= = = = . Now we just plug in x=0, and we see that zero to the zero is one!
Cleverer student: No, you’re wrong! You’re not allowed to divide by zero, which you did in the last step. This is how to do it:
= = = =
which is true since anything times 0 is 0. That means that
Cleverest student : That doesn’t work either, because if then
so your third step also involves dividing by zero which isn’t allowed! Instead, we can think about the function and see what happens as x>0 gets small. We have:
So, since = 1, that means that = 1.
High School Teacher: Showing that approaches 1 as the positive value x gets arbitrarily close to zero does not prove that . The variable x having a value close to zero is different than it having a value of exactly zero. It turns out that is undefined. does not have a value.
Calculus Teacher: For all , we have
That is, as x gets arbitrarily close to (but remains positive), stays at .
On the other hand, for real numbers y such that , we have that
That is, as y gets arbitrarily close to , stays at .
Therefore, we see that the function has a discontinuity at the point . In particular, when we approach (0,0) along the line with x=0 we get
but when we approach (0,0) along the line segment with y=0 and x>0 we get
. Therefore, the value of is going to depend on the direction that we take the limit. This means that there is no way to define that will make the function continuous at the point .
Mathematician: Zero raised to the zero power is one. Why? Because mathematicians said so. No really, it’s true. Let’s consider the problem of defining the function for positive integers y and x. There are a number of definitions that all give identical results. For example, one idea is to use for our definition:
where the y is repeated x times. In that case, when x is one, the y is repeated just one time, so we get
= . However, this definition extends quite naturally from the positive integers to the non-negative integers, so that when x is zero, y is repeated zero times, giving
which holds for any y. Hence, when y is zero, we have
. Look, we’ve just proved that ! But this is only for one possible definition of . What if we used another definition? For example, suppose that we decide to define as
In words, that means that the value of is whatever approaches as the real number z gets smaller and smaller approaching the value x arbitrarily closely.
[Clarification: a reader asked how it is possible that we can use in our definition of , which seems to be recursive. The reason it is okay is because we are working here only with , and everyone agrees about what equals in this case. Essentially, we are using the known cases to construct a function that has a value for the more difficult x=0 and y=0 case.]
Interestingly, using this definition, we would have
= = =
Hence, we would find that rather than . Granted, this definition we’ve just used feels rather unnatural, but it does agree with the common sense notion of what means for all positive real numbers x and y, and it does preserve continuity of the function as we approach x=0 and y=0 along a certain line.
So which of these two definitions (if either of them) is right? What is really? Well, for x>0 and y>0 we know what we mean by . But when x=0 and y=0, the formula doesn’t have an obvious meaning. The value of is going to depend on our preferred choice of definition for what we mean by that statement, and our intuition about what means for positive values is not enough to conclude what it means for zero values.
But if this is the case, then how can mathematicians claim that ? Well, merely because it is useful to do so. Some very important formulas become less elegant to write down if we instead use or if we say that is undefined. For example, consider the binomial theorem, which says that:
= where means the binomial coefficients.
Now, setting a=0 on both sides and assuming we get
where, I’ve used that for k>0, and that . Now, it so happens that the right hand side has the magical factor . Hence, if we do not use then the binomial theorem (as written) does not hold when a=0 because then does not equal .
If mathematicians were to use , or to say that is undefined, then the binomial theorem would continue to hold (in some form), though not as written above. In that case though the theorem would be more complicated because it would have to handle the special case of the term corresponding to k=0. We gain elegance and simplicity by using .
There are some further reasons why using is preferable, but they boil down to that choice being more useful than the alternative choices, leading to simpler theorems, or feeling more “natural” to mathematicians. The choice is not “right”, it is merely helpful.