Thank you so very much. This explanation is one that is so needed in order to understand this concept. If only middle school teachers would explain how to arrive at the answer of "one" when a number to the power of 0 is 1, then students would have logical procedures in arriving at an answer that makes sense instead of just repeating or memorizing that the answer is 1. Excellent. Loved the video.
It’s nice to see someone explain something simple in the most convoluted way possible. Usually teachers try to do the opposite, so it’s refreshing to see something a little different.
Thank you. Nobody has ever explained it so clearly and simply as you have, and for the reason that it has to be so. If they had, 40+ years ago, when I was at school, I might have had a more positive attitude to maths. As it wasn't explained to me, my attitude to maths, was akin to my attitude to religion; skeptical of anything that did not prove itself and was the only possible choice available. Thanks again, Gary.
No one in the field of mathematics is debating whether 0^0 is 0 or 1. We are all sure that 0^0 is in fact not defined. Also, exponents are closely related to logarithms. In fact, in the history of mathematics you will find that logarithms have been around longer. Once one has defined the natural log function via the an integral (which is how many believe it should be done), then proving a^0 = 1 for all a > 0 is a trivial task.
I'm not sure what a "real mathematician" is, hence I did not use that terminology. Maybe I am, maybe I'm not. I've heard many math educators (at the college level, even) say that everyone is a mathematician. Through my undergraduate and master's in math, and now as a teacher of calculus for 22 years, I've never come across anyone, or read any math text book, or other related math book, that has entertained the idea of 0^0 being equal to 1. I could certainly why one would want to DEFINE 0^0 as being one, from the basis of limits, but I could define anything I want and it doesn't necessarily make it correct. I'd love to hear more about this debate (other than on UA-cam!, or read where people want this is being discussed, so if you could, please direct me to the some reputable literature on this. I'd be curious if these (people) are debating that 0/0 - 1, also. I suppose this would explain a lot. I recall two instances that got me riled up, albeit early in my teaching. One was an elementary school teacher telling me that the sqrt(4) = +/-2. I tried explain that it wasn't, it was only 2, as the sqrt symbol implied only the principal (positive) root, but they were having nothing of it. I tried my best to explain why x^2 = 4 has two solutions, one positive, one negative, and showed two different ways to see that. Alas, it was a futile effort, so I gave up. Next, was someone who was convinced that sqrt(x^2) was x and not absval(x). Even after I showed examples of negative x's they just would not let their false thinking go. But, that doesn't mean one couldn't define sqrt(x^2) as x...they'd just be living in a completely different world (ie, their own) of mathematics:). Oh, I appreciate being referred to as a "kid." Many have said I look young for my age (I am over 50)...and I'm not a billy goat, either:).
@@bryan9587 I know these are old comments but. Indeed, there is no debate about whether 0^0 is 0 or 1. It's considered to be an indeterminant form. You can construct examples where 0^0 becomes 1, or 0, or anything actually. Take for instance, these two functions, both become 0^0 at x=0, but one of them goes towards 0, and the other towards 1. www.desmos.com/calculator/box5zwu1w8
I would agree that no one in the field of mathematics is debating whether 0^0 is 0 or 1. On the other hand, I disagree with the conclusion that we are all sure that 0^0 is in fact not defined. It all depends on context - in particular, what does "exponentiation" mean? Depending on what exponentiation means, either 0^0 = 1 or 0^0 is undefined. For example, you can look up the set theoretic construction of the natural numbers (which includes 0 as a natural number) and look up the set theoretic definition of natural number exponentiation. For two natural numbers n and m, n^m is defined as the natural number in bijection with the set of functions from m to n. In the case that m = 0 and n = 0, there vacuously exists precisely 1 function from the empty set to itself. Hence, by this definition 0^0 = 1. Generally, whether 0^0 = 1 or 0^0 is undefined depends on whether exponentiation is viewed as a discrete or continuous operation. In virtually every discrete context, 0^0 = 1. The reason for this is that, in the discrete context, exponentiation represents repeated multiplication. As such, x^0 is a product with no factors, i.e., the empty product, regardless of the value of x. The empty product is defined based on the associative property of multiplication, and hence, has a value of 1. On the other hand, if you're in a continuous context, then exponentiation is defined in terms of limits or logarithms, as you suggest. In such contexts, the definition of exponentiation does not allow 0 as a base to be raised to virtually any power. As such, 0^0 is left undefined. If you teach a calculus course, it makes sense to state that 0^0 is undefined, since you don't want students to say that their limit is 1 when they get the indeterminate limiting form of 0^0. Of course, things then become awkward when you get to power series and have to use 0^0 = 1 there. Of course, you _could_ try to explain why 0^0 should be replaced with 1 in the context of power series in a number of ways, but it fits nicely into the discrete context there, since the exponents of x in a power series are discrete exponents representing repeated multiplication.
This was extremely helpful. I'm currently teaching myself Electrical Engineering and the book I'm using did not explain this and I was very confused. Thank you so much.
Not only that it was explained clear and concise, but It was also explained in a simple way, apart from what I've seen till now (and I have watched some videos in this particular subject). Love it, I am looking forward on this channel if I encounter another misconception. Thank you!
