the imaginary unit

On the real number line there is no number whose square is negative — square anything real, positive or negative, and you land on something positive or zero. So the equation x^2 = -1 has no real answer. The imaginary unit, written i, is the new number we invent precisely to fill that gap: it is a symbol that obeys the single rule i^2 = -1. You do not have to picture it sitting anywhere on the ordinary number line; you just agree to treat i as a quantity that, when squared, gives -1, and then do ordinary algebra with it.

Once you accept i^2 = -1, the powers of i cycle in a tidy four-step loop: i^1 = i, i^2 = -1, i^3 = i^2 times i = -i, i^4 = (i^2)^2 = 1, and then it repeats — i^5 = i, i^6 = -1, and so on. To simplify any power of i, divide the exponent by 4 and keep the remainder: i^27, for instance, has 27 = 4 times 6 + 3, so i^27 = i^3 = -i. This little cycle is the whole arithmetic engine behind everything that follows.

It is worth being honest about the name. Calling i 'imaginary' is a historical accident — these numbers are no less real, no less useful, than the negatives or the fractions once were. They are simply a second dimension of number, and the moment you let i join the reals you can solve every polynomial equation, not just some. The word 'imaginary' was originally a slur (from Descartes) and stuck; do not let it fool you into thinking i is a trick.