Page 10 - Calculus with Complex Numbers

C o m p I ex n u m b e rs

I . I The square root of m i nus one
Complex numbers originate from a desire to extract square roots of negative
numbers. They were lirst taken seriously in the eighteenth centul'y by mathemati-
ciarls such as de Moivre who proved the lirst theorem in the subject in 1722. Also
Huler, who introduced the notation i for - 1 and who discovered the mysterious
formula ei0 = cos 0 + i sin 0 in 1748 . And third Gauss, who was the lirst to prove
the fundamental theorem of algebra concerning existence of roots of polynomial
equations in 1799. The nineteenth centul'y saw the constnlction of the lirst model
for the complex numbers by Argand in 1806 later known as the Argand diagram,
and more recently as the complex plane. Also the lirst attempts to do calculus with
complex numbers by Cauchy in 1825. Complex numbers were lirst so called by
Gauss in 183 1. Previously they were known as imaginal'y numbers or impossible
numbers. lt was not until the twentieth centul'y that complex numbers found appli-
cation to science and teclmology, particularly to electrical engineering and lluid
dynamics.
lf we want square roots of negative numbers it is enough to introduce i = - 1
since then, for example, -2 = - 1sV2 = if-. Combining i with real numbers
l
by addition and multiplication cannot produce anything more general th= .x + iy
where .x y are real. This is because the sum and product of any two numbers of
this forrn are also of this forrn. For exarnple

= 3 + 10ï - 8
= - 5 + 1Oï.

Subtraction produces nothing new since for example