Page 11 - Calculus with Complex Numbers

Neither does division since for example

1 + li (1 + 2ï)(3 - 4ï) 3 + li - 8ï2
3 + 4/ (3 + 4ï)(3 - 4ï) 9 - 16/2
3 + li + 8 11 + li 11 2
= = = + i .
9 + 16 25 V V

The number 3 - 4/ is called the conjugate of 3 + 4i. For any .x + iy we have
(x + iy) (x - iy) = .r2 + y2

so division can always be done except when .x = y = 0 that is when .x + iy = 0.
lt is also possible to extract square roots of numbers of the form .x + iy as
numbers of the same form. For example suppose

1 + li = .4 + iB,

then we have
1 + li = (.4 + iB)l = .42 + IiAB - Bl.

So we require

42 - p 2 = j
.
.4# = 1.

The second equation gives B = l/.4 which on substitution in the lirst equation
gives

Solving this quadratic equation in .42 by the formula we obtain

1 + ./S
42 =
.
(! '
For real .4 we must take

x'S + 1
42 =
.
(2 5
which gives