Meet i

Try to solve x^2 = -1. A positive number times itself is positive, a negative number times itself is positive too, and 0 squared is 0. No real number has a negative square, so on the number line this equation has no solution.

Mathematicians answered with a new number. The imaginary unit i is defined by one rule: i^2 = -1. So i is a square root of -1, and √(-1) = i. It is not on the number line at all. Descartes called such numbers "imaginary" in 1637, and he did not mean it kindly.

In algebra, i behaves like a letter with one extra rule: whenever i^2 appears, replace it with -1. So 3i + 2i = 5i, just as 3x and 2x make 5x. And -1 has two square roots, i and -i, because (-i)^2 is also -1, just as 9 has the two square roots 3 and -3.