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.

Code: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
--- = n^(x-y)
n^y

for all n, x, and y. So for example,
3^4
--- = 3^(4-2) = 3^2
3^2


3^4
--- = 3^(4-3) = 3^1
3^3

Now suppose we have the fraction:
3^4
---
3^4

This fraction equals 1, because the numerator and the denominator are the same. If we apply the law of exponents, we get:
3^4
1 = --- = 3^(4-4) = 3^0
3^4

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^4
x^0 = x^(4-4) = --- = 1
x^4

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)

Code:Q: Why does any number to the zero power equal one?

Consider this.

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.

Furthermore, a^1=e^(1*ln(a))=e(ln(a))=a.

And a^2=(e^(ln(a))^2)=a^2.

Code:
Question :
Why is it that any number raised to the power zero is equal to 1 and not zero ?

Answer :
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.


The definition of a number to a power that I have been using is: the number multiplied times itself that [the power] number of times.

Just follow the pattern:

...
2^3 = 8
2^2 = 4
2^1 = 2
2^2 = 0
2^-1 = 1/2
2^-2 = 1/4
2^-3 = 1/8
...

And then ... if you have 2^3 * 2^-3 ... you have to add the exponents, so the most reasonable value for 2^0 is 1. So it is with any other base

OK, forget that definition; it's not very good. Try this one instead - it works better!

The definition of a number to a power is: one multiplied by the number that [the power] number of times.

So:
10 to the power 2 = 1 multiplied by 10 twice = 1 x 10 x 10 = 100.
10 to the power 1 = 1 multiplied by 10 once = 1 x 10 = 10.
10 to the power 0 = 1 not multiplied by 10 at all = 1.

10 to the power -1 = 1 'un-multiplied' [i.e. divided] by 10 once = 1/10 = 0.1
and so on.

Does that make it clearer?

Math only works for finite numbers...
That's new. Never heard of that rule before.
These are philosophy forums. The only rule is that there are no rules.

You can use 0 and infinity in multiplication if you want to. It would just be good to keep in mind that you're not really making sense.... which is maybe more obvious when you try to divide by 0 or infinity.