Where multiplication is taken apart

The fact that positive and negative numbers are really right- and left-facing arrows, and that there is a whole continuum of arrows in between is truly beautiful. It's beautiful because it's true. For example, it's not an invention -- inventions can be different depending on who's inventing -- this continuum is really a discovery. If two people discover the same thing, they end up in full agreement despite their best efforts at being original, because there is something out there in the intangible space of ideas that is built in a very particular way, and it can be discovered by thinking.

Let's get back to the continuum between positive and negative numbers. Is it really there? Can we make it visible?

When you add two numbers, like 2 + 2, you fully expect that one big addition can be split into two smaller ones, for example 2 + 1 + 1. And if you wanted to look at things even more finely, you could say 2 + 1/2 + 1/2 + 1/2 + 1/2. This ability to do addition in whatever size pieces you like gives it a physical feel: you know that if you keep adding little bits of one quantity to another, eventually you'll add the whole thing.

But multiplying, say, a by -1, involves a jump -- there are no intermediate steps to be seen, and all of a sudden you're at -a; a long way from where you started. What's the continuous process that multiplies a by -1 in little steps? Does it exist? If we started multiplying a by -1, then stopped the process halfway, and looked at the result, where would we be?

Yes, the process exists, and it's visualized quite easily. When you multiply a by -1, you are adding 180° to a's arrows' angle. The arrow peels off the positive axis, rotates all the way to the diametrically opposite point, and stops there. If you stop halfway between a(0°) and a(180°), you're not nowhere, you're at a(90°), or a*i, as the illustration below shows:


Another way of saying "half way in the process of multiplying by -1" is to write -1^1/2, which is read as "minus one to the power 1/2". Using power 1/2, we gave -1 a multiplying power half as strong as it originally had. -1 raised to the power 1/2 multiplies other numbers half as much as it used to, in terms of this continuous process. And if we now wanted to restart the process from the midpoint, we could multiply the number we had so far by another -1^1/2, and we'd end up at -a.

But what if we stopped the process even earlier? We could stop halfway between a and ai, that would be a * -1^1/4 or a * i ^1/2. $We\; could\; keep\; going\; with\; ever-smaller\; powers.\; The\; smaller\; the\; power,\; the\; less\; powerful\; the\; multiplication.\; In\; fact,\; if\; we\; stopped\; the\; multiplication\; right\; at\; the\; start,\; we\text{'}d\; be\; at\; a\; *\; -1$⁰$.\; So\; you\; see,$$-1$⁰$$ is a powerless multiplier. In fact, any number to the power zero is a powerless multiplier, and is equal to 1.

What can be subdivided once, can be subdivided many times. We can now do any amount of multiplication. For example, we can perform half a multiplication and save the rest for later:
a * b^1/2
We can multiply twice:
a * b².
We can multiply 0.35 times, and then another 0.65 times, and the the original:
a * b = a * b^0.35 * b^0.65.
We can multiply zero times, which does nothing:
a * b⁰ = a.
We can multiply three times, and then undo one multiplication:
a * b³ * b^-1 = a * b²

If you like taxonomy, you'll be happy. There is a whole zoo of labels for these operations.

• b squared refers to b²
  
• square root of b refers to b^1/2, because b^1/2 * b^1/2 = b.
• division refers to b^-1
• exponentiation refers to all of these -- any time you change the power of multiplication.
  
• the word power refers to x in b^x
• the word base refers to b in b^x
• the word logarithm refers to power when it's not known. for example when you try to figure out which x satisfies b^x = c, given b and c, x is the logarithm of c.
• i refers to -1^1/2