Thank you so much for this video, It was briefly talked about in my math glass but for me to understand something I need to know WHY, and this video explained it very well. Have a great day!
2^0 =1 which could stand for the number of points in a dimension. For example 2^0=1, so 0 is the dimension and 1 is the point in the zeroth dimension. Then, 2^1=2 which would be the 1st dimension that has 2 points on a line. 2^2=4 which equals the 4 points on a 2D square for the second dimension. 2^3=8 which would be the 8 points on a 3d cube in the third dimension. 2^4 = 16 which would be the 16 vertices on a 4 dimensional cube or tesseract....then, a 5d cube has 32 vertices (2^5=32). etc
Thank you! It's one of things I've always wondered about. It's like, you can go through entire courses and never have to know WHY it's this way (which usually means just taking someone's word for it). But I don't like to ever do that. I have to see it for myself. Thanks again!
Amazing!!!😅 This Video has single-handedly answered my lifetime's question or one of them, and my answer is that invisible 1 that no other video told me about. 😭Bravo *claps*
@@H3Vtux I absolutely second MysticCyber's re both the applause and the importance of mentioning the "invisible 1". Could you please put a link to some of the articles/debate you mentioned at the end of the video though? I'd love to learn about them and also help me understand why negative exponents results in fractions of 1. Thank you so much!
Fantastic! I needed to explain this to my teenage son learning about exponents (his teacher just did the hand wave and called it good), this video is a perfect explanation for him. Thanks!
Another way to prove that a^0=1: a^n / a^n = 1 because anything divided by itself is 1. But, if you apply one of your exponent rules...: a^n / a^n = a^(n-n)= a^0 = 1, because of the first line.
Have you ever seen things on TV starting as one but then becomes two and then four and then eight and so on? If so, then the 0th power always being 1 makes a lot more sense. In this case, the number 2 is being risen to a power. So if you start from 5 for example and keep raising it to a power, you must therefore keep multiplying it by 5. And we start from 1, multiply it by 5, then it becomes 5 because 1x5=5. After that, the 1st power is just the number itself. Then, for the second power and so on, just take that many of the base number and multiply them together. The base is the number we want to multiply by starting from 1 and the exponent is how many times we do that. So we multiply 1 by the base number the same number of times as the exponent. Hence, the 1st power is 1 because 1 times any number is always just the latter itself.
Incredible!! I dont seem to remember these math rules if they dont make sense to me and its also not any fun that way. I wish everything could be explained like in this video. I would never forget a thing. Thank you so much, this was fascinating 😁😁
1:32 minutes in and I finally understood why, I guess it's the same as why (-2)^2 isn't the same as -2^2, on both cases there is an invisible multiplication with (-1) so the first one means ((-1)(2))^2 while the second one means (-1)(2)^2 which by order of operations we always do what is inside () first, then the exponents 2nd, and since there is nothing inside the second case we do not multiply (-1)(-1).
The video should say any non-zero number raised to the 0 power is always equal to 1. 0^0 is indeterminate, which is useful in calculus to compute limits of indeterminate forms using l’Hopital’s rule.
When n is not equal to 0, n^0 is 1 BECAUSE: 1. The limit where x approaches 0 in n^x will approach 1 2. n^0 = n^0/x where x is greater than 0. This results in the xth root of n^0, or the xth root of 1, which is always 1 3. n^-0 = 1/n^0 which is 1/1 which is 1. As -0 is 0, n^-0 = n^0 and n^-0 equalled 1 so n^0 is 1.
if y>0, y^x=exp(x ln(y)) then : y^x=exp(0 ln(y))=exp(0)=1 and expo(0)=1 because ln(1)=0 and ln(1)=0 because ln is the unique primitive of 1/x that cancel in 1. and if y
a^0 = 1 (such that a does not equal to 0) why? proof: a^0 = a^n-n (n-n=0) but a^n-n = a^n/a^n which is equal to 1 clearly(anything divided by itself is equal to 1) hence a^0=1
I find 0 wierd at times because sometimes it defines the laws of maths. Like 0/0 or 2/0 which doesn't happen with any other number. Even -1^1/2 Aka "i" is easier to understand at times. There isn't a single number where x(x) = x except for 0 or even complex fractions (1/x) / ( (1+x)/x) 1/x ÷ (1+x)/x 1/x × x/(1+x) x/x(1+x) 1/(1+x) If we say x = 2 then (1/2) / (1+2) / 2 (1/2) / (3/2) = 1/3 1/(1+2) = 1/3 But if x = 0 1/1+0 = 1 (1/0) / (1+0/0) So complex fractions are wierd. So what if 0 was also a concept Edit: As much as I did prove to myself that 0^0 =1 (0^1 = 1x0, 0^0 = 1, 0^-1 = 1÷0) But it's hard to comprehend at times. So Zero multiplied by itself zero times is 1. Or at least close to 1 since 0.000001^0.000001 = 0.999... Also yes I did say that there isn't a number where x^2 = x but I forgot that 1^2 = 1 . Sometimes one for me is wierd but really useful especially with fractions. In co-ordinate geometry "0" isn't an issue. I meant that it's an issue with solving some equations. There are also some wierd things with 0 like 0! = 1 (and I still believe that 0/0 = 1 since 0.0001/0.0001 = 1) Lots of times 0 is understandable but it's still different thiugh if it would be different maths would be possibly wrong. Eg : xy =x yes 1 would fit is as y =1 and y = x 1=x but it's only true with 1x1. If y =2 then 2x =! x unless x = 0 What i mean is that 1x1 = 1 but only there while 0 works with everything. 0x2 = 0 0×34553566347899 = 0. It's not changing since 0 + 0 = 0 again. Makes sense but almost none of the numbers do that. Best example is with fractions 20/1 = 20-1 19-1 18-1 17-1 .... x20 20/2 = 20-2 18-2 16-2 14-2.... x10 20/0 = 20-0 20-0 20-0 20-0... xoo But also... 20/-2=-20-2 18-2 16-2 14-2 x-10 20/ -1 = 20-1 19-1 18-1 17-1... x-20 So when it finally reaches 0 we get +/- oo
Yes, it all stems from the facts that humans designed math to be used for counting things, not for "doing math". It's hard to imagine having 0 of a thing, which mathematically is different from having nothing.
