Complex Numbers

Complex numbers

Luca Zaffonte

Set of complex numbers

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 ). \( \mathbb{R} \times \mathbb{R} \).2 {(a, b) : a ∈ R, b ∈ R}

Operations on complex numbers

• Sum: (a, b) + (c, d) = (a + c, b + d)
• Product: (a, b) · (c, d) = (ac − bd, ad + bc)

Algebraic form of complex numbers

If x, y ∈ R:

• (x, 0) + (y, 0) = (x + y, 0)
• (x, 0) · (y, 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 \( R \) on the numbers of the type (x, 0).

The imaginary unity

The complex number \( i = (0, 1) \) is called an imaginary unit.

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 \( a + ib \) (called algebraic form).

\( C = \{a + ib : a, b ∈ R\} \)

Properties of complex numbers

In the equation \( x^2 + 1 = 0 \), there is at least one solution. \( i^2 = i \cdot i = (0, 1) \cdot (0, 1) = (-1, 0) \)

Observation: \( a + ib = c + id \) if (a, b) = (c, d), hence a = c and b = d.

The algebraic representation of complex numbers is unique. Just because it is unique, I can name as the real part and imaginary part. If the real part of a complex number equals 0, then it is purely imaginary (or pure imaginary).

Theorem: Field properties of complex numbers

\( (C, +, \cdot) \) is a field.

\( C \) is not an ordered field because the equation \( x^2 + 1 = 0 \) has a solution. In an ordered field, it cannot have it. Hence, it is not possible to introduce an order relation that makes it an ordered field.

Algebraic calculus in \( C \)

• Sum: (a + ib) + (c + id) = (a + c) + i(b + d)
• Product: (a + ib) · (c + id) = ac + aid + cib + i bd = (ac − bd) + i(ad + cb)
• Reverse: if \( z = a + ib \in C, z ≠ 0 \) ((a, b) ≠ (0, 0))

\(\frac{1}{z} = \frac{1}{a + ib} = \frac{a - ib}{a^2 + b^2}\)

• Quotient: \(\frac{a + ib}{c + id} = \frac{(a + ib)(c - id)}{(c + id)(c - id)} = \frac{(ac + bd) + i(bc - ad)}{c^2 + d^2}\)

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).

Geometric interpretation of operations in \( C \)

1. Given \( a + ib, c + id \in C \), their sum is on the vertex of the parallelogram constructed on the sides identified by the two complex numbers.
2. Multiplication by i of \( x + iy \in C \) is equivalent to a rotation of \( \pi/2 \).

Conjugate complex

Given the complex number \( z = x + iy \), \( \overline{z} = x - iy \) is called the conjugate of \( z \).

Properties of the conjugate complex

• It is symmetrical with respect to the real axis.
• For all \( z_1, z_2 \in C \), \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \).