Why do numbers to the power of 0 equal 1?

Patreon: / h3vtux
My attempt at making an instructional visual explanation. Music is by Eric Prydz.

[@cakeandicecream1582]

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.

[@H3Vtux]

You're welcome!

[@esthersandoval1345]

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.

[@thenetsurferboy]

I never saw a number to power 0 in my 16 years studying maths

[@bahmannosratollah7017]

unbelievably easy explanation. easy to understand. Thank you.

[@H3Vtux]

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!

[@garyfinch7337]

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.

[@RahulShah-oh7oh]

Me bhi maths ke vedio banata hu aap dekhe or comment kre youtube.com/@RKEVEDIO?si=dSkOtJxP-JiNfI4v

[@bradleystoll6911]

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.

[@bryan9587]

Bradley Stoll Oh ffs. Just because YOU are not debating that topic, doesn't mean real mathematicians aren't either. Calm down, kid.

[@bradleystoll6911]

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 KZbin!, 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:).

[@Vaaaaadim]

@@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

[@careydedo3621]

@@bryan9587 What are you even doing here?

[@MuffinsAPlenty]

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.

[@slimshady4life689]

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.

[@xNiGHT96]

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!

[@RaaZVan99]

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;

[@oDuckology]

so basically it's a geometric sequence where the values get divided by the base

[@ItzPinecone]

excuse me WHAT

[@kumoki]

no I don't understand whaaat

[@andressstyle6390]

@Lukas yea, it should ... Anyway, it's the same demonstration showed in the video .

[@lumen2705]

weird flex but okay lol

[@DigilusionStudios]

Sometimes you just need to find someone that can explain things in different ways to learn it, Thanks.

[@H3Vtux]

Thanks man, i'm glad it helped!

[@princegupta8140]

Amazing.....make more video like this...plz

[@mollymolster7586]

This was sooo helpful .I feel like sharing this with everyone I know but that would make me SERIOUSLY nerdy. lol.

[@BoahYo]

I was thinking about this and came to the same conclusion as the video. Just wanted confirmation

[@starpaw2000]

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!

[@cancionesdeyu]

0: OMG!!! I was stuck on this for so long! I FELT SO DUMB BUT THANK YOU SO MUCH FOR CLEARING IT OUT!!😭

[@RealityCheck6T9]

I can't compliment your videos enough, they're wonderfully explained.

[@mysticcyber3840]

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]

Thanks I appreciate the feedback, I'm glad it helped!

[@dzee127]

@@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!

[@TheZmoliver]

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!

[@buttermilkpancakes916]

Omg thank you so much !!! 😭 this helped me out greatly.

[@user-zf8gy1yw1p]

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.

[@dragoncat5767]

Thanks,much easier explained

[@why8778]

damn

[@artemisiavelvet2046]

Nice. In that case "a" must be > 0

[@princegonzales4807]

If isn't here im gonna go to bed with my anxiety again

[@indyjoe6]

the breakdown at the end was excellent

[@hello-yk4hn]

Another way of proving it: 1 =64/64 =4^(3)/4^(3) =4^(3-3) =4^(0)

[@TheVideoMaker1999]

This would be the turning point for him

[@xaviercomelli]

Thanks, great explanation!

[@adriasorensen2249]

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

[@marka.200]

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!

[@mayattv4986]

Wtf. Sue the teacher lol

[@handleisntfkinavailable]

The teacher probably didnt know why, he prob just accepted that anything ^0 is 1 without questions

[@geet.ikagupta]

Thanks a lot...u solved my confusion in just 10 sec ❤️ !!!

[@theloveofreading3563]

I bet the first person to figure this out was excited !

[@kevincassidy7233]

Any number divided by itself is 1. It's essentially the base unit for multiplication.

[@subscreibe_productions]

OMG ITS SO HELPFUL I COULDNT LIFE OF ME FIGURE THIS OUT

[@z-inkp6478]

4:50 is when it all made sense. great video though.

[@bramk6720]

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 😁😁

[@B.W.Bricks_productions]

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.

[@jallipallisaipraveenkumar1801]

Thank you very much sir We still hope more videos from you of such questions

[@brianskellenger9344]

Thanks, I was reading Algebra the Very Basics and had a question on the first page 😂 good work 👍

[@jounoun3600]

Thanks for this video. Now i understand this rule perfectly.

[@petermcgrandle8298]

Thank you very much - one of the things which made me think a lot of Maths was voodoo has now been very well explained.

[@DylanLCutshall]

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?

