Complex numbers
Complex numbers

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Complex numbers

3.3 Section summary

In this section we have seen a correspondence between complex numbers and points in the plane using Cartesian coordinates; the real part of the complex number is represented on the real axis (“horizontal”) and the imaginary part on the imaginary axis (“vertical”). We can also use polar coordinates (r,θ) in which case r, the modulus of a non-zero complex number z is positive and θ is an argument of z, defined only to within an additive integer multiple of 2. The principal value of the argument is that value of θ in (− ], and is denoted by Arg z.

Multiplication and division are particularly easy operations to carry out when the numbers are in polar form. We discussed the conjugate of a complex number and how to find the nth roots of a complex number.

Example 3

(i) If z = i, then |z| = 1 and

(ii) If z = −1, then |z| = 1 and Arg z = .

(iii) If z = i then |z| =  = 2 and Arg = −

It is usually a good idea to scribble a little diagram like Figure 11 to make sure that you get the correct angle for Arg z.

Figure 11
Figure 11

(iv)

In Figure 12 we have taken α (0, /2).

Figure 12
Figure 12

Example 4

(i)

(Figure 13).

Figure 13
Figure 13

(ii)

(Figure 14).

Figure 14
Figure 14

Example 5

(i)

(ii)

Example 6

M332_1

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