@@H3Vtux I was also asking myself on why is it better to say that a circle has infinite edges then it has 0 edges. But over time I answered it myself. (I love writing things down when no one cares and it's too obvious) . Square has 90°. Pentagon has 72°. Hexagon has 60°... More sides and lower (but more) angles. Keep on making new sides and angles. Do it oo many times and you get a circle
Well adding in an ‘invisible 1’ makes zero sense to just add a brand new component, but by that logic, is 0^0 then not 0? Because apparently if there’s always an invisible 1, it would be 1 x 0 = 0. Yet apparently it’s ‘proven’ that 0^0=0? Which I disagree with, if I gestured to the air and said ‘take an object’ there would be no object, because there’s no container, there’s literally nothing there. But once it’s put in some maths equation, apparently people defy real world logic and it becomes something?
Sometimes I like to think dividing by zero practically would mean an object is warped into it's own non-existence. But if you start out with nothing, does the reverse happen? Are things spontaneously created at once?
One more: x^y/x^y=1 ----------1 [same numerator and denominator will make 1] also, x^y/x^z = x^(y-z) But here, z=y Therefore, x^y/x^y = x^(y-y) =x^0 ----------2 [y-y=0] Now, x^y/x^y=x^y/x^y From eqn 1 and 2 1=x^0 Hence proved that x^0=1.
It was great . But all the math teachers told us there are many ways to ascertain this subject. Which way we should always use it? which one is better?
There is one more thing about 0 I was playing with calculator and saw that tan89.999 and 180/pi aren't that different 57.2957795131... Well I did estemate that tan90 starts with 572957795131.... TanA= sinA/cosA Sin90 = 1 Cos90 = 0 1/0 = 572957795131 Also I said to myself 10 ÷ 2 is how many times can 2 go into 10. So 0 can go into 1 infinite times. But how can a negative number divide negative times? 20/-2 = -10. I kind of get that -20/2 = -10 because 2 can go into neg 20 neg 10 tines
That limit will go to e. There are actually 3 indeterminate forms that you put in this equation. At x=0 and x=1 and as x gets infinitely large, you get the form 0^0, 1^infinity, and infinity^0, all of the limits will go to e.
If the reasoning simply doesn't make sense then either the reasoning or the expression of the math needs to change, this is something a whole lot of teachers don't like to hear. Even if a concept works you must find a way to show that it makes sense! If you can't then you can't teach math properly.
Honestly I learned my 7 year old son both positive and negative exponents and I did not even have to explain 0 because without power 0 you can not explain 3 ^ -1.
For Fractional or irrational exponents things get very complicated and there's unfortunately no way I can explain that in a comment section. I would imagine other youtubers have covered this in videos, probably kahn academy.
But if you have 2 to the power of 1 it equals 2, because the base is 2 not 1, there for the base is what it is, it is not 1. The base of 2 is 2, the base of 3 is 3, the base of 4 is 4 and so on...... it is not 1, unless of cause you write a base of 1. So 2 to the power of 0 means I have a base number of 2 and they want it to be 0 times that means 0. No matter how they want to mess with it, it still means I started with a base of 2. Or it could even mean that I’m not multiplying it at all 2 to the power of 0, just means 2, I stay with the 2 because I’m not doing anything at all to it as it’s a power of and not really a multiplication, it’s just 2.
Raising the exponent of a positive integer tells you how many times you need to multiply itself (e.g. 2^3=2*2*2=8). For negative integer exponents, it tells you how times you need to divide (e.g. 2^-3=1/(2*2*2)=1/8). For a 0 exponent as long as the base is not 0, there is no special rule for this. It will always equal to 1. This doesn't mean you multiply/divide 0 times. It doesn't work that way. You can think of it of as 2^(3-3)=2^3/2^3=8/8=1). For non-integer exponents. you will have to learn the rules of rational exponent. It doesn't make sense to say 1/2 times or so. That's not how it works. x^(m/n) is the same as nth root of x^m or nth root of x and then raised this to the mth power. For example, 8^(2/3)=4. If an exponent is irrational, you can't do anything about that. We just leave it as an answer there (e.g. 2^pi).
