What is the imaginary unit i?

The imaginary unit i is defined by the property i² = −1. Powers of i repeat with period 4: i⁰ = 1, i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and so on.

How do you perform arithmetic on complex numbers?

Addition and subtraction operate on the real and imaginary parts separately. Multiplication uses the distributive law with i² = −1. Division multiplies numerator and denominator by the conjugate of the denominator to make the denominator real.

What is the relationship between the conjugate and the absolute value?

For z = a + bi, the conjugate is z̄ = a − bi. The product z·z̄ = a² + b² = |z|² holds, where |z| = √(a² + b²) is the distance from the origin in the complex plane.

Chapter 1: Definition and Arithmetic of Complex Numbers

Imaginary Unit, Four Operations, Conjugates, Absolute Value

Motivation

Within the real numbers, no number squares to a negative value. For example, the equation

$$x^2 + 1 = 0$$

has no real solution, because $x^2 \geq 0$ for every real $x$, so $x^2 + 1 \geq 1 > 0$.

To resolve this, mathematicians introduced a new kind of number: the imaginary unit $i$.

Definition of the Imaginary Unit

Definition: Imaginary Unit

The imaginary unit $i$ is defined as a number satisfying:

$$i^2 = -1$$

That is, $i = \sqrt{-1}$.

Powers of $i$

The powers of $i$ follow a cyclic pattern:

$i^0 = 1$
$i^1 = i$
$i^2 = -1$
$i^3 = i^2 \cdot i = -i$
$i^4 = (i^2)^2 = (-1)^2 = 1$
$i^5 = i^4 \cdot i = i$
...

The powers of $i$ repeat with period 4. In general, $i^n = i^{n \bmod 4}$.

Definition of Complex Numbers

Definition: Complex Number

A complex number is a number of the form:

$$z = a + bi$$

where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

$a$ is called the real part of $z$, written $\mathrm{Re}(z) = a$
$b$ is called the imaginary part of $z$, written $\mathrm{Im}(z) = b$

The set of all complex numbers is denoted $\mathbb{C}$.

Examples

$z = 3 + 2i$: $\mathrm{Re}(z) = 3$, $\mathrm{Im}(z) = 2$
$z = -1 + 4i$: $\mathrm{Re}(z) = -1$, $\mathrm{Im}(z) = 4$
$z = 5$: $\mathrm{Re}(z) = 5$, $\mathrm{Im}(z) = 0$ (a real number is a special case of a complex number)
$z = 3i$: $\mathrm{Re}(z) = 0$, $\mathrm{Im}(z) = 3$ (called a purely imaginary number)

Equality of Complex Numbers

Two complex numbers $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$ are equal if and only if

$$a_1 = a_2 \quad \text{and} \quad b_1 = b_2$$

That is, two complex numbers are equal precisely when their real parts and imaginary parts are respectively equal.

Arithmetic of Complex Numbers

Addition and Subtraction

For $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$:

$$z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i$$ $$z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2)i$$

In other words, add or subtract the real parts and imaginary parts separately.

Example: Addition and Subtraction

$(3 + 2i) + (1 - 4i) = (3+1) + (2-4)i = 4 - 2i$

$(3 + 2i) - (1 - 4i) = (3-1) + (2-(-4))i = 2 + 6i$

Multiplication

For $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$:

\begin{align*} z_1 \cdot z_2 &= (a_1 + b_1 i)(a_2 + b_2 i) \\ &= a_1 a_2 + a_1 b_2 i + b_1 a_2 i + b_1 b_2 i^2 \\ &= a_1 a_2 + a_1 b_2 i + b_1 a_2 i - b_1 b_2 \quad (\because\; i^2 = -1) \\ &= (a_1 a_2 - b_1 b_2) + (a_1 b_2 + b_1 a_2)i \end{align*}

Example: Multiplication

$(2 + 3i)(1 - i) = 2 \cdot 1 + 2 \cdot (-i) + 3i \cdot 1 + 3i \cdot (-i)$

$= 2 - 2i + 3i - 3i^2 = 2 + i + 3 = 5 + i$

Division

When $z_2 \neq 0$, $\displaystyle\frac{z_1}{z_2}$ is computed by multiplying by the conjugate of the denominator:

$$\frac{a_1 + b_1 i}{a_2 + b_2 i} = \frac{(a_1 + b_1 i)(a_2 - b_2 i)}{(a_2 + b_2 i)(a_2 - b_2 i)} = \frac{(a_1 a_2 + b_1 b_2) + (b_1 a_2 - a_1 b_2)i}{a_2^2 + b_2^2}$$

Example: Division

Compute $\displaystyle\frac{3 + 2i}{1 - i}$. Multiply numerator and denominator by the conjugate $1 + i$:

$$\frac{3 + 2i}{1 - i} = \frac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)} = \frac{3 + 3i + 2i + 2i^2}{1 - i^2} = \frac{3 + 5i - 2}{1 + 1} = \frac{1 + 5i}{2} = \frac{1}{2} + \frac{5}{2}i$$

Complex Conjugate

Definition: Complex Conjugate

For a complex number $z = a + bi$, its complex conjugate $\bar{z}$ is defined as:

$$\bar{z} = a - bi$$

That is, the sign of the imaginary part is reversed.

Properties of the