Multiplication and beyond

Let's add some numbers, like this:

a + b + c + ...

Now let's start chopping things off the end of this list:

a + b +

Then

a +

What remains if we chop off the a? The original expression was really

0 + a + b + c + ...

So if we remove all the pieces, we get 0. And it makes sense, if we never add anything, we're left with nothing.

OK, let's now consider something very similar:

a * b * c * ...

We know that if we start chopping things off, at some point there will be nothing left. Is that nothing zero? Surprisingly, no, because our expression was really this:

1 * a * b * c * ...

So at the end, what's left is 1. In fact, 1 is to multiplication what 0 is to addition. Adding zero does the same as multiplying by 1 -- nothing. If you want an equation, here is one: a*1 = a+0.

We can write 1 * a * a using the shortcut notation as 1 * a², which is read as "a to the power of two". And as we chop things off the end, we first have 1 * a¹, then 1 * a⁰ -- which is another way at arriving at the fact that a⁰ = 1.

Note how each next series of operations squeezes more work into the same amount of space.
First, we had a+a+a+..., and we replaced that with the shortcut a*3 in order to understand the process better, to be able to think about things that are common to all such summations.
Then, we started having having multiplications like a*a*a*..., where each of the * signs stands the summation (a+a+a+...).
We replaced that with the shortcut a³.
The next step would be a^aa..., where the placing of one number above another stands for several multiplications (a*a*a*... etc).