Numbers Real and Imaginary

Subtraction is not the only close relative of addition. Addition repeated is multiplication: 3 × 5 is shorthand for 3+3+3+3+3. The integers, positive and negative, take well to multiplication, always resulting in another integer.

However, once this convenient operation is put to use, it quickly reveals shortcomings in our new number system. New kinds of questions can be phrased, the most troubling of which is:

or equivalently:

There is no number among the integers to satisfy this question.

Once again an extraordinary solution has to be imagined: something that is not a number, but falls between existing numbers. At least this value is not as outlandish as the negative numbers were; we can actually find a place on the number line where it would reasonably fit. Once this new number is accepted, it leads to a whole family of similar fractions that divide 1 into smaller and smaller parts: one-third, one-fourth, ad infinitum.

 

In order for the integers to accoomodate division we must extend them not with just one new value, but a whole family of them. If i is any integer, then the missing pieces are their inverses, a set we may as well call i:

All possible multiples of these fractions (call them ratios) now become part of a new number system, the rational numbers. They will be a number plus some multiple of a number's inverse:

where a and b are any of the integers.

Again, because fractions are second nature to us after the fourth grade, we typically abbreviate them by omitting the the addition. Instead of:

we let 1 ^2/₃ suffice. The outcome is the same: a more sophisticated notation for a more powerful number system, one now robust enough to close over all four of the conventional mathematical operations.

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