What this video really does is explain to me that calling 2^2 "2 squared" is kind of misleading, since what is being done has nothing to do with geometry. If it were about geometry, then 2^2 would be representative of a line segment of length two being made into a square, which would result in 4 units. 2^1 would just result in a line segment, which would be undefined in area, and 2^0 would be a point, also undefined in area. So it's not really about geometry. Geometry just lines up nicely with any exponent higher than 1.
OK. I have 2 comments and a query please: First of all, thank you. Everything about your video is spot on, and I appreciate the presentation. I'm gonna share it with my Facebook community. Someone is sure to be intrigued or at the most, thankful that this little birdy was dropped in their lap. Now for the query, are you sharing this because it (and all my basic math that I thought I never would use anywhere) has something to do with understanding computers and their operations? HINT: PLEASE say NO!
I really am lost.If 0^0=1,then it follows that (0+0)^0=0^0+0^0=2,In that case (1+1)¹=?. We know that (1+1)^1=1¹+1¹=2.so (0+0)^0=2=(1+1)^1.This conclusion leaves us to infer 0=1,which is an absurd conclusion. 0:08
0^0= 1 and 0; two values. Please look the paper: International Conference on Differential & Difference Equations and Applications ICDDEA 2017: Differential and Difference Equations with Applications pp 293-305 | Cite as log0=log∞=0 and Applications Authors Authors and affiliations
0^0 if anything clearly is more towards 1 and far from 0. If you don't believe me, take your calculator and do this: 0.0000000000000001 ^ 0.00000000000000001 =? See if the answer is closer to 0 and closer to just about 1. Right nibbling it at its tail.
No, raising zero to any non zero integer means we're multiplying zero by itself that many times. 0^3=0*0*0 0^0 is the result of two conflicting mathmatical rules n^0=1 0^n=0 So depending on which of those rules we follow we get either 1 or 0 with 0^0...
Thank you so very much. This explanation is one that is so needed in order to understand this concept. If only middle school teachers would explain how to arrive at the answer of "one" when a number to the power of 0 is 1, then students would have logical procedures in arriving at an answer that makes sense instead of just repeating or memorizing that the answer is 1. Excellent. Loved the video.
I never saw a number to power 0 in my 16 years studying maths
It’s nice to see someone explain something simple in the most convoluted way possible. Usually teachers try to do the opposite, so it’s refreshing to see something a little different.
You're welcome!
I think an easier way to explain this would be this
a^x/a^y = a^(x-y) => a^(n-1) = a^n/a;
Let a=2 and let's start with 2^4 = 16;
2^4 = 16;
2^3 = 2^4/2 = 16/2 = 8;
2^2 = 2^3/2 = 8/2 = 4;
2^1 = 2^2/2 = 4/2 = 2;
2^0 = 2^1/2 = 1;
And moving forward, that's why 2^-1 = 2^0/2 = 1/2, etc;
so basically it's a geometric sequence where the values get divided by the base
excuse me WHAT
no I don't understand whaaat
@Lukas yea, it should ...
Anyway, it's the same demonstration showed in the video .
weird flex but okay lol
Thank you. Nobody has ever explained it so clearly and simply as you have, and for the reason that it has to be so. If they had, 40+ years ago, when I was at school, I might have had a more positive attitude to maths. As it wasn't explained to me, my attitude to maths, was akin to my attitude to religion; skeptical of anything that did not prove itself and was the only possible choice available. Thanks again, Gary.
Me bhi maths ke vedio banata hu aap dekhe or comment kre youtube.com/@RKEVEDIO?si=dSkOtJxP-JiNfI4v
No one in the field of mathematics is debating whether 0^0 is 0 or 1. We are all sure that 0^0 is in fact not defined. Also, exponents are closely related to logarithms. In fact, in the history of mathematics you will find that logarithms have been around longer. Once one has defined the natural log function via the an integral (which is how many believe it should be done), then proving a^0 = 1 for all a > 0 is a trivial task.
Bradley Stoll Oh ffs. Just because YOU are not debating that topic, doesn't mean real mathematicians aren't either. Calm down, kid.
I'm not sure what a "real mathematician" is, hence I did not use that terminology. Maybe I am, maybe I'm not. I've heard many math educators (at the college level, even) say that everyone is a mathematician. Through my undergraduate and master's in math, and now as a teacher of calculus for 22 years, I've never come across anyone, or read any math text book, or other related math book, that has entertained the idea of 0^0 being equal to 1. I could certainly why one would want to DEFINE 0^0 as being one, from the basis of limits, but I could define anything I want and it doesn't necessarily make it correct. I'd love to hear more about this debate (other than on UA-cam!, or read where people want this is being discussed, so if you could, please direct me to the some reputable literature on this. I'd be curious if these (people) are debating that 0/0 - 1, also. I suppose this would explain a lot. I recall two instances that got me riled up, albeit early in my teaching. One was an elementary school teacher telling me that the sqrt(4) = +/-2. I tried explain that it wasn't, it was only 2, as the sqrt symbol implied only the principal (positive) root, but they were having nothing of it. I tried my best to explain why x^2 = 4 has two solutions, one positive, one negative, and showed two different ways to see that. Alas, it was a futile effort, so I gave up. Next, was someone who was convinced that sqrt(x^2) was x and not absval(x). Even after I showed examples of negative x's they just would not let their false thinking go. But, that doesn't mean one couldn't define sqrt(x^2) as x...they'd just be living in a completely different world (ie, their own) of mathematics:). Oh, I appreciate being referred to as a "kid." Many have said I look young for my age (I am over 50)...and I'm not a billy goat, either:).
