Numbers Real and Imaginary

Of course, we don't typically use the symbol i for the value 0 - 1. We write instead -1, just as we abbreviate:

as:

And we have grown so familiar with these that we don't think of them as imaginary at all, but simply as the negative numbers.

The genius of the negative numbers is not just in their conception, but in harmonizing the rules of arithmetic to accommodate them. The negative numbers only joined the integers fully when it was realized that they formed a mirror world of numeration that took the old number line:

and extended it symmetrically in the opposite direction:

This not only clearly represents the relation between the two kinds of numbers, it allows them to be arithmetically combined in a satisfying way. Addition is "movement" to the right (the direction of increase), and subtraction is movement leftward (the direction of decrease).

Notice also the powerful metaphor that has been put in place. The abstract idea of numbers has become a geometric idea of points, lines and movement. In particular, multiplication by -1 has the effect of converting between the two realms of numbers by "reflecting" a segment around the central zero, as in 3 × −1:

Multiply by -1 again, and we cycle back to our original point — a full trip of 360 degrees (equivalent to 2π radians) around the number line.

That's a long trip to arrive at a familiar place, but this exercise in building the complete set of integers carries with it all the major concepts that used in higher realms. And the work has paid off: we have now a beautifully symmetric number system that's properly closed under addition as well as subtraction.

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