2018 | OriginalPaper | Chapter

# 7. Complex Numbers

Published in:
The Discrete Math Workbook

## Abstract

Complex number z is an ordered pair of real numbers (a, b), where \(a, b\in \mathbb {R}\). The first number a is called the real part of the complex number \(z=(a, b)\) and is denoted by symbol \(\mathrm {Re}\, z\), while the second number of the pair b is called the imaginary part z and is denoted by \(\mathrm {Im}\, z\).

A complex number of the form (a, 0), where the imaginary part is zero, is identified with the real number a, i.e., \((a, 0)\equiv a\). This allows considering the set of all real numbers \(\mathbb {R}\) as s subset of set of complex numbers \(\mathbb {C}\).

Two complex numbers \(z_1 = (a_1,b_1)\) and \(z_2 = (a_2,b_2)\) are considered equal if and only if their

real and imaginary parts are pairwise equal: \(z_1=z_2\Leftrightarrow a_1=a_2\), \(b_1=b_2\).

On the set \(\mathbb {C}\) the operations of addition and multiplication of complex numbers are defined. Sum of the complex numbers \(z_1 = (a_1,b_1)\) and \(z_2 = (a_2,b_2)\) is the complex number z, equal to \(z_1+z_2 = {(a_1+a_2,b_1+b_2)}\). Product of the numbers \(z_1 = (a_1,b_1)\) and \(z_2 = (a_2,b_2)\) is such a complex number \(z=(a, b)\), that \(a=a_1 a_2-b_1 b_2\), \(b=a_1 b_2+a_2 b_1\).

Fundamental theorem of algebra states that any polynomial of a zero degree with complex coefficients has a complex root. This is why an arbitrary polynomial with real (or complex) coefficients always has some root \(z\in \mathbb {C}\).