@@bryan9587 I know these are old comments but.
Indeed, there is no debate about whether 0^0 is 0 or 1. It's considered to be an indeterminant form. You can construct examples where 0^0 becomes 1, or 0, or anything actually.
Take for instance, these two functions, both become 0^0 at x=0, but one of them goes towards 0, and the other towards 1.
www.desmos.com/calculator/box5zwu1w8
@@bryan9587 What are you even doing here?
I would agree that no one in the field of mathematics is debating whether 0^0 is 0 or 1. On the other hand, I disagree with the conclusion that we are all sure that 0^0 is in fact not defined.
It all depends on context - in particular, what does "exponentiation" mean? Depending on what exponentiation means, either 0^0 = 1 or 0^0 is undefined.
For example, you can look up the set theoretic construction of the natural numbers (which includes 0 as a natural number) and look up the set theoretic definition of natural number exponentiation. For two natural numbers n and m, n^m is defined as the natural number in bijection with the set of functions from m to n. In the case that m = 0 and n = 0, there vacuously exists precisely 1 function from the empty set to itself. Hence, by this definition 0^0 = 1.
Generally, whether 0^0 = 1 or 0^0 is undefined depends on whether exponentiation is viewed as a discrete or continuous operation. In virtually every discrete context, 0^0 = 1. The reason for this is that, in the discrete context, exponentiation represents repeated multiplication. As such, x^0 is a product with no factors, i.e., the empty product, regardless of the value of x. The empty product is defined based on the associative property of multiplication, and hence, has a value of 1. On the other hand, if you're in a continuous context, then exponentiation is defined in terms of limits or logarithms, as you suggest. In such contexts, the definition of exponentiation does not allow 0 as a base to be raised to virtually any power. As such, 0^0 is left undefined.
If you teach a calculus course, it makes sense to state that 0^0 is undefined, since you don't want students to say that their limit is 1 when they get the indeterminate limiting form of 0^0. Of course, things then become awkward when you get to power series and have to use 0^0 = 1 there. Of course, you _could_ try to explain why 0^0 should be replaced with 1 in the context of power series in a number of ways, but it fits nicely into the discrete context there, since the exponents of x in a power series are discrete exponents representing repeated multiplication.
This was extremely helpful. I'm currently teaching myself Electrical Engineering and the book I'm using did not explain this and I was very confused. Thank you so much.
Not only that it was explained clear and concise, but It was also explained in a simple way, apart from what I've seen till now (and I have watched some videos in this particular subject). Love it, I am looking forward on this channel if I encounter another misconception. Thank you!
unbelievably easy explanation. easy to understand. Thank you.
Thanks man, this was actually my first teaching video so it's nice to see people stumble upon it every now and then. I'm glad it helped!
At 1:18 you said that it's being multiplied by "well, 1" and I don't follow. It seems like it should be they should be multiplied by each other.
I can't compliment your videos enough, they're wonderfully explained.
Sometimes you just need to find someone that can explain things in different ways to learn it, Thanks.
Thanks man, i'm glad it helped!
Thank you so much for this video, It was briefly talked about in my math glass but for me to understand something I need to know WHY, and this video explained it very well. Have a great day!
This was sooo helpful .I feel like sharing this with everyone I know but that would make me SERIOUSLY nerdy. lol.
Amazing.....make more video like this...plz
0: OMG!!! I was stuck on this for so long! I FELT SO DUMB BUT THANK YOU SO MUCH FOR CLEARING IT OUT!!😭
2^0 =1 which could stand for the number of points in a dimension. For example 2^0=1, so 0 is the dimension and 1 is the point in the zeroth dimension. Then, 2^1=2 which would be the 1st dimension that has 2 points on a line. 2^2=4 which equals the 4 points on a 2D square for the second dimension. 2^3=8 which would be the 8 points on a 3d cube in the third dimension. 2^4 = 16 which would be the 16 vertices on a 4 dimensional cube or tesseract....then, a 5d cube has 32 vertices (2^5=32). etc
Thank you! It's one of things I've always wondered about. It's like, you can go through entire courses and never have to know WHY it's this way (which usually means just taking someone's word for it). But I don't like to ever do that. I have to see it for myself. Thanks again!
4:50 is when it all made sense. great video though.
Amazing!!!😅 This Video has single-handedly answered my lifetime's question or one of them, and my answer is that invisible 1 that no other video told me about. 😭Bravo *claps*
Thanks I appreciate the feedback, I'm glad it helped!
@@H3Vtux I absolutely second MysticCyber's re both the applause and the importance of mentioning the "invisible 1". Could you please put a link to some of the articles/debate you mentioned at the end of the video though? I'd love to learn about them and also help me understand why negative exponents results in fractions of 1. Thank you so much!
