1.1 Powers of $i$ and Complex Representation | Complex Numbers | Mathematics 11

What is a Complex Number?

A complex number $z$ is any number of the form:

$z = a + bi$

where:

$a, b \in R$ (real numbers)
$i = - 1$ is the imaginary unit

Alternatively, $z$ can be written in ordered pair notation as $z = (a, b)$ .

Real and Imaginary Parts

Symbol	Name	Value
$Re (z)$	Real part	$a$
$Im (z)$	Imaginary part	$b$

Example: For $z = - 7 + 4 i$ : $Re (z) = - 7$ and $Im (z) = 4$ .

Note: $b$ is the imaginary part, NOT $bi$ . The imaginary part is always a real number.

Powers of $i$

Since $i = - 1$ , successive powers of $i$ cycle with period 4:

Power	Value
$i^{0}$	$1$
$i^{1}$	$i$
$i^{2}$	$- 1$
$i^{3}$	$- i$
$i^{4}$	$1$
$i^{5}$	$i$
$⋮$	$⋮$

Rule for Large Powers

To evaluate $i^{n}$ for any large positive integer $n$ :

Divide $n$ by $4$ and find the remainder $r$ .
Then $i^{n} = i^{r}$ .

Example: $i^{23}$ : $23 = 4 \times 5 + 3$ , so $r = 3$ and $i^{23} = i^{3} = - i$ .

Example: $i^{4 k + 2} = (i^{4})^{k} \cdot i^{2} = 1^{k} \cdot (- 1) = - 1$ .

Equality of Complex Numbers

Two complex numbers $z_{1} = a + bi$ and $z_{2} = c + d i$ are equal if and only if:

$a = c AND b = d$

That is, their real parts are equal and their imaginary parts are equal.

Example: If $(2 x - 1) + (y + 3) i = 5 + 7 i$ , then:

$2 x - 1 = 5 \Rightarrow x = 3$
$y + 3 = 7 \Rightarrow y = 4$

Basic Operations on Complex Numbers

Let $z_{1} = a + bi$ and $z_{2} = c + d i$ .

Addition and Subtraction

$z_{1} \pm z_{2} = (a \pm c) + (b \pm d) i$

Multiplication

$z_{1} \cdot z_{2} = (a c - b d) + (a d + b c) i$

Use $i^{2} = - 1$ when expanding.

Example: $(3 + 2 i) (1 - 4 i) = 3 - 12 i + 2 i - 8 i^{2} = 3 - 10 i + 8 = 11 - 10 i$

Division

To divide $\frac{a + bi}{c + d i}$ , multiply numerator and denominator by the conjugate of the denominator $(c - d i)$ :

$\frac{a + bi}{c + d i} = \frac{( a + bi ) ( c - d i )}{( c + d i ) ( c - d i )} = \frac{( a c + b d ) + ( b c - a d ) i}{c ^{2} + d ^{2}}$

The denominator becomes real because $(c + d i) (c - d i) = c^{2} + d^{2}$ .

Example: $\frac{2 + 3 i}{1 - 2 i} = \frac{( 2 + 3 i ) ( 1 + 2 i )}{( 1 - 2 i ) ( 1 + 2 i )} = \frac{2 + 4 i + 3 i + 6 i ^{2}}{1 + 4} = \frac{2 + 7 i - 6}{5} = \frac{- 4 + 7 i}{5} = - \frac{4}{5} + \frac{7}{5} i$

Complex Conjugate

The complex conjugate of $z = a + bi$ is defined as:

$\overset{z}{ˉ} = a - bi$

Only the sign of the imaginary part is changed.

Key property: $z \cdot \overset{z}{ˉ} = (a + bi) (a - bi) = a^{2} + b^{2}$

Examples:

$z = 3 + 5 i \Rightarrow \overset{z}{ˉ} = 3 - 5 i$
$z = - x - i y \Rightarrow \overset{z}{ˉ} = - x + i y$

Modulus (Absolute Value) of a Complex Number

The modulus (or absolute value) of $z = a + bi$ is defined as:

$∣ z ∣ = a^{2} + b^{2}$

Geometrically, $∣ z ∣$ is the distance from the origin to the point $(a, b)$ in the complex plane.

Example: $z = 3 + 4 i \Rightarrow ∣ z ∣ = 3^{2} + 4^{2} = 9 + 16 = 25 = 5$

Note: $∣ z ∣^{2} = z \cdot \overset{z}{ˉ} = a^{2} + b^{2}$

Summary Table

Concept	Definition
Complex number	$z = a + bi$ , also $(a, b)$
Real part	$Re (z) = a$
Imaginary part	$Im (z) = b$
Equality	$a + bi = c + d i \Leftrightarrow a = c, b = d$
Conjugate	$\overset{z}{ˉ} = a - bi$
Modulus	$
Powers of $i$	Cycle: $1, i, - 1, - i$ (period 4)

What is a Complex Number?

