What is 0 to the power of 0?
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- Опубліковано 23 чер 2014
- Near the end of the lesson, one of my students asks a question about why the values start turning around 0.4 - I made a couple of videos that explain this phenomenon: • A Turning Point Myster... (Part 1) and • A Turning Point Myster... (Part 2)
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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 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.
"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.
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 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.
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😢😂🎉🎉🎉🎉🎉🎉🎉😢😢😢😮😮
This is a rare type of teacher wherein the students can actually learn from
Not in this case
so true
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?
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.
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 LOVE this man. I was very lucky to have had an amazing teacher like him for trigonometry with just as much energy and passion, and he made me love math. I wish i was one of his students!
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.
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….
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. ;/
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.
“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
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
I’m 40 years now. And never thought I would spend 14 minutes on a math video. This guy is such a great teacher. Lucky who have a teacher like this.
I love this energy while teaching. You convey the subject really good and keep the students interested and it really sticks.
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?
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?
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'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.
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?
“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
i studied limits 20 years ago.
I memorized most of the limit stuff cus nobody summarized it so brilliantly as you did.
Thank you.
Thanks to whoever that recorded this session, we know there are great passionate teachers around
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
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
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.
I am over 50 and I do teach at Univ at times BUT I wish I had a teacher like him in school/univ. I watched the entire video just out of curiosity and I was amazed by the journey of learning (so to speak) he took me on.
Best part was in the end when a student asks a questions and he replies that he didn't know.
I am from India and the education culture largely (changing though nowadays) is not to question the teacher. This emanates from the culture of respect to elders and the highest is accorded to a teacher who is supposed to be wisest of all. But this approach is misplaced when it comes to knowledge sharing.
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
'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.
This actuary approves of the technique and passion used here. WAY TO GO!!!
first time in my life I've watched a math lesson for fun
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.
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).
It may not be the same I'm working on but the class was great for today's exercises and projects thank you
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!
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
@@Jennyispoop yes... because the actual value of any no. Say 'x' is x - 0.9999999999999999....
i.e 5=4.999999999999999999999999999.......
My father asked me this question and my answer to this was "Undefined". Now that I see this, it is very interesting.
you were not wrong, it is only 1 when you apply a limit, otherwise the value is pretty much indeterminate
u r right 0^0 means 0/0 its the same thing
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....
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.
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 now can appreciate what I put my teacher through at school.. Real value is on that board and within this lesson. Thankyou Sir Woo
Eddie is amazing! I'm very fascinated by how why and perhaps where exactly the pattern shifts.
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
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.
Brilliant teacher, if only most teachers were as competent and inspiring.
Fantastic Eddie. Such an inspiring math prof. I had fun and imagine those students in real time. Blessed and learning to the élan.
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
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"
This showed up in my recommendations and I thought, I'll watch a little bit of this, but I ended up seeing the whole thing. I wish I had a teacher with your enthusiasm in my school years.
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.
+
@@Mr6Sinner 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
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
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.
*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
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.
@@MKRM27 tf bro
Just came across this video. An excellent teacher and an intriguing question from a student. So I wanted to find the minimum of the function, and because I finished my college 44 years ago I googled it and calculated the minimum to be 1/e as many wrote in the comments, but I enjoyed the exercise. Thanks for the teacher and the student, I wonder what he became to be
I wish every student has a teacher like you.
Sacred trust.
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'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.
That was the best explanation I have heard of that question. Nicely done. When it was taught to me, it was simply given as 1 is the answer.
Sir, I just addicted to your videos. You are awesome sir. Now I am actually interested in maths. And in my mind my thoughts are that none of my classmates know this basic concepts they are only just doing limits, integration etc. without understanding it's actual meaning. Thank you sir for your great effort.👍
"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?
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…
What a brilliant teacher who actually makes math interesting and enjoyable in my opinion.
The difference between a good student and a poor one is no double influenced/determined by the teacher.
Never have I been so entranced listening to a mathematical explanation. Brilliant & loved it!
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.
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
Great idea to look at limits and exploring how a calculator deals with 0^0. The other problem not explored is the mathematical idea behind zero power, namely division by zero creating something undefined. It would be interesting to consider how this sits with an answer from limit of 1. Really interesting problem in maths here. Great presentation!
Good teaching style. Students are lucky and Eddie shows things clearly.
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 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*
An example of the perfect maths lesson. It even got them asking questions to get more information.
This is a really good explanation on 0^0, Nice job giving your students a (close) answer to this!
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 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
Well, here I am, during Christmas break, enjoying a video about math....... I may be a nerd😂😂
Same :)
An excellent teacher through and through. An absolute legend! ❤
“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
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.”
His teaching reminded me to my math teacher. Ngl, he had graduated in Japan and got a lot experience. He's the one who make me become smart thinking and analysis the number which is hard for some stu
I love maths and this teacher just reminds me of that❤
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
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
One thing I love from this lesson is by looking at both methods of reading the division of exponents, also helps prove the law that anything to the power of 0 is 1. Because on the left you have a^m/a^n, and to the right you have that equals a^(m-n). Now, if you set both m and n to 1, it becomes a^1/a^1=a^(1-1) Now, obviously, anything to the power of 1 is itself, because you have 1 unit of a so the left side simplifies to a/a, and anything divided by itself becomes 1. Then on the right side, we know that 1-1=0, so the right side simplifies to a^0. Now we have 1=a^0, and there's the proof. However, it's quite a bit trickier when it comes to a=0. Because you start with 0^1/0^1=0^(1-1). So, like we proved before, the right side simplifies to 0^0, while the left sides simplifies to 0/0, which is undefined, because you cannot divide by 0. So we're left with 0^0=0/0 And while normally anything divided by itself would equal 1, you cannot divide by 0. However, when you enter 0^0 into a calculator, you get 1.
P
I never liked anything to do with numbers but this guy is making it so interesting that I am actually looking to see what the values are. very cool.
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
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.
Nothing not multiplied is always one. Nothing is always sufficient for everything.
Best part is that he clearly told his student that he don't have answer right now ,but we would try to find out ,love from India.❤
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.
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 ...?
Great problem-based introduction to limits. The problem is that to answer the last question, one would have to use the same example to introduce calculus, i.e., how the slope is negative to the left of the minimum and positive to the right and that at some point it is zero -- and how you can take approximations of the slope and then arrive at the idea of taking the derivative, setting it to zero, and solving for x.
Thank you!
You always help me to find my questions's answer ! Your students are sooo lucky . I love mathematical so much but my teacher makes it boring , we need teachers like you!
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?
How can students not pay attention to such an awesome teacher...
Even me, he got my attention