Fantastic! I needed to explain this to my teenage son learning about exponents (his teacher just did the hand wave and called it good), this video is a perfect explanation for him. Thanks!
Wtf. Sue the teacher lol
The teacher probably didnt know why, he prob just accepted that anything ^0 is 1 without questions
Another way to prove that a^0=1:
a^n / a^n = 1 because anything divided by itself is 1.
But, if you apply one of your exponent rules...: a^n / a^n = a^(n-n)= a^0 = 1, because of the first line.
Thanks,much easier explained
damn
Nice. In that case "a" must be > 0
Thanks, great explanation!
Have you ever seen things on TV starting as one but then becomes two and then four and then eight and so on? If so, then the 0th power always being 1 makes a lot more sense. In this case, the number 2 is being risen to a power. So if you start from 5 for example and keep raising it to a power, you must therefore keep multiplying it by 5. And we start from 1, multiply it by 5, then it becomes 5 because 1x5=5. After that, the 1st power is just the number itself. Then, for the second power and so on, just take that many of the base number and multiply them together. The base is the number we want to multiply by starting from 1 and the exponent is how many times we do that. So we multiply 1 by the base number the same number of times as the exponent. Hence, the 1st power is 1 because 1 times any number is always just the latter itself.
Omg thank you so much !!! 😭 this helped me out greatly.
Incredible!! I dont seem to remember these math rules if they dont make sense to me and its also not any fun that way. I wish everything could be explained like in this video. I would never forget a thing. Thank you so much, this was fascinating 😁😁
the breakdown at the end was excellent
I bet the first person to figure this out was excited !
1:32 minutes in and I finally understood why, I guess it's the same as why (-2)^2 isn't the same as -2^2, on both cases there is an invisible multiplication with (-1) so the first one means ((-1)(2))^2 while the second one means (-1)(2)^2 which by order of operations we always do what is inside () first, then the exponents 2nd, and since there is nothing inside the second case we do not multiply (-1)(-1).
Thank you very much sir
We still hope more videos from you of such questions
I was thinking about this and came to the same conclusion as the video. Just wanted confirmation
Thanks for this video. Now i understand this rule perfectly.
OMG ITS SO HELPFUL I COULDNT LIFE OF ME FIGURE THIS OUT
The video should say any non-zero number raised to the 0 power is always equal to 1. 0^0 is indeterminate, which is useful in calculus to compute limits of indeterminate forms using l’Hopital’s rule.
When n is not equal to 0, n^0 is 1
BECAUSE:
1. The limit where x approaches 0 in n^x will approach 1
2. n^0 = n^0/x where x is greater than 0. This results in the xth root of n^0, or the xth root of 1, which is always 1
3. n^-0 = 1/n^0 which is 1/1 which is 1. As -0 is 0, n^-0 = n^0 and n^-0 equalled 1 so n^0 is 1.
0! is not equal to 1 in this scenario.
okay, but what i’d you kept going down. So what’s 3^-1 and 3^-2 etc. Would the division by the base strategy keep working?
Thank you sir love ❤️ from India
if y>0,
y^x=exp(x ln(y))
then : y^x=exp(0 ln(y))=exp(0)=1
and expo(0)=1 because ln(1)=0
and ln(1)=0 because ln is the unique primitive of 1/x that cancel in 1.
and if y
Thanks a lot...u solved my confusion in just 10 sec ❤️ !!!
This was very helpful to me in understanding this principle while studying for the GMAT. Thank you for making this video!
Thanks, I was reading Algebra the Very Basics and had a question on the first page 😂 good work 👍
Thank you very much - one of the things which made me think a lot of Maths was voodoo has now been very well explained.
Another way of proving it:
1
=64/64
=4^(3)/4^(3)
=4^(3-3)
=4^(0)
How about:
Let a>0 ,then
a^0=a^1 * a^-1
=a^1/a^1 = 1
I think this is a much simpler way to understand this since if a=0
0^0= 0^1 *0^-1= not defined
Thank you very much, subscribed to you👍😊😊
a^0 = 1 (such that a does not equal to 0) why?
proof: a^0 = a^n-n (n-n=0)
but a^n-n = a^n/a^n which is equal to 1 clearly(anything divided by itself is equal to 1)
hence a^0=1
Fantastic. This was making my brain hurt. Thank you for explaining it in a way that didn't make my brain hurt even more.
I find 0 wierd at times because sometimes it defines the laws of maths. Like 0/0 or 2/0 which doesn't happen with any other number.
Even -1^1/2 Aka "i" is easier to understand at times.
There isn't a single number where x(x) = x except for 0 or even complex fractions
(1/x) / ( (1+x)/x)
1/x ÷ (1+x)/x
1/x × x/(1+x)
x/x(1+x)
1/(1+x)
If we say x = 2 then
(1/2) / (1+2) / 2
(1/2) / (3/2) = 1/3
1/(1+2) = 1/3
But if x = 0
1/1+0 = 1
(1/0) / (1+0/0)
So complex fractions are wierd. So what if 0 was also a concept
Edit: As much as I did prove to myself that 0^0 =1 (0^1 = 1x0, 0^0 = 1, 0^-1 = 1÷0)
But it's hard to comprehend at times. So Zero multiplied by itself zero times is 1. Or at least close to 1 since 0.000001^0.000001 = 0.999...
