The Chemistry Maths Book, Second Edition

226 Chapter 8Complex numbers

Apart from their role in extending the concept of number, complex numbers occur in

several branches of mathematics that are important in the physical sciences; for

example, the solutions of the differential equations of motion in both classical and

quantum mechanics often involve complex numbers. Complex numbers also occur

in the formulation of physical theory; the basic equations of quantum mechanics

necessarily involve.

8.2 Algebra of complex numbers

Complex numbers can be added, subtracted, multiplied, and divided in much the

same way as ordinary real numbers. It is only necessary to remember to replacei

2

by

− 1 whenever it occurs. Let two complex numbers be

z

1

1 = 1 x

1

1 + 1 iy

1

, z

2

1 = 1 x

21

• 1 iy

2

(8.5)

Equality

Two complex numbers are equal if their real parts are equal andif their imaginary

parts are equal:

z

1

1 = 1 z

2

if x

1

1 = 1 x

2

and y

1

1 = 1 y

2

(8.6)

Addition

z

1

1 + 1 z

2

1 = 1 (x

1

1 + 1 x

2

) 1 + 1 i(y

1

1 + 1 y

2

) (8.7)

The real parts ofz

1

andz

2

are added to give the real part of the sum, the imaginary

parts are added to give the imaginary part of the sum.

EXAMPLES 8.1Addition and subtraction

(i)(3 1 + 12 i) 1 + 1 (4 1 − 13 i) 1 = 1 (3 1 + 1 4) 1 + 1 (2 1 − 1 3)i 1 = 171 − 1 i

(ii) (3 1 + 12 i) 1 − 1 (3 1 − 12 i) 1 = 1 (3 1 − 1 3) 1 + 1 (2 1 + 1 2)i 1 = 14 i

(iii)(3 1 + 12 i) 1 + 1 (4 1 − 12 i) 1 = 17

0 Exercises 1–3

Multiplication

z

1

z

2

1 = 1 (x

1

1 + 1 iy

1

)(x

2

1 + 1 iy

2

)

= 1 x

1

(x

2

1 + 1 iy

2

) 1 + 1 iy

1

(x

2

1 + 1 iy

2

) 1 = 1 (x

1

x

2

1 + 1 ix

1

y

2

) 1 + 1 (iy

1

x

2

1 + 1 i

2

y

1

y

2

) (8.8)

= 1 (x

1

x

2

1 − 1 y

1

y

2

) 1 + 1 i(x

1

y

2

1 + 1 y

1

x

2

)

usingi

2

1 = 1 − 1.

i=− 1