Mathematical Methods for Physics and Engineering : A Comprehensive Guide

3.2 MANIPULATION OF COMPLEX NUMBERS

Rez

Imz

|z|

x

y

argz

Figure 3.4 The modulus and argument of a complex number.

3.2.2 Modulus and argument

The modulus of the complex numberzis denoted by|z|and is defined as

|z|=

√

x^2 +y^2. (3.4)

Hence the modulus of the complex number is the distance of the corresponding

point from the origin in the Argand diagram, as may be seen in figure 3.4.

The argument of the complex numberzis denoted by argzand is defined as

argz=tan−^1

(y

x

)

. (3.5)

It can be seen that argzis the angle that the line joining the origin tozon

the Argand diagram makes with the positivex-axis. The anticlockwise direction

is taken to be positive by convention. The angle argzis shown in figure 3.4.

Account must be taken of the signs ofxandyindividually in determining in

which quadrant argzlies. Thus, for example, ifxandyare both negative then

argzlies in the range−π<argz<−π/2 rather than in the first quadrant

(0<argz<π/2), though both cases give the same value for the ratio ofytox.

Find the modulus and the argument of the complex numberz=2− 3 i.

Using (3.4), the modulus is given by

|z|=

√

22 +(−3)^2 =

√

13.

Using (3.5), the argument is given by

argz=tan−^1

(

−^32

)

.

The two angles whose tangents equal− 1 .5are− 0 .9828 rad and 2.1588 rad. Sincex=2and
y=−3,zclearly lies in the fourth quadrant; therefore argz=− 0 .9828 is the appropriate
answer.