Also yes I did say that there isn't a number where x^2 = x but I forgot that 1^2 = 1 . Sometimes one for me is wierd but really useful especially with fractions.
In co-ordinate geometry "0" isn't an issue. I meant that it's an issue with solving some equations. There are also some wierd things with 0 like 0! = 1 (and I still believe that 0/0 = 1 since 0.0001/0.0001 = 1)
Lots of times 0 is understandable but it's still different thiugh if it would be different maths would be possibly wrong.
Eg : xy =x yes 1 would fit is as y =1 and y = x 1=x but it's only true with 1x1.
If y =2 then 2x =! x unless x = 0
What i mean is that 1x1 = 1 but only there while 0 works with everything. 0x2 = 0 0×34553566347899 = 0. It's not changing since 0 + 0 = 0 again. Makes sense but almost none of the numbers do that.
Best example is with fractions
20/1 = 20-1 19-1 18-1 17-1 .... x20
20/2 = 20-2 18-2 16-2 14-2.... x10
20/0 = 20-0 20-0 20-0 20-0... xoo
But also...
20/-2=-20-2 18-2 16-2 14-2 x-10
20/ -1 = 20-1 19-1 18-1 17-1... x-20
So when it finally reaches 0 we get +/- oo
Yes, it all stems from the facts that humans designed math to be used for counting things, not for "doing math". It's hard to imagine having 0 of a thing, which mathematically is different from having nothing.
@@H3Vtux
I was also asking myself on why is it better to say that a circle has infinite edges then it has 0 edges. But over time I answered it myself.
(I love writing things down when no one cares and it's too obvious)
. Square has 90°. Pentagon has 72°. Hexagon has 60°...
More sides and lower (but more) angles. Keep on making new sides and angles. Do it oo many times and you get a circle
Yo Eric Pyrdz personally produced the music for your video? Amazing! 😂
Good video thanks, it explains the concept clearly and concisely
This would be the turning point for him
Well adding in an ‘invisible 1’ makes zero sense to just add a brand new component, but by that logic, is 0^0 then not 0? Because apparently if there’s always an invisible 1, it would be 1 x 0 = 0. Yet apparently it’s ‘proven’ that 0^0=0? Which I disagree with, if I gestured to the air and said ‘take an object’ there would be no object, because there’s no container, there’s literally nothing there. But once it’s put in some maths equation, apparently people defy real world logic and it becomes something?
If isn't here im gonna go to bed with my anxiety again
Thank you very much. You explained it very well.
Love you for this man
Ah thanks,... I've read on this and watched other instructional videos but this was the first time it made sense.
Sometimes I like to think dividing by zero practically would mean an object is warped into it's own non-existence. But if you start out with nothing, does the reverse happen? Are things spontaneously created at once?
I actually did a video on this very topicua-cam.com/video/1XPIWRXdSMI/v-deo.html
Hi, your vids are great! Thank You!
Any number divided by itself is 1. It's essentially the base unit for multiplication.
One more:
x^y/x^y=1 ----------1 [same numerator and denominator will make 1]
also, x^y/x^z = x^(y-z)
But here, z=y
Therefore,
x^y/x^y = x^(y-y) =x^0 ----------2 [y-y=0]
Now, x^y/x^y=x^y/x^y
From eqn 1 and 2
1=x^0
Hence proved that x^0=1.
It was great . But all the math teachers told us there are many ways to ascertain this subject. Which way we should always use it? which one is better?
By these calculations we can 0^0 is coming as infinity and not 0 OR 1 as expected. Even 0^x is coming as infinity.
Interstellar has taught us that the solution to 0^0 is in a black hole
Thank you for the explanation
There is one more thing about 0
I was playing with calculator and saw that
tan89.999 and 180/pi aren't that different
57.2957795131...
Well I did estemate that tan90 starts with 572957795131....
TanA= sinA/cosA
Sin90 = 1
Cos90 = 0
1/0 = 572957795131
Also I said to myself 10 ÷ 2 is how many times can 2 go into 10.
So 0 can go into 1 infinite times.
But how can a negative number divide negative times?
20/-2 = -10. I kind of get that -20/2 = -10 because 2 can go into neg 20 neg 10 tines
At this point I don't take the 0 as a number. It does wierd things
please make a video on quantum physics
what a nice and simple explanation
For y=x^(1/ln(x)), at x=0, I would like to consider two values y=0 and y=1.
That limit will go to e. There are actually 3 indeterminate forms that you put in this equation. At x=0 and x=1 and as x gets infinitely large, you get the form 0^0, 1^infinity, and infinity^0, all of the limits will go to e.
If the reasoning simply doesn't make sense then either the reasoning or the expression of the math needs to change, this is something a whole lot of teachers don't like to hear. Even if a concept works you must find a way to show that it makes sense! If you can't then you can't teach math properly.
Great video! Could you please share the source of this information?
Thank you. Why books don’t explain this is beyond me
Thanks once again Jr High!
