Math in minutes 200 key concepts explained in an instant

i i is a “number” used to represent the square root of −1. This otherwise unrepresentable concept is not really a number in the sense of counting, and is known as an imaginary number.

The concept of i is useful when we are trying to solve an equation like x2 + 1 = 0, which can be rearranged as x2 = −1. Since squaring any positive or negative real number always gives a positive result, there can be no real-number solutions to this equation. But in a classic example of the beauty and utility of mathematics, if we define a solution and give it a name (i), then we can reach a perfectly consistent extension of the real numbers. Just as positive numbers have both a positive and negative square root, so −i is also a square root of −1, and the equation x2 + 1 = 0 has two solutions.

Armed with this new imaginary number, a new world of complex numbers, with both real and imaginary components, opens out before us (see pages 288–311).