N’th roots of 1: video clip


With ordinary (“real”) numbers, the equation z3=1 has only one root. With complex numbers, it has three – shown in the picture above.

With complex numbers, every quadratic has two roots, every cubic has three roots, every quartic has four roots, and so on (so long as we count a double root α, when the equation can be factorised with a term [x−α]2 as a factor, as two roots). (This is the Fundamental Theorem of Algebra).

So 1 has 3 cube roots, 4 4th-roots, 5 5th-roots, 6 6th-roots, 7 7th-roots, and so on.

This video clip shows how to find all those roots. It’s good and clear, and especially good for showing how the roots are spread out like the spokes of a wheel; but just note two things before you start.

The guy doing the video uses j for √−1 where we would use i. Electrical engineers usually do that, because they already use i for current.

He writes complex numbers in polar form as r e, where in FP1 we write them as r cis θ (check here for what we mean by cis θ).

If you just read “cis θ” every time he writes or says “e“, then you’ll be fine. The stunning thing which you’ll learn in FP2 is that for all angles φ (in radians):

cis φ = cos φ + j sinφ = e

where e is a special number, about 2.71828. Wow!