Date: Mon, 21 Sep 1998 15:10:15 -0400

From: Alex Bogomolny

Dear Nathan:

> Real numbers are numbers on the number line.

What makes the points on the line numbers is that you can add and multiply them.

> That is why I don't understand what unreal numbers are. Where are they if > they are not on the number line?

The numbers you call unreal are known as "complex numbers" and are located in the plane. Every point in the plane is interpreted as a complex number the moment you define operations of addition and multiplication.

To every point in the plane there relates a vector - a directed line segment - from the origin to that point. Vectors have a length which is a real nonnegative number. (The length is only zero for the complex number located at the origin. This number is known as the complex zero.) The length of a complex number is its distance from the origin or the length of the corresponding line segment. Vectors (other than zero) also have an angle which is measured from the positive x-axis to the vector's line.

Define now the two operations:

### Addition:

Given to complex numbers (two points in the plane), form a parallelogram on the two corresponding vectors. The parallelogram has 4 vertices: the two numbers, the origin and the fourth as yet unnamed one. The fourth vertex is the sum of the two numbers.

### Multiplication:

Given two complex numbers. Let L be the product of their lengths. Let A be the sum of their angles. Draw a vector of length L and the angle from the positive x-axis equal to A. Its end point is, by definition, the product of the two given complex numbers.

That is all it takes to define complex numbers. The reason for this definition is manifold. Most important is that there is a number (usually denoted as i) whose length is 1 and angle is 90 degrees. This is the unit vector in the direction of the positive y-axis. When you square this number, you get a vector of length 1 (1x1=1) with angle 180 = 90 + 90 degrees. This number lies on the negative x-axis and corresponds to the real number -1. Therefore, among complex numbers there is a number whose square is -1. No real number has this property. This is why number i is called imaginary. However, it is not more unreal than any of the real numbers. It just lies on the y-axis.

For more see Algebraic Structure of Complex Numbers.

All the best,

Alexander Bogomolny