Page 12 - Calculus with Complex Numbers

Hence we obtain

x''l + 1 s?s - 1
1 + li = + i .
2 2

This lastproperty of numbers of the form .x + iy represents a bonus over what might
reasonably have been expected. lntroducing square roots of negative real numbers
is one thing. Creating a number system in which square roots can always be taken
is asking rather more. But this is precisely what we have achieved. Hxistence of
square roots means that quadratic equations can always be solved. W e shall see
shortly that much more is true namely that polynomial equations of any degree
can be solved with numbers of the form .x + iy. This is the fundamental theorem
of algebra (see Chapter 8).

I .2 N otation and term i nology
lf i = - 1 then numbers of the form .x + iy are called complex numbers. W e
write z = .x + iy and call .x the real part of z which we abbreviate to Re z, and y
the imaginary part of z which we abbreviate to lm z.
N.B. Re z, lm z are lnoth real.
For z = .x + iy we write (by delinition) i = .x - iy, and call i the conjugate of z.

For z = .x + iy we write (by delinition) I zl= .r2 + y2, and call I zlthe
modulus of z.
For example, if z = 3 + 4/ we have Re z = 3, lm z = 4, i = 3 - 4ï, and

z
I .3 P roperties of i, I.I
We list the fundamental properties of @, Iz I.

zi = (.r + iyllx - iy) = .r2 + ,2 = IcI2.