Theory

Complex numbers

What complex numbers are, how they are represented in algebraic, trigonometric and exponential form, how powers and roots are computed, and how equations are solved in the complex field.

On this page

Definition and algebraic form
Operations

Definition and algebraic form

1. Complex number

A complex number is an ordered pair $(a, b)$ of real numbers, with sum and product defined by

(a, b) + (c, d) = (a + c, b + d), (a, b) \cdot (c, d) = (a c - b d, a d + b c) .

The set of complex numbers is denoted by $C$ . Setting $i = (0, 1)$ , the definition of product gives $i^{2} = (0, 1) \cdot (0, 1) = (- 1, 0)$ , and identifying $(a, 0)$ with the real number $a$ one obtains the notation

z = (a, b) = a + ib, i^{2} = - 1,

called the algebraic form. In algebraic form one computes as with polynomials in $i$ , replacing $i^{2}$ with $- 1$ every time it appears.

2. Real part and imaginary part

For $z = a + ib$ with $a, b \in R$ we set

Re (z) = a, Im (z) = b .

The imaginary part is the real number $b$ , not $ib$ . Two complex numbers are equal if and only if they have the same real part and the same imaginary part; equations solved by setting $z = x + i y$ are based on this property, because an equality in $C$ is equivalent to a system of two equalities in $R$ .

Remark

In $C$ there is no ordering compatible with the operations: expressions such as $z_{1} < z_{2}$ have no meaning between non-real complex numbers. Inequalities in $C$ always concern real quantities, such as the modulus or the real part.

Operations

Sum, product, reciprocal and quotient

For $z = a + ib$ and $w = c + i d$ :

z + w = (a + c) + i (b + d), z \cdot w = (a c - b d) + i (a d + b c), \frac{1}{z} = \frac{a - ib}{a ^{2} + b ^{2}} (z \neq = 0), \frac{z}{w} = z \cdot \frac{1}{w} (w \neq = 0) .

The product is obtained by expanding $(a + ib) (c + i d)$ with the distributive property and substituting $i^{2} = - 1$ . For the reciprocal and the quotient, numerator and denominator are multiplied by $c - i d$ , called the conjugate of the denominator $c + i d$ , so that the denominator becomes the real number $(c + i d) (c - i d) = c^{2} + d^{2}$ . With these operations $C$ is a field: the same algebraic rules as in $R$ hold.

Quotient in algebraic form

\frac{2 + i}{1 - 3 i} = \frac{( 2 + i ) ( 1 + 3 i )}{( 1 - 3 i ) ( 1 + 3 i )} = \frac{2 + 6 i + i + 3 i ^{2}}{1 + 9} = \frac{- 1 + 7 i}{10} = - \frac{1}{10} + \frac{7}{10} i

Numerator and denominator are multiplied by $1 + 3 i$ , the conjugate of the denominator; the denominator becomes $1^{2} + 3^{2} = 10$ , which is real, so the result splits into real part and imaginary part.

The theory continues with 7 more sections (conjugate, the complex plane, trigonometric form…), plus all worked examples, the solutions and the PDF.

Unlock now