Complex Numbers

C

Complex numbers

Luca Zaffonte

From a set point of view, the set of complex numbers is the set of ordered pairs of real numbers (Cartesian product

of for himself ).

2 2

× {(a, ∈ ∈

R R = R R R = b) : a R, b R}

On this set we define the following operations:

• Sum: (a, b) + (c, d) = (a + c, b + d)

• Product: · −

(a, b) (c, d) = (ac bd, ad + bc)

0.1 Algebraic form of complex numbers

Se ∈ ⇒

x, y R

• (x, 0) + (y, 0) = (x + y, 0)

• · −

(x, 0) (y, 0) = (xy 0, 0 + 0) = (xy, 0)

I identify these complex numbers (with the second null component) with real numbers, hence the operations of C

operate as the operations of on the numbers of the type

R (x, 0).

1 The imaginary unity

The complex number it is called an imaginary unit.

i = (0, 1)

If ∈ ⇒

a, b R

• · ·

i b = (0, 1) (b, 0) = (0, b)

• a + ib = (a, 0) + (0, b) = (a, b)

Each ordered pair can be written as the sum of (called So:

a + ib algebraic form).

{a ∈

C = + ib : a, b R}

In the equation has at least one solution.

2 2

· −1.

i = i i = (0, 1)(0, 1) = (−1, 0) = C x + 1 = 0

Observation: e

⇐⇒ ⇐⇒

a + ib = c + id (a, b) = (c, d) a = c b = d

Observation:

Then the algebraic representation of complex numbers is unique. Just because it is unique, I can name as the

a

real part and imaginary part. If the real part of a complex number equals 0, then is purely imaginary (or

ofz b z z

pure imaginary). it is a field.

·)

(C, +,

Theorem:

is not an ordered field because the equation has a solution. in an ordered field it cannot have it.

2

C x + 1 = 0

is a field but it is not possible to introduce an order relation that makes it an ordered field.

C

Observation: C

2 Algebraic calculus in

• Sum: (a + ib) + (c + id) = (a + c) + i(bd)

• Product: 2

· −

(a + ib) (c + id) = ac + aid + cib + i bd = (ac bd) + i(ad + cb)

|{z}

=−1

• Reverse: if ∈ ̸ ̸

z = a + ib C, z = 0((a, b) = (0, 0)) 1

then

− − a b

1 a ib a ib = + i

= = 2 2 2 2 2 2

−

a + ib (a + ib)(a ib) a + b a + b a + b

• Quotient: − −

a + ib (a + ib)(c id) (ac + bd) + i(bc ad) =

= = 2 2

−

c + id (c + id)(c id) c + d

−

ac + bd bc ad

= + i

2 2 2 2

c + d c + d

2.1 Graphical representation of complex numbers

Complex numbers can be put in one-to-one correspondence with the points of a Cartesian plane (Argand-Gauss

plane). C

2.2 Geometric interpretation of operations in

1. Given e their sum is on the vertex of the parallelogram constructed on the sides identified

∈

a + ib c + id C,

by the two complex numbers.

2. Multiplication by given which is equivalent to a rotation of π

∈ · −y

i, x + iy C i (x + iy) = + ix, 2

2.3 Conjugate complex

Given the complex number is called conjugate of

−

z = x + iy z̄ = x iy z

3 Properties of the conjugate complex

• It is symmetrical with respect to the real axis

• ∀z ∈

z + z = z + z , , z C

1 2 1 2 1 2

•