I suck at math and I've thought this same question myself. I've wondered if it's just a convention, rather than something rigorously and mathematically correct. (in other words, did someone one day say "Hey guys! When I put a number to the zero power, I really just mean 1! I know, it doesn't make any sense, but just humor me, alright!) But math doesn't normally work like that as far as I know, so there's probably a good (and likely difficult to understand) reason for it being like that.
I did some internetting to try to figure this out, but I guess b/c I suck at math the answers I found made no sense to me. Here's what I found.
First of all - BOOO to Theaetetus for copying and pasting directly from a website and not citing the source. At any rate, what he/she posted is supposed to explain why any number to the zero power equals one, according to a "Dr. Math."
Math Forum: Ask Dr. Math FAQ: N to Zero Power
Let's first look at an example. Let's look at the list of numbers
3^1, 3^2, 3^3, 3^4, ....
Finding the actual values, we get 3, 9, 27, 81, ....
So what is the pattern in the bottom sequence? Well, every time you move to the right in the list you multiply by 3, and every time you move to the left in the list you divide by 3. So we could take the bottom sequence and keep going to the left and dividing by 3, and we'd have the sequence that looks like this:
..., 3^-3, 3^-2, 3^-1, 3^0, 3^1, 3^2, 3^3, 3^4, ....
..., 1/27, 1/9, 1/3, 1, 3, 9, 27, 81, ....
So now we know what all the powers of 3 are! Actually, we just did the integer powers of 3. But that's probably enough for now.
However, the page there goes on with further explanations. I'll copy and paste them here, but it seems they will be in a scroll box in my post, which may make them difficult to read, so of course you should feel free to go to the actual site and take a look at them there.
While the above argument might help convince your intuitive side that any number to the zero power is 1, the following argument is a little more rigorous.
This proof uses the laws of exponents. One of the laws of exponents is:
--- = n^(x-y)
for all n, x, and y. So for example,
--- = 3^(4-2) = 3^2
--- = 3^(4-3) = 3^1
Now suppose we have the fraction:
This fraction equals 1, because the numerator and the denominator are the same. If we apply the law of exponents, we get:
1 = --- = 3^(4-4) = 3^0
So 3^0 = 1.
We can plug in any in number in the place of three, and that number raised to the zero power will still be 1. In fact, the whole proof works if we just plug in x for 3:
x^0 = x^(4-4) = --- = 1
Here's another explanation from Aldo Daniel Completa, a contributor from Argentina:
In the FAQ there is an explanation about " n^0 (any number to zero power) ." I think another is this:
Let's begin with examples.
Ex.1: 5^3 / 5^2 = 125 / 25 = 5
5^3 / 5^2 = 5 ^(3-2) = 5^1 = 5
Ex.2: 5^3 / 5^3 = 125 / 125 = 1
5^3 / 5^3 = 5 ^(3-3) = 5^0 =
... and the result must be 1. So 5^0 =1.
The rule is: x^b / x^c = x^(b-c).
In order to generalize this rule for the case b=c, it must be defined that x^0 = 1 (x is any number different from 0). In mathematics, usefulness and consistency are very important. This convention allows us to extend definitions of power that would otherwise require treating 0 as a special case.
This method can also explain the definition: x^(-b) = 1 / x^b.
Ex. 5^2 / 5^4 = 25 / 625 = 1 / 5^2
5^2 / 5^4 = 5 ^(2-4) = 5^(-2)
Here's another attempt at an answer; it also does nothing for me.
WikiAnswers - Why does any number to the zero power equal one
Q: Why does any number to the zero power equal one?
a^b. it is natural to restrict a > 0, but we'll only assume that number b is any real number.
We'll use the natural exponential function defined by the derivative of the exponential function.
Now we have a^r=e^rln(a). And we know that e^rln(a)=e^((ln(a))^r), where a >0 and r is in the domain of all real numbers negative infinity to infinity.
We can apply this definition to any number a to any power r.
Particularly, a^0. By the provided definition, a^0=e^(0*ln(a))=e^0=1.
Yet another attempt to explain is here:
Math Logic - Number raised to the power zero is equal to 1
Why is it that any number raised to the power zero is equal to 1 and not zero ?
When a number is raise to the power 0, we are not actually multiplying the particular number by 0. For example, let us take 20. In this case we are not actually multiplying the number 2 by 0.
We define 20 = 1, so that each power of 2 is one factor of 2 larger than the last, e.g., 1,2,4,8,16,32...
This involves the rules of exponents particularly division.
If a is a number and x and y are also numbers, then according to the rule of division for powers with the same base,
a^x/a^y = a^(x - y).
It says the quotient of two powers with the same base is equal to the common base raise to the exponent equal to the difference between x and y.
So, if x = y, then a^x/a^y = a^(x -y) = a^0
But a^x is equal to a^y, since x = y; hence a^x/a^y = 1
Therefore, by Transitive property of Equality,
a^0 = 1
Thus, this result says that number raised to the power zero is equal to 1.
You may also want to look at what Wikipedia has to say about this - Exponentiation - Wikipedia, the free encyclopedia
- it seems like it might be a relatively simple and accurate answer, but what do I know, b/c (yes, I'll say it again) I suck at math.
For more on this question just google somthing like "why is a number raised to the zero power always equal to one"; it turns out that particular search returns 2,750,000 results. :Not-Impressed: