 This mathematics video will be about the question, what is zero to the power of zero? Well, there's an old mathematics joke about someone who asked their lawyer, what is two plus two? And the lawyer said, well, what do you like it to be equal to? And the lawyer in fact gave the correct answer because no symbols in mathematics of any meaning until you have defined the meaning and you have the freedom to define them to be anything you like. So a very simple answer about what is zero to the zero is that it is either undefined or you can define it to be anything you like. It is in fact the wrong question. The correct question is, what is the most useful definition of zero to the power of zero? And it turns out there are two answers to this. The problem is that although we write x to the y for exponentiation, this is really two different functions and we get a different answer for zero to the power of zero depending on which of these two different functions you're talking about. Now, you don't usually notice that this is ambiguous and refers to two different functions because these functions are nearly always the same. So the two different functions are x to the y for y an integer and x to the y for y a real number. Let's take x to be greater than or equal to zero in both cases. And you may say that the function x to the y for y an integer is really a special case of the function x to the y for y real because integers are special cases of reals and nearly all the time you would be right but every now and then this isn't quite true. So let me give you an example when it's not quite true. So suppose you're trying to calculate 2.0 to the power of two on a computer and you may actually get a different answer depending on whether this exponent two is a real number or an integer because computers usually store integers and real numbers in different ways. So an integer will be a 64 bit words and a real number is encoded in some complicated way with an exponent and nums off the decimal point and the computer calculates exponentiation in two different ways. So if we have 2.0 to the two where this number is an integer the computer will work it out by just multiplying 2.0 times 2.0 and getting the answer 4.0 unless there is something badly wrong with your floating point unit. On the other hand, if you ask a computer to calculate 2.0 to the 2.0 for this number is a real number what the computer will usually do is it will use the rule that x to the y for x and y real is given by x of y times the natural logarithm of x. So it will calculate 2.0 to the 2.0 as x of 2.0 to the natural logarithm of 2.0. Now when it calculates a natural logarithm it can't get the answer exact because it only worked for so many decimal places or binary places or whatever. So there may be a small error introduced in this and it might give you the answer 4.0 but it might give you the answer 3.9999999 or it might give the answer 4.0001. So as far as a computer is concerned the two functions x to the y where y is an integer or a real are very slightly different and may in fact give you very slightly different answers. Well, of course that's only works on a computer and surely in mathematics these two numbers are going to be the same whether or not this is an integer or a real number and yes, they are the same in mathematics. However, in the special case of zero to the power of zero you do get a slightly different result even in mathematics. So let me explain. So let's first of all look at x to the y for y an integer. Now if let me write I get confused writing y for an integer. So let me write x to the n for n an integer because it's traditional to use n for an integer. Now if n is greater than zero x to the n is normally defined as x times x times x and so on where you take n times obviously. And this has the rule that x to the n plus one is equal to x to the n times x. And now we want to extend this to n equals zero and if we want this rule to hold for x none zero this implies that x to the zero equals one so that x to the zero plus one equals x to the zero times x. Well, for x not if x is equal to zero then this still leaves zero to the zero ambiguous but this shows for integer exponentiation we should at least define x to the zero being one for at least when x is not zero. Well, what about when x is zero? Well, we want to choose the most useful definition so we want to choose the definition that gives a nice answer most of the time. So let's look at some examples. So the first example m to the power of n here I'm taking m n to be integers is the number of functions from an n element set to an m element set. I hope I've got m and n the right way around because it's incredibly easy to get these the wrong way around so let's just check. So for instance three squared should be the number of functions from a two element set to a three element set. And you can see that any function has to take the first point to one of three elements there are three choices and the second to one of three elements there are three choices there are three times three functions. So I actually managed to get it the right way around for once. So zero to the power of zero should be functions from a nought element set to a set with nought elements. And this is kind of really confusing how many functions are there from a set with zero elements to zero elements? And if you think about it a bit you'll realize there's exactly one function from a zero element set to the zero