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 $\sqrt{-1} = i$. 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 ${\left(-i\right)}^{2}$ is also -1, just as 9 has the two square roots 3 and -3.