Numbers Real and Imaginary

The impossiblility of certain subtractions is an ancient crossroads in numeration. One choice is to stick to natural arithmetic, ignoring such meaningless inquiries. The other is to set aside practical implications and build a new and more complete system of unnatural numbers, closed under all possible operations. The latter route is the revolutionary one: we leave arithmetic behind and begin to do mathematics.

Completing the Integers

Since no existing number satisfies:

? + 1 = 0

We can invent one that does, and add it by decree to the system of natural numbers we already have. It's a number not found in nature, but in our imaginations; call it i. So:

i + 1 = 0

We claim that this imaginary quantity can be added it to a "real" one to get an real result, since it must be that:

i + 2 = 1
i + 3 = 2
...

In fact, this is philosophically dangerous ground, debated hotly in Europe through the seventeenth century. Nevertheless we can't stop there. By the rules of multiplication, there must also exist an infinitude of other imaginary values:

2i, 3i, 4i, ...