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19.1 Geometry and Arithmetic of Complex Numbers 671

This is the distance from the origin to the point (x, y) in the complex plane or the length of
the arrow representing the vector xi + yj (Figure 19.1(c)). |z − w| is the distance between the
complex numbers z and w or, equivalently, between these points in the plane (Figure 19.1(d)).

The complex conjugate,orjust conjugate of x + iy is the complex number x − iy with the
sign of the imaginary part reversed. Denote the conjugate of z as z.

In the complex plane, z is the reflection of z across the real axis (Figure 19.2). We have
Re(z) = Re(z) and Im(z) =− Im(z).
Conjugation (the operation of taking a conjugate) and magnitude have the following properties.
1. z = z.
2. z + w = z + w.
3. zw = (z)(w).
4. z/w = z/w if w = 0.
5. |z|=|z|.
6. |zw|=|z||w|.
1
1
7. Re(z) = (z + z) and Im(z) = (z − z).
2 2i
8. |z|≥ 0, and |z|= 0 if and only if z = 0.
2
9. If z = x + iy, then |z| = zz.
2
2
These are established by routine calculations. For property (5), observe that x + y remains
the same if y is replaced with −y. Equivalently, z and z are the same distance from the origin.
For property (9), compute
2
2
2
|z| = x + y = (x + iy)(x − iy) = zz.
Conjugates are often used to compute a complex quotient z/w. Multiply the numerator and
denominator of this quotient by the conjugate of the denominator:
z z w zw 1
= = = (zw).
w w w ww |w| 2
This converts a division problem z/w into one of computing a product zw, which is a simpler
operation. For example,
2 − 7i 2 − 7i 8 + 3i (2 − 7i)(8 − 3i) 5 62
= = =− − i.
8 + 3i 8 + 3i 8 + 3i 64 + 9 73 73

y
z

x

z

FIGURE 19.2 Complex
conjugate.

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October 15, 2010 18:5 THM/NEIL Page-671 27410_19_ch19_p667-694