The completeness of the complex number system becomes clearer when we
think of what are called ‘the nth roots of unity’ (for mathematicians ‘unity’ means
‘one’). These are the solutions of the equation zn = 1. Let’s take z^6 = 1 as an
example. There are the two roots z = 1 and z = −1 on the real number line
(because 1^6 = 1 and (−1)^6 = 1), but where are the others when surely there
should be six? Like the two real roots, all of the six roots have unit length and
are found on the circle centred at the origin and of unit radius.
More is true. If we look at w = ½+ √3/2 i which is the root in the first
quadrant, the successive roots (moving in an anticlockwise direction) are w2, w^3 ,
w^4 , w^5 , w^6 = 1 and lie at the vertices of a regular hexagon. In general the n roots
of unity will each lie on the circle and be at the corners or ‘vertices’ of a regular
n-sided shape or polygon.
Extending complex numbers
Once mathematicians had complex numbers they instinctively sought
generalizations. Complex numbers are 2-dimensional, but what is special about
2? For years, Hamilton sought to construct 3-dimensional numbers and work out
a way to add and multiply them but he was only successful when he switched to
four dimensions. Soon afterwards these 4-dimensional numbers were themselves
generalized to 8 dimensions (called Cayley numbers). Many wondered about 16-