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 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.
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!
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 RU-vid!, 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.
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.
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.
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!
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.
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!
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!
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
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 😁😁
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.
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?
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.
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).
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.
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?
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
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
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.
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
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.
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!
Serouj Ghazarian they’re not. 0^0 can have other values. Let’s say for example y=x^(1/ln(x)). Do you see what is happening at x=0 since it’s not defined? You will see that the limit is e.
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?
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).
Sometimes it is easier for the mathematicians to label something they don't fully understand as "undefined" and move on. But this doesn't mean no body thinks about it. Someone who stumbled upon it will think the same as we do and wonder what really 0^0 is. There is one video I would like to share to you that have another idea regarding this topic: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-d0kr2d64TfA.html You can also find other videos that introduces concept not found on mainstream mathematics.
As we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible.
@@AlbertTheGamer-gk7sn You're overcomplicating things... they labeled it as undefined not because they were afraid but because for most practical applications 0^0 doesn't come up. It's just a tiny tiny slice of the huge pie that is math, and not choosing to obsess over it doesn't make you afraid.
