The Fourfold Number Line
Again, here's the cyclic behavior of the three fundamental values 1, −1 and i:
1×1 = 1
1×−1 = −1 ; −1×−1 = 1
1×i = i ; i×i = −1 ; −1×i = −i ; −i×i = 1
1×−1 = −1 ; −1×−1 = 1
1×i = i ; i×i = −1 ; −1×i = −i ; −i×i = 1
Positive 1 has a cycle period of one, just as the natural numbers form a line extending in one direction.
−1 has a cycle period of two. It's part of the integers, lying on a line extending in two directions.
And i? The pattern suggests splitting the universe of numbers into four parts: the familiar real numbers and the new imaginary ones of i and its multiples. Once we do, the geometric interpretation of multiplication by i becomes obvious.