[@roger72715]

Hi, your vids are great! Thank You!

[@simenhansenbeitesimen]

My teachet denied this, im so mad

[@tchikoumahmoud4665]

Why explain pls why does he deny this

[@walterbrownstone8017]

Because the 0th degree of an exponent is not a mathematical operation. It's a definition. The base unit of the function is 1.

[@tico6028]

Love you for this man

[@alexanderbreeding4008]

This was very helpful to me in understanding this principle while studying for the GMAT. Thank you for making this video!

[@neonWHALE002]

Interstellar has taught us that the solution to 0^0 is in a black hole

[@acktack3578]

Good video thanks, it explains the concept clearly and concisely

[@juanrodriguez9971]

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).

[@adambakas13]

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.

[@jXd]

Ah thanks,... I've read on this and watched other instructional videos but this was the first time it made sense.

[@BTS_ARMY1997]

Thank you very much. You explained it very well.

[@lohasingh982]

Thank you sir love ❤️ from India

[@hastimak8682]

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?

[@gailmargret23]

Fantastic. This was making my brain hurt. Thank you for explaining it in a way that didn't make my brain hurt even more.

[@justabunga1]

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.

[@minkademko2335]

Thanks for taking it back in history to how this was originally figured out.

[@Ongshimi_Mandu]

thankful for this video

[@JackRumer]

Thanks, my man

[@ricardoblikman2676]

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.

[@i_gaurav_kr.]

Thanks buddy ❤

[@Theyadoresharvari._]

Let the mathematicians lose their mind over that 0, I'm just glad that now I know why anything to the power 0 is one

[@donsal.t.1765]

Thanks once again Jr High!

[@aaronrajinthrakumar7390]

what about the rationale behind e.g. y^(1/2) is equal to the square root

[@gugulethukheswa4605]

Thank you for the explanation

[@dayanandt3016]

please make a video on quantum physics

[@jeremy5602]

Yo Eric Pyrdz personally produced the music for your video? Amazing! 😂

[@jrjr1313jrjr]

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?

[@H3Vtux]

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.

[@serologism]

what a nice and simple explanation

[@juliapdanderson3891]

Absolutely brilliant! Thank you very much, thorough and clear 😊

[@witchstriker5380]

Thank you very much, subscribed to you👍😊😊

[@manjurhasanchisti2264]

Many thanks.

[@alexman97412]

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

[@nicolastelfyr6792]

Thank you!

[@Kleermaker1000]

The explanation was good, the music bad. :)

[@user-jt7vn4pq5c]

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

[@Dreamkilled]

Thank you!!

[@quty6936]

OHHH this makes so much sense!!! THank You so much

[@H3Vtux]

No problem, I'm glad it helped!

[@jonahsousa1986]

very cool, great explanation

[@patricksusulin9335]

this is truly big brain time.

[@captainb.mccrea5480]

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?

[@ashishop9233]

Thanks bro I was finding this and nobody was answering me

[@WisdomCalls]

Thank you. Why books don’t explain this is beyond me

[@JJFsixxx]

Perfectly said my man

[@gamingkids8262]

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.

[@AlbertTheGamer-gk7sn]

0! is not equal to 1 in this scenario.

[@mukhtarahmad3532]

*Amaaaaaaazing Explanation Sir* *Your videos always be Awesome*

[@gunterstunter]

Great job on this video!

[@NiteshBahekar]

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.

[@BramVanhooydonck]

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?

[@H3Vtux]

I actually did a video on this very topickzbin.info/www/bejne/Z4mzeoqIjcmGg6s

[@technoultimategaming2999]

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

[@kingkirby8960]

Nice job then!

[@karenvickery6070]

2 / 2 = 1 not 2

[@segayanmx4442]

Hi! Nice demonstration !

[@PWR_ADA]

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

[@xxJomoxx]

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 ?

[@H3Vtux]

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...

[@kacperzy92]

pure genius

[@thewatcher8657]

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.

[@clearlypellucid]

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.

[@giorgosg8760]

a^n/a^n=1, a^n/a^n=a^(n-n)=a^0 => a^0=1 Simple

[@RealYukiSan]

How about 2^(-1) ?

[@kandimegahan7844]

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!

[@technoultimategaming2999]

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

[@technoultimategaming2999]

At this point I don't take the 0 as a number. It does wierd things

[@NoahR1620]

Thanks for this. 😅

[@renzo9661]

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

[@mohammedviso2269]

I really love it ...Thanks alot