What is Exponentiation definition and how it explains the base to power 0 is 1?

Question

I tried to understand exponentiation, but I can't understand that why any base raised to the power of 0 becomes 1.

As per Wikipedia the definition of b raise to power n is multiplications of base n times. Shouldn't it make more sense if we say a unit quantity is increased/decreased by base n times. If not then how would you explain anything raise to power 0 as 1 in layman/practical terms, like say counting bananas.

Addition: By increase decrease I mean multiplied and divided. And please help me understand not by math formulas, as I am trying to imagine it in real world scenarios.

Answer 1

In mathematics there is something called " recursive definition ". Just as the name states, you define "sequences " recursively. So lets motivate the idea behind the notion of taking, lets say a number to an arbitrary power ( in this case to the power of a natural number ). What you are actually doing is : take an arbitrary "number" lets call it $a$, define $a^0$ as $1$ ( more generally you define it to be the "neutral element", with respect to the given (binary) operation (which lets say we assume to be associative as well), if there is any). Then assume inductively that you have already defined what $a^n$ is for a given $n$ where n denotes a natural number, and then define $a^{n+1}$ to be $a^{n}$ times $a$. What do i mean by times. I did not give you the exact operation for which we have defined this notion, so you could just exchange it with lets say multiplication or addition ( for real numbers or rationals, etc., or any set equipped with a binary operation, which is associative and has a neutral element). (!So in conclusion we could just say that this is a convention in mathematics.)( as one is not always interested in real world applications)

Answer 2

NO, "the number $b$ multiplied by itself $n$ times" is the definition of $b^n$ if $n$ is a positive integer; for other $n$ we need a different definition.

First thing to note about definitions is we can make any definition that we want; the only question is is it useful. There's no mathematical problem with the definition $$b^n=17$$ for all $b$ and $n$, but it would make exponentiation sort of useless.

So. How do we want to define $b^0$? We note that if $n$ and $m$ are both positive integers then $$b^{n+m}=b^nb^m\quad(*).$$That's a useful thing, so we try to preserve it when we extend the definition to $n=0$. Think about it for a second; defining $b^0=1$ is the only way to get $(*)$ to work for $n\ge0$.

Similarly for why we define $b^{-n}=1/b^n$.