Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

3.2 MANIPULATION OF COMPLEX NUMBERS


Rez

Imz
iz

−iz

z

−z

Figure 3.5 Multiplication of a complex number by±1and±i.

multiplyzby a complex number then the argument of the product is the sum


of the argument ofzand the argument of the multiplier. Hence multiplying


zby unity (which has argument zero) leaveszunchanged in both modulus


and argument, i.e.zis completely unaltered by the operation. Multiplying by


−1 (which has argumentπ) leads to rotation, through an angleπ, of the line


joining the origin tozin the Argand diagram. Similarly, multiplication byior−i


leads to corresponding rotations ofπ/2or−π/2 respectively. This geometrical


interpretation of multiplication is shown in figure 3.5.


Using the geometrical interpretation of multiplication byi, find the producti(1−i).

The complex number 1−ihas argument−π/4 and modulus




  1. Thus, using (3.10) and
    (3.11), its product withihas argument +π/4 and unchanged modulus




  1. The complex
    number with modulus



2 and argument +π/4is1+iand so
i(1−i)=1+i,

as is easily verified by direct multiplication.


The division of two complex numbers is similar to their multiplication but

requires the notion of the complex conjugate (see the following subsection) and


so discussion is postponed until subsection 3.2.5.


3.2.4 Complex conjugate

Ifzhas the convenient formx+iythen the complex conjugate, denoted byz∗,


may be found simply by changing the sign of the imaginary part, i.e. ifz=x+iy


thenz∗=x−iy. More generally, we may define the complex conjugate ofzas


the (complex) number having the same magnitude aszthat when multiplied by


zleaves a real result, i.e. there is no imaginary component in the product.

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