Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

Rez

Imz

z 1

z 2

z 1 +z 2

Figure 3.3 The addition of two complex numbers.

or in component notation

z 1 +z 2 =(x 1 ,y 1 )+(x 2 ,y 2 )=(x 1 +x 2 ,y 1 +y 2 ).

The Argand representation of the addition of two complex numbers is shown in

figure 3.3.

By straightforward application of the commutativity and associativity of the

real and imaginary parts separately, we can show that the addition of complex

numbers is itself commutative and associative, i.e.

z 1 +z 2 =z 2 +z 1 ,

z 1 +(z 2 +z 3 )=(z 1 +z 2 )+z 3.

Thus it is immaterial in what order complex numbers are added.

Sum the complex numbers1+2i, 3 − 4 i,−2+i.

Summing the real terms we obtain

1+3−2=2,

and summing the imaginary terms we obtain

2 i− 4 i+i=−i.

Hence

(1 + 2i)+(3− 4 i)+(−2+i)=2−i.

The subtraction of complex numbers is very similar to their addition. As in the

case of real numbers, if two identical complex numbers are subtracted then the

result is zero.