why does 0 to the power of 0 equal 1?
In: 97
I think the main confusion is thinking x^y means x times itself y times. This is a property of the exponential function when x and y are non zero positive integers, but using this as a definition quickly becomes nonsense. For example what does it mean to do 2^(-3)=1/8. Are we multiplying 2 negative 3 number of times? What does it mean to multiply negative times. Or how about 2^1.5, what does it mean to multiply it one and a half time? What is a “half” multiplication. Or 3^pi and dealing with irrationals, or with imaginary numbers, or as you mentioned things to the exponent 0. For this reason, the “actual” definition of exponents is different in mathematics and it just so happens we can prove this property holds with the “actual” equation for positive integers. The real equations are more confusing and need a deeper knowledge of math to understand, which is why schools simplify it with teaching a wrong definition that works in some cases and is more “intuitive”.
The “ELi5” reason:
Any number to the power of 0 is equal to 1: So, 0^0=1
Any number multiplied by 1, is itself: 12 * 1 = 12
Zero to the power of zero is really equalling a multiplicity of one: 12 * (0^0) = 12
This is done in order to make sure that *any number to the power of zero* is equal to no value in multiplication (in other words, 1)
If you want a reason beyond 9th grade math or a proof, then it gets a lot more complicated. There is also a lot more to discuss about 0^0… but, we’re doing ELi5 here.
edit: I accidentally did some algebra… and that’s not ELi5
It is undefined generally, however it is very common to define it as 1 for most purposes. One way of looking at a^b is as the number of functions from a set of size b to a set of size a. When both are 0 there is a single function, the empty function.
Anything to the 0th power equals 1 because it can’t completely disappear. He’s 5 people not 50
When a and b are natural numbers, a^b is defined as the number of functions from a set of size b to a set of size a. There is exactly one function from the empty set to itself, so 0^0 = 1.
(It is also true, but irrelevant, that the function f(x, y) = x^y has an essential discontinuity at (0,0).)
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