Numbers Real and Imaginary

The evolution from the natural numbers to the rationals is a lesson in how much we take for granted in the numbers we're used to dealing with every day. Neither simple nor obvious, they are a profound system at work yoking widely divergent concepts.

The device that imposes order on this numeric zoo is the number line. Though no more than a spatial analogy, it's a powerful invention that lets numbers positive, negative and fractional freely interchange — a feat not to be taken for granted.

Other methods, for instance, exist for dealing with negative values. The two-column system of credits and debits is one that strictly categorizes the two number types. It's not unsuccessful, but it's a special-purpose solution at best. Because it permits arithmetic only within the same category, and not across them (there is no way to put 2 debits and 9 credits together to get 7) it falls well short of the mathematical power of the line.

The number line, then, is an invention at least as important as the new kinds of numbers we've been arranging on it, and one that is called upon once more in the next mathematical step.

Repeated addition creates multiplication. Repeated multiplication is exponentiation: raising a number to a power. We make the abbreviation:

This is a handy notation. But as usual new notation brings with it the ability to phrase questions with no apparent answers. Trouble arises here when we try to mix exponentiation and negative numbers:

Or:

The square root of -1 is clearly not part of our number system; indeed it's hard to see that it's any kind of number at all. Mathematicians rejected it as soon as it began to appear, uninvited, in solutions to quadratic equations. We can do what they did eventually: call it imaginary, let it into the number system provisionally, and explore its properties to see just what kind of number it might be.

Therefore let us write:

Or equivalently:

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