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Complex Numbers

Appendix B

Complex Numbers

P. Danziger

Some Useful Sets

1.1

The Empty Set

Deﬁnition 1 The empty set is the set with no elements, denoted by φ.

1.2

Number Sets

N = 0, 1, 2, 3, . . . - The natural numbers.

Z = . . . , 3, 2, 1, 0, 1, 2, 3, . . . - The integers.

Q = x y x

- The rationals.

R = (

) - The Real numbers.

I = R

Q (all real numbers which are not rational) - The irrational numbers.

C = x + yi x, y

R - The Complex numbers.

Note: There are many real numbers which are not rational, e.g. π,

2 etc.

Complex Numbers

2.1

Introduction

We can’t solve the equation x

+ 1 = 0 over the real numbers, so we invent a new number i which

is the solution to this equation, i.e. i

Complex numbers are numbers of the form

z = x + iy, where x, y

The set of complex numbers is represented by C. Generally we represent Complex numbers by z

and w, and real numbers by x, y, u, v, so

z = x + iy, w = u + iv, z, w

C, x, y, u, v

Numbers of the form z = iy (no real part) are called pure imaginary numbers.

Complex numbers may be thought of as vectors in R

with components (x, y). We can also represent

Complex numbers in polar coordinates (r, θ) (θ is the angle to the real (x) axis), in this case we

write

z = re . Thus x = r cos θ, y = r sin θ,

and we have Demoivre’s Theorem.

Theorem 2 (Demoivre’s Theorem)

re = r(cos θ + i sin θ)

Complex Numbers