What is the intuition behind n^0 and why is it equal to 1?

[iScience]

i wasn't sure which section to post this topic so I'm just posting it here.

i understand that there is a pattern with exponents regardless of the base number..

n^-3 n^-2 n^-1 n^0 n^1 n^2 n^3 corresponds to...

(1/nnn) (1/nn) (1/n) (n/n) (n) (nn) (nnn)

so then using this pattern i see that n^0 will always equal 1. but could someone please provide me some intuition as to what n^0 actually means? (intuitively?)

 

[BucketOfFish]

An intuitive way to think about it is as follows: If you have a quantity which doubles every second, then after 1 second you have 2 times as much. After two seconds you have 2^2=4 times as much. After three seconds you have 2^3=8 times as much. At time t=0 you have your starting amount, which is 1 time as much. 1 second ago, assuming the trend held, you had 2^-1=1/2 as much. And so on. So we see that:

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

 

[dextercioby]

To make exponents work, 0-th power must be 1: 7^(5-5)=7^(5)/7^(5) = 1. To make factorials work, 0! must be 1.

 

[HallsofIvy]

It is easy to show, with n a positive integer and a any positive number, that $(a^n)(a^m)= a^{n+m}$ and $(a^n)^m= a^{nm}$. If we want those very nice properties to be true even if n= 0 (and, extending, n negative, n a fraction) we want $a^na^0= a^{n+0}= a^n$. Since a is positive, $a^n> 0$ and we can divide both sides by it: $a^0= a^n/a^n= 1$.

That is, if we want $(a^n)(a^m)= a^{n+m}$ to be true we must define $a^0= 1$.

 

[Millennial]

It is easy to show, with n a positive integer and a any positive number, that $(a^n)(a^m)= a^{n+m}$ and $(a^n)^m= a^{nm}$. If we want those very nice properties to be true even if n= 0 (and, extending, n negative, n a fraction) we want $a^na^0= a^{n+0}= a^n$. Since a is positive, $a^n> 0$ and we can divide both sides by it: $a^0= a^n/a^n= 1$.

That is, if we want $(a^n)(a^m)= a^{n+m}$ to be true we must define $a^0= 1$.

This is a very important point, and it applies to many other topics. Many things in mathematics are the way they are because we want them to be, because they are nicer to work with and perhaps more intuitive. These questions are very common:

Why is $a^0$ 1?
Why is 0! 1?
Why is 0,9999999... equal to 1?
...and so on and so forth.

The answer to all of those are simply because we defined them that way! We ask ourselves, "What is the best value for $a^0$?", and it is 1 for the reasons explained above. We could have made $a^0$ equal to 100 just as easily, but we would need to rework all of our rules and add exceptions. Occam's razor: The simplest explanation or solution to a problem is the best one.

This may seem a bit abstract to you at the moment, but think of it this way: Normally, exponents as taught in primary schools is iterated multiplication and only exists for positive integers. But we wanted to expand that definition and allow exponentiation with every number you can think of, and we wanted it to be as natural as possible. To do that, we use existing equations that hold true like $a^n a^m = a^{n+m}$ and we use those to find new values. Do they have to be like that? No. But it is more natural and easier to work with if they are.