A complex number $z$ is any number of the form:

$z = a + bi$

where:

$a, b \in R$ (real numbers)
$i = - 1$ is the imaginary unit

Alternatively, $z$ can be written in ordered pair notation as $z = (a, b)$ .

Real and Imaginary Parts

Symbol	Name	Value
$Re (z)$	Real part	$a$
$Im (z)$	Imaginary part	$b$

Example: For $z = - 7 + 4 i$ : $Re (z) = - 7$ and $Im (z) = 4$ .

Note: $b$ is the imaginary part, NOT $bi$ . The imaginary part is always a real number.

Powers of $i$

Since $i = - 1$ , successive powers of $i$ cycle with period 4:

Power	Value
$i^{0}$	$1$
$i^{1}$	$i$
$i^{2}$	$- 1$
$i^{3}$	$- i$
$i^{4}$	$1$
$i^{5}$	$i$
$⋮$	$⋮$

Rule for Large Powers

To evaluate $i^{n}$ for any large positive integer $n$ :

Divide $n$ by $4$ and find the remainder $r$ .
Then $i^{n} = i^{r}$ .

Example: $i^{23}$ : $23 = 4 \times 5 + 3$ , so $r = 3$ and $i^{23} = i^{3} = - i$ .

Example: $i^{4 k + 2} = (i^{4})^{k} \cdot i^{2} = 1^{k} \cdot (- 1) = - 1$ .

Equality of Complex Numbers

Two complex numbers $z_{1} = a + bi$ and $z_{2} = c + d i$ are equal if and only if:

$a = c AND b = d$

That is, their real parts are equal and their imaginary parts are equal.

Example: If $(2 x - 1) + (y + 3) i = 5 + 7 i$ , then:

$2 x - 1 = 5 \Rightarrow x = 3$
$y + 3 = 7 \Rightarrow y = 4$

Basic Operations on Complex Numbers

Let $z_{1} = a + bi$ and $z_{2} = c + d i$ .

Addition and Subtraction

$z_{1} \pm z_{2} = (a \pm c) + (b \pm d) i$

Multiplication

$z_{1} \cdot z_{2} = (a c - b d) + (a d + b c) i$

Use $i^{2} = - 1$ when expanding.

Example: $(3 + 2 i) (1 - 4 i) = 3 - 12 i + 2 i - 8 i^{2} = 3 - 10 i + 8 = 11 - 10 i$

Division

To divide $\frac{a + bi}{c + d i}$ , multiply numerator and denominator by the conjugate of the denominator $(c - d i)$ :

$\frac{a + bi}{c + d i} = \frac{( a + bi ) ( c - d i )}{( c + d i ) ( c - d i )} = \frac{( a c + b d ) + ( b c - a d ) i}{c ^{2} + d ^{2}}$

The denominator becomes real because $(c + d i) (c - d i) = c^{2} + d^{2}$ .

Complex Conjugate

The complex conjugate of $z = a + bi$ is defined as:

$\overset{z}{ˉ} = a - bi$

Only the sign of the imaginary part is changed.

Key property: $z \cdot \overset{z}{ˉ} = (a + bi) (a - bi) = a^{2} + b^{2}$

Examples:

$z = 3 + 5 i \Rightarrow \overset{z}{ˉ} = 3 - 5 i$
$z = - x - i y \Rightarrow \overset{z}{ˉ} = - x + i y$

Modulus (Absolute Value) of a Complex Number

The modulus (or absolute value) of $z = a + bi$ is defined as:

$∣ z ∣ = a^{2} + b^{2}$

Geometrically, $∣ z ∣$ is the distance from the origin to the point $(a, b)$ in the complex plane.

Example: $z = 3 + 4 i \Rightarrow ∣ z ∣ = 3^{2} + 4^{2} = 9 + 16 = 25 = 5$

Note: $∣ z ∣^{2} = z \cdot \overset{z}{ˉ} = a^{2} + b^{2}$

Summary Table

Concept	Definition
Complex number	$z = a + bi$ , also $(a, b)$
Real part	$Re (z) = a$
Imaginary part	$Im (z) = b$
Equality	$a + bi = c + d i \Leftrightarrow a = c, b = d$
Conjugate	$\overset{z}{ˉ} = a - bi$
Modulus	$
Powers of $i$	Cycle: $1, i, - 1, - i$ (period 4)