who else is watching this instead of Netflix during their Covid isolation?
Me too 🤫
Me too
Mee too iGuess
me
Ya me pn
This is the mathematical interpretation of the phrase "it gets worse before it gets better".
Albeit later being fallacy.
You are the most absolutely hilarious person I have ever heard
0^0
I know fuck all about maths, but you know you may actually be right! This is how this happens...
Can you express the function? It seems involving 1/e.
This video addresses a problem I've been dealing with for some time. Although it could be demonstrated using the Taylor series of
e^x , this method is amazing. Thank you for sharing it. I greatly appreciate it.
Mr. Woo, I just came across this and want you to know how much I appreciate your teaching style. Math has fascinated me my entire life (which began in 1951) and it's teachers like you that made it fascinating. Bravo!
Best of everything,
Mike
THIS IS THE BEST WAY to teach Maths. What make Maths difficult for some is that it has to follow detailed rules and a lot of sets of logic that human invented while observing patterns in numbers and in nature. When you get to associate the reason of these rules; like what Eddie Woo is doing, then Maths become easier to understand. At younger age it is quite challenging to understand Maths because few teacher get to explain that every operation must follow its own set of rules. Most teacher don't bother to explain them because it demands time, effort and most importantly a fun way to get students attention. If we explain to children that traffic lights exist to enforce the law, then they will never appreciate their role. If you convince children that these traffic lights make traffic organized, give everyone his share of the road and save lives, then the children will definitively appreciate them. This is exactly what Eddie Woo is doing and I hope the other Maths teachers and professors will adopt his enjoyable way of teaching.
If I may make a comment that has nothing to do with the video, but supports your argument: Gerald Wilde published an e-book on traffic, and his conclusion (one of them) was that traffic rules are not about rules (or for that matter safety), but rather to allow more mobility, i.e. easy of movement.
That lesson gave the kids a chance to express their understanding of maths and air their conceptions...many of which were partly wrong. You can't drag people up to a level of understanding...they have to think it out for themselves. That lesson was good for them.
@@danmybusiness1167 yes, because traffic accidents hold up the traffic!
I had a math teacher at the university who was so passionate about his teachings. He always made time for his students. we could knock on his door and ask for help and he'd sit with you one on one for a while and really help you and explain things. Thanks to him I managed to go from a C in the first year to an A in the last year. We need more people like this :)
I now can appreciate what I put my teacher through at school.. Real value is on that board and within this lesson. Thankyou Sir Woo
This teacher is very cool. The students don't know it, but they are actually learning calculus.
YES! They're learning the concepts of 'limits,' which calculus is better suited to address. Excellent teacher!
yep. i'm currently taking ap calculus and limits were the first thing we learned
wait limits are part of calculus, GEEZ. I never realized how simple cal could be.
yep
Yeah I'm also confused, because the students sound much older than I am and they have never been introduced to limits somehow. I suppose they have more knowledge on other mathematical subjects than I do?
When you realise that competitive exams has removed a thing called "exploring deeply".
True bruh
agreed
That's the sad truth...
Competitive exams me aata h calaclus jo use kha hota h real life me pta nhi???? 🤪🤪🧐🤓
@@lucminff7796 well it has its uses but for really specific science/engineering related fields
Physicists invented it since they needed it
Im literally in uni doing engineering but no one's ever really delved into stuff like these as deeply as Eddie Woo, so whoever his students are must be so lucky. Also what class is this? like MAN, no one ever really teaches cool math theories like this in high school or uni unless you're taking a course in pure mathematics.
I think he teaches at his alma mater James Ruse, which is the most elite high school in Australia.
I don't know, for me it's not real maths, it's just using your calculator and get the intuition of what's happening. There's no proof nor any understanding of the concepts behind exponentiation, just an observation that anyone can make using a calculator. A limit has a rigorous definition, it has to be more than "it looks like it's 1". I feel like this way of approaching the problem teaches nothing, it only gives you the answer with no explanation of why that is true. The real way of getting to this result is to calculate the limit of e^(x ln x) when x approaches 0. Since the exponential function is continuous everywhere, this is the same as the exponential of the limit of x ln x when x approaches 0. And for that there's Hospital's rule, which, to be honest, I just found out about. I just remember when I was doing math we knew a bunch of limits and didn't have to re-demonstrate them (x ln x when x -> 0 being one of them). Anyway, that would be a more mathematical approach than using a calculator.
Same here
@@dicidicee Its important to highlight that this is a high school class, whilst it does not dwell into the advanced mathematics, Eddie did a great job on teaching student to think. To consider why 0 to the power of 0 is different than what they are expecting. He's teaching them how to think flexibly which in my opinion is a very important traits.
@@dicidicee the point is to first introduce the the concept in a easily digestible way(less things to consider like e and natural log), then expand on it without drawing up a table of values. It would be an issue if there's no continuation after this, but I'm pretty sure that's what he's doing off camera.
idk, it would be like using exponents to teach kids multiplication. Quite a bit more extreme but I hope you get the point.
Zero to the power of ain't paying attention, all with a smile on his face. Made class interact, let them see logic and start to understand without being "told". Amazing teacher :)