imaginary unit

The imaginary unit is a brand-new number, written i, invented to answer a question that real numbers refuse to answer: what squares to give a negative? No real number works, because a positive times a positive is positive and a negative times a negative is also positive. So mathematicians simply declared a new number i and gave it one defining rule: i times i equals negative one. Think of it as a fresh ingredient added to the number system, not a number you can find on the ordinary number line.

Formally, i satisfies i^2 = -1. From this single rule everything else follows. Powers of i cycle in a pattern of four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then it repeats. Once i is allowed, the equation x^2 = -1 finally has solutions, namely x = i and x = -i.

A common confusion: people write i = sqrt(-1), and that is a useful shorthand, but be careful. The familiar rule sqrt(a) times sqrt(b) = sqrt(ab) does not hold for negative numbers, so writing sqrt(-1) times sqrt(-1) = sqrt(1) = 1 is wrong — the correct answer is -1. Always go back to the defining rule i^2 = -1 when in doubt.