Honestly I learned my 7 year old son both positive and negative exponents and I did not even have to explain 0 because without power 0 you can not explain 3 ^ -1.
These definitions work for integer powers, but how do you multiply 1 times 2 to the 1/2 times? or 1 times 2 to the pi times?
For Fractional or irrational exponents things get very complicated and there's unfortunately no way I can explain that in a comment section. I would imagine other youtubers have covered this in videos, probably kahn academy.
very cool, great explanation
*Amaaaaaaazing Explanation Sir*
*Your videos always be Awesome*
OHHH this makes so much sense!!! THank You so much
No problem, I'm glad it helped!
0^1=0 and to see what 0^0 is you can divide the number you got with the first number so 0÷0=0
Thank you so much!!!!!!!!!
The inverse/reverse to any index is by dividing by its base, would that be right to say?
thankful for this video
Absolutely brilliant! Thank you very much, thorough and clear 😊
But if you have 2 to the power of 1 it equals 2, because the base is 2 not 1, there for the base is what it is, it is not 1. The base of 2 is 2, the base of 3 is 3, the base of 4 is 4 and so on...... it is not 1, unless of cause you write a base of 1. So 2 to the power of 0 means I have a base number of 2 and they want it to be 0 times that means 0. No matter how they want to mess with it, it still means I started with a base of 2.
Or it could even mean that I’m not multiplying it at all 2 to the power of 0, just means 2, I stay with the 2 because I’m not doing anything at all to it as it’s a power of and not really a multiplication, it’s just 2.
Raising the exponent of a positive integer tells you how many times you need to multiply itself (e.g. 2^3=2*2*2=8). For negative integer exponents, it tells you how times you need to divide (e.g. 2^-3=1/(2*2*2)=1/8). For a 0 exponent as long as the base is not 0, there is no special rule for this. It will always equal to 1. This doesn't mean you multiply/divide 0 times. It doesn't work that way. You can think of it of as 2^(3-3)=2^3/2^3=8/8=1). For non-integer exponents. you will have to learn the rules of rational exponent. It doesn't make sense to say 1/2 times or so. That's not how it works. x^(m/n) is the same as nth root of x^m or nth root of x and then raised this to the mth power. For example, 8^(2/3)=4. If an exponent is irrational, you can't do anything about that. We just leave it as an answer there (e.g. 2^pi).
Thanks, my man
pure genius
What this video really does is explain to me that calling 2^2 "2 squared" is kind of misleading, since what is being done has nothing to do with geometry. If it were about geometry, then 2^2 would be representative of a line segment of length two being made into a square, which would result in 4 units. 2^1 would just result in a line segment, which would be undefined in area, and 2^0 would be a point, also undefined in area. So it's not really about geometry. Geometry just lines up nicely with any exponent higher than 1.
Thanks buddy ❤
Hi! Nice demonstration !
Many thanks.
OK. I have 2 comments and a query please: First of all, thank you. Everything about your video is spot on, and I appreciate the presentation.
I'm gonna share it with my Facebook community. Someone is sure to be intrigued or at the most, thankful that this little birdy was dropped in their lap.
Now for the query, are you sharing this because it (and all my basic math that I thought I never would use anywhere) has something to do with understanding computers and their operations? HINT: PLEASE say NO!
Thank you!!
excellent explanations
YESSSS. This IS litteraly what I thought!!!
2^0=? ÷2
2^1=2 ÷2
2^2=4 ÷2
Then did
2^-2= 1/4
So if 2^2 = 2*2 =4
2^1 = 2
2^0 = 1
2^ -1 = 2÷2
Wait no!
2^2 = 1*2*2
2^1= 1*2
2^0=1
2^-1= 1÷2 = 1/2
Nice job then!
2 / 2 = 1 not 2
This was helpful. Good explanations and i like the eric prydz but it was too loud.
I really am lost.If 0^0=1,then it follows that (0+0)^0=0^0+0^0=2,In that case (1+1)¹=?.
We know that (1+1)^1=1¹+1¹=2.so (0+0)^0=2=(1+1)^1.This conclusion leaves us to infer 0=1,which is an absurd conclusion. 0:08
0^0 does not equal one, as explained at the end of this video it's the exception. It's undefined.
0^0= 1 and 0; two values.
Please look the paper:
International Conference on Differential & Difference Equations and Applications
ICDDEA 2017: Differential and Difference Equations with Applications pp 293-305 | Cite as
log0=log∞=0 and Applications
Authors
Authors and affiliations
this is truly big brain time.
0^0 if anything clearly is more towards 1 and far from 0. If you don't believe me, take your calculator and do this:
0.0000000000000001 ^ 0.00000000000000001 =?
See if the answer is closer to 0 and closer to just about 1. Right nibbling it at its tail.
Great job on this video!
But doesn't that mean 0 to the power of anything is undefined since 0^2 would be equal to 0^3 devided by zero ?
No, raising zero to any non zero integer means we're multiplying zero by itself that many times.
0^3=0*0*0
0^0 is the result of two conflicting mathmatical rules
n^0=1
0^n=0
So depending on which of those rules we follow we get either 1 or 0 with 0^0...
what about the rationale behind e.g. y^(1/2) is equal to the square root