The geometry of numbers and the quaternions

In my previous article I explained what the main different sets of numbers are — natural , integers, rational, real, and complex.

A brief recap — the natural numbers ℕ are the whole positive numbers, the integers ℤ are the whole negative and positive numbers, rational numbers ℚ include the fractions, the real numbers ℝ include irrational numbers — i.e., numbers that cannot be expressed in fraction form, and the complex numbers ℂ are the numbers obtained from real plus the number i which is characterised by the property that the square of i is -1.

All numbers listed above, except the complex, can be represented as points on a line. We know their order — we know which one is larger and which one is smaller. When we work with the integers and the larger groups, we have the number zero in the middle of the line. We also know that the numbers decrease to the left and increase to the right (as shown on the picture below). Since we need only one measure to specify each number, i.e., the number itself, we can think of numbers as being a one-dimensional entity. This is also consistent with the fact that they are represented on a line which is a one-dimensional space.

We are in a slightly different scenario when it comes to the complex numbers. As we can see from the way they are written, to specify the number a+bi we need two…