The Chemistry Maths Book, Second Edition

238 Chapter 8Complex numbers

The number can therefore be written as

with complex conjugate and inverse

Also,

0 Exercises 30–33

EXAMPLE 8.10Express in cartesian formx 1 + 1 iy:

(i)

(ii)e

−iπ 22

1 = 1 cos(−π 2 2) 1 + 1 i 1 sin(−π 2 2) 1 = 1 −i

0 Exercises 34–36

EXAMPLE 8.11The numbere

iπ

.

By Euler’s formula,

e

iπ

1 = 1 cos 1 π 1 + 1 i 1 sin 1 π

Becausecos 1 π 1 = 1 −1andsin 1 π 1 = 10 it follows that

e

iπ

1 = 1 − 1 (8.39)

This relation, involving the transcendental numbers eand π, the negative unit − 1 ,

and the imaginary unit i, is probably the most remarkable relation in mathematics.

0 Exercises 37–39

0 Exercises 40–42

22 ei i 2123213

iπ

ππ

3

=+ 33













=+













cos sin =+ ii

cos sin

ππ

ππ ππ

4

1

24

1

2

44 4

=+, = −

−













−

ee

i

ee

ii ii 44













ze i

−−i

== −













14

1

2

1

2

44

π

ππ

cos sin

ze i

i

*c== −ossin













−

22

44

π 4

ππ

ze

i

= 2

π 4