element set which is the sort of empty function which takes no values on the domain but that's okay because the domain doesn't have any values in it. So the number of functions from a zero element set to a zero element set is in fact one if you're really sought out your definitions carefully. So this suggests that zero to the zero should be one at least if this is an integer as well. Now let's look at a second example. Let's look at the following geometric series one plus X plus X squared plus X cubed and so on. Now if X is absolute value less than one you know this converges and is equal one over one plus one over one minus X. Well, we can rewrite this series in a slightly neater form. We can write as X to the zero plus X to the one plus X squared plus X cubed and so on. So here we're writing this as sum of n equals nought to infinity of X to the n. So now we've got this value X to the zero and in order to make this equal to one over one minus X even for X equals zero we see well all these terms here are gonna be zero and this term is going to be one. So zero to the zero equals one. So if we want, if we want, if for X equals zero we want X to the zero plus X to the one plus and so on to be equal to one over one minus X. This forces zero to the zero equals one. So the point is it is convenient to define X, to define nought to the n to be one, sorry, nought to the nought equals one if the exponent is always an integer. So if the exponent of X to the Y is an integer because of course zero is an integer, but anyway. So that's not a proof that nought to the nought equals one because you can't prove zero to the zero equals one. It's just zero to the zero is anything you want it to be. What I'm saying is that the most useful value is one. I'm not saying it's true, it's one and just saying I'm going to choose it to be one. So that's done the case when you're raising things to integer exponents. Now let's look at the case X to the Y where we allow Y to be real. And again, I'm going to take X to be greater than zero because taking negative numbers to real exponents is occupied opens a whole can of worms. So what would we like this to be true? What would we like zero to the zero to be equal to? So what's the most convenient value? Well, suppose you've got a function F of X to the G of X and you want to look at the limit of this as X tends to zero, say. So what does this tend to? Let's suppose that the limit as X tends to zero of F of X equals zero and the limit as X tends to zero of G of X is zero. Then the limit of this should be, let's put a question mark, zero to the power of zero. So that's the limit of F and the limit of G. So in order to figure out what zero to the zero should be, all we have to do is to work out the limit of this expression where F and G both tend to zero. Well, let's figure this out. F of X to the G of X is equal to the exponent of G of X times the logarithm of F of X. This is if F of X and G of X are greater than zero. Now we've said that G of X is going to tend to zero. And F of X tends to zero. Well, F of X is tending to zero. That means this bit tends to minus infinity. So this is X of zero times minus infinity. Well, this is meaningless. The problem is zero times minus infinity conventionally means what is the limit if you multiply something very close to zero by something very close to minus infinity? In other words, a large negative number. And the answer is the limit can be anything you like. If you tend to zero very rapidly into minus infinity, slowly this limit will be zero. And if you tend to minus infinity rapidly in zero slowly, the limit with minus infinity, and it can be anything in between. So this just isn't defined in general. Well, most of the time the limit will be one. The reason for this is that although the logarithm of F of X will tend to minus infinity, it will tend to minus infinity very, very, very slowly. So most of the time G of X will tend to zero faster than log of F of X tends to minus infinity, if you see what I mean. So the limits of this will very often be zero. The limits of this bit will very often be zero. So the limits of the exponential of it will usually be one. So we could say that zero to the zero is usually one, but sometimes not. So let's give an example where it's not. Suppose you take e to minus 1 over Y squared, the power of Y squared. So here I'm taking F of X equals e to the minus 1 over Y squared. Sorry, I seem to have switched X and Y, F of Y equals that. And G of Y equals Y squared. You can see G of Y has limit zero as Y tends to zero, and F of Y also has limit zero as Y tends to zero. But this expression is always e to the minus 1. So as Y tends to zero, and you think of the limit of this as being zero to the zero, we find zero to the zero is now e to the minus 1. So you can have an argument for zero to the power of zero being any real number you like by choosing suitable F of X and G of X. So let me summarize recommended values. So recommended values for zero to zero is one if exponents are integers and undefined if exponents are allowed to be real. If you have to define it when the exponents are real, the least worst choice is one. But this is really, really, really not recommended. It's defining zero to the zero to be one when the exponents are real is kind of like cycling around without a cycling helmet. Most of the time you will get away with it. Every now and then if something goes wrong, you will really, really regret doing this.