Seeing complex numbers

A number line is a picture that represents numbers as points on a straight horizontal line. As can be seen in the image, the numbers increase as you move to the right along the line and decrease as you move to the left. Zero is represented in the middle of the line, bisecting the positive and negative numbers. Fractions and irrational numbers are also representable on this line between the whole number points.

Operations such as addition, subtraction, multiplication and division are simply transformations of a number on this line. For example, to add 2 to the number 3 means to start at the number three point and then move two positions to the right along the line (in geometrical terms, this is called translation). To multiply 5 by 2 is to start at the number five point and then move right until you are twice the distance from the origin (geometrically, this is called scaling).

In the sixteenth century, a new kind of number called a complex number was introduced by Italian mathematician Gerolamo Cardano. It is expressed as a + bi where a and b are real numbers and i is termed the imaginary unit which satisfies the equation i ² = -1. As negative numbers have applications in areas such as temperature measurement and finance, so complex numbers apply in areas such as physics and fractal geometry.

But is it possible to have an intuitive understanding of complex numbers? Didn't we learn in school that any number (including a negative number) multiplied by itself is always a positive number? So how can i ² = -1 make any sense? What could i be?

The first step to understanding complex numbers is to extend our concept of the number line to a number plane by adding a second dimension that cuts through the number line origin. As shown in the image, this is called the imaginary dimension with i as the unit and it is perpendicular to the real dimension.

The second step is to introduce rotation around the origin as a new transformation operation. When we rotate a number on the real number line by 90^o in an anti-clockwise direction, we can see that it ends up on the imaginary number line. This operation is what it means to multiply by i. If we rotate a further 90^o in an anti-clockwise direction (that is, multiply by i again), then we end up back on the real number line. But we are now on the opposite side of the origin. That is, if we start at 3, multiply by i, then multiply by i again, we end up at -3. That is, 3i ² = -3. The effect of multiplying by i ² is simply to negate our original number by rotating 180^o to the other side of the origin. If we multiply by i ² again, then we pass through -3i and end up back where we started at 3.

Transforming a negative number into a positive number requires translation on the real number line. Transforming a complex number into a real number requires rotation on the real-imaginary plane. So what is i? The real number one rotated by 90^o onto the imaginary number line. That's all it means.