Complex Numbers

Abstract

If x is any positive or negative number, the square of x is always positive. Therefore, no real number satisfies the quadratic equation

$${x^2} = - 1$$

((15.1.1))

However, nobody likes a result that states “it is impossible”. Mathematicians began early to search for a new sort of numbers. One could formally write $x = \sqrt { - 1}$, but it is not possible to state whether $\sqrt { - 1}$ is greater or smaller than a given real number. For a long time people thought that it is a necessary attribute of numbers to have a “size” with a specific order. Consequently, $\sqrt { - 1}$ could not be called a number. On the other hand, algebraic operations with $\sqrt { - 1}$ could be performed easily. The situation finally led to a compromise: $\sqrt { - 1}$ was called an imaginary number. The first letter of “imaginary” was proposed as a notation:

$$i = \sqrt { - 1} $$

((15.1.2))

.