Zero to the power of zero
Zero is probably the most mysterious number and the most counterintuitive. After all, there is no real-life analog for it. Zero is the absence of something. But why is zero raised to the power of zero equal to one? And the main question, is it really true? You can check it on your calculator before reading further…
00=1 ?
Zero to the power of zero
How can this be? Here’s how: 10=1, 20=1…. x0=1. Any number raised to the zero power equals one. Why would zero be any different? But it’s not that simple.
What does it mean to raise to a power? For example, “two squared.” What do we do, we multiply two by itself two times (2*2=4), “two cubed,” we multiply two by itself three times (2*2*2=8). But what if the power is “zero”? You need to take a number and multiply it by itself… zero times? That sounds strange.
Here is how the graph of the function y=xx looks:
As you can see, when the value of X decreases, the value of Y first decreases, then starts to increase and turns into… one when X reaches very small (almost zero) values. It would be logical to assume that when the value decreases to zero, it would still be one.
Again, let’s return to simple numbers:
32=9
What does this mean? To get nine, you need to multiply three by itself two times. Right?
30=1
How many times do you need to multiply three by itself to get one? What if you divide 1 by 3? There is no simple answer, right? Logically, the larger the power, the bigger the result, and the smaller the power, the smaller the result.
But the graph above shows that the curve “levels off” at a limit, at one. More precisely, the function value becomes equal to 1 when zero is not even reached yet. And if you decrease X even more, it will still not move beyond one.
Context of the Problem
How can it be that when multiplying zero by itself, something greater than zero comes out?
If, in real life (not mathematics), we ate all the apples and now have zero apples, how can multiplying the missing apples by other “zero” fruits result in a whole apple? If this question seems simple to you, it is.
From one point of view, this strange expression would be equal to one, but from another point of view, it would be “undefined.” That is, there can be no one resulting from multiplying zero by zero, right?
Mathematics says:
32×32 is the same as 32+2 = 34 = 3 × 3 × 3 × 3 = 81
or
45÷43 is the same as 45–3 = 42 = 4 × 4 = 16
Then, if the powers are the same:
32÷32 is the same as 32-2 = 30 = Oops?!
But we can also not subtract the powers and just perform the two operations separately:
32÷32 is the same as 32-2 = 30, but 32=9, then 32÷32 = 32-2 or 32÷32=9÷9=1
What happens if you divide a number by itself? One!
Mathematical Analysis
From the perspective of mathematical analysis, everything is both complicated and simple. Zero to the power of zero = undefined. Which, agree, is more logical. After all, if we have nothing and multiply nothing by itself, can something come from this emptiness?
Set Theory
Let’s look at it from the perspective of set theory. Suppose we have two sets.
The first set is the number of characters in the password that locks access to your social media page, or, even better, the PIN code of a bank card — let’s say it’s 4 characters.
****
The second set is the number of values that each character can take. Suppose it’s just digits, meaning there are 10 digits.
The question is, how many possible combinations are there? How many tries do you need to make a random combination to guess the password? Each character:
104=10,000 possible combinations.
You could say that the set of digits (10) maps to the set of possible characters (4). But there are also “empty” sets. For example, if you didn’t set a password at all, and have zero characters to guess, how many attempts would it take to access the account? Exactly one.
So, 100=1, but the same thing would happen if there is no password and no values — 00=1. In simple terms, zero raised to the zero power means that the password is not set and every value is also zero. Then only one such “combination” could exist.
But what is the reality?
As you can easily guess, this mathematical expression has no practical application. No engineer or economist would think of multiplying zero by zero zero times. It is simply an inapplicable construct. So, the question remains in the realm of mathematics and perhaps philosophy.
This is probably the only case where, as a mathematician, you can freely decide for yourself what “0 to the power of 0” equals.
