The Most Controversial Number in Math

i'm probably about to start a battle in the comments section but i think it's worth it because i want to tell you what zero to the zero power is you may have been told zero to the zero is undefined but i think we can do better than that by the end of this video you'll know exactly why we should define zero to the zero and what it actually is you might have some reasons why you want to leave zero to the zero undefined i'm going to debunk those a common objection is that zero to the zero is the same as zero to the one minus one that's true one minus one is zero and if you use your properties for exponents you might conclude wrongly that this is the same as zero to the first over zero to the first and everyone knows zero to the first power is zero and what we have here is zero over zero which is undefined in most contexts but we just as easily could have done the same trick with zero to the first i think you would agree that zero to the first is zero but if we try to do the same thing one in the exponent one is the same as two minus one and if we do this same exponent trick well we're getting the same result here this would be zero squared over 0 to the first and this is still 0 over 0 which you're claiming is undefined well which is it is it 0 or is it 0 over 0. i think most people would agree that 0 to the first is zero so this argument doesn't work and what's going wrong here is that you're trying to use a law of exponents this one in fact x to the a minus b is x to the a over x to the b but this is only true if x isn't zero and x is 0 in this argument so this line of reasoning doesn't work using limits are another way of arguing against 0 to the 0. think about the functions x to the 0 power and 0 to the x power we typically think anything to the zero power is one most people would agree with that provided x isn't zero also zero to any power is zero you think of zero times itself as many times as you like you're always going to get zero unless maybe x is zero if we take x to the zero and we try to get x as close to zero as we possibly can trying to get to our answer zero to the zero this is like taking the limit back in calculus the limit as x approaches 0 from the right of this function x to the 0. and if you imagine traveling along the graph of this function it's going to be 1 1 1 1 1 all the way to the right of 0 and so this limit is end up being 1. if we try to do the same thing with 0 to the x if we try to take x getting closer and closer to 0 well this graph is entirely along the x-axis and so what's happening is we're getting closer and closer to zero and so it looks like although these functions are very similar and they're both approaching the value zero to the zero we're getting different answers one answer is one one answer is zero and we can't have two different results for this but the problem with this argument against zero to the zero is that it's limits it's not actually at the value it's approaching the value we're getting very very close to zero to the zero via a couple functions but we're not actually computing the arithmetic number zero to the zero power so even if the arguments against zero to the zero are faulty you might think well who cares we never use zero to the zero power anyway actually you probably do if you've taken calculus you've probably run into zero to the zero hidden in your functions without even realizing it there are a number of series and formulas that really benefit from having the definition zero to the zero power take the binomial theorem for example this is an extremely famous and important theorem that absolutely requires zero to the zero power to be defined otherwise we would have to put a whole bunch of restrictions on it like x and y can't be zero or x can't be minus y how about the power series definition of e to the x i bet calculus students like using this one if we don't allow for zero to the zero power well we're going to have to change our series definition by re-indexing it you see this sum starts at zero so when we plug in the very first value for n assuming we're calculating e to the zero using its series definition well you're running into a zero to the zero power right there so how do we determine what zero to the zero power really should be all we need to do is go back to how we actually define how exponents work take three cubed for example you know this means multiply the base three by itself exponent three times or in other words three times three times three and if we do three squared of course three to the first would be three times itself one time and if we keep going along this pattern well three to the zero would mean hmm the pattern doesn't quite follow or does it i've heard it said we continue this pattern by dividing by three every single time to get from here to here we've done a division by three to get from here to here we've done a division by three and if we did it once more we get to the usual definition 3 divided by 3 is 1. it's not that we're dividing by 3 each time it's that we're multiplying by 1 less three each time so instead rely upon the simple algebra rule that one times x is x for any real number so all this means is that one times three cubed is the same as one times three times three times three one times three squared is the same as one times three times three one times three to the first is one times three and one times three to the zero is one times zero threes and we get our usual definition that three to the zero is one apply this same logic to zero to the zero one times zero to the zero is one times no zeros and there you have it zero to the zero power is one you'll even see this result if you plug it into a good enough calculator and zero to the zero works just fine for the formulas mentioned before if you plug in zero to the exponential function's power series well e to the zero that's certainly one and if we start writing out all the terms of the series we see they all go to zero except the first term which is zero to the zero and it must be one zero to the zero is kind of like a distant cousin to zero over zero and dividing by zero if you wanna know how to deal with those click the video on the screen right here i'll see you in that one

Channel: BriTheMathGuy

Views: 1,221,541

Keywords: math, maths, 0^0, 0^, zero to the power of zero, zero to the power of zero proof, 0^0=1, 0^0=1 proof, 0^0 proof, 0^0 undefined, 0^0 power, 0^0=1 why, 0^0 meaning, why is 0^0 undefined

Id: jNhjB4UfR9A

Length: 6min 45sec (405 seconds)

Published: Mon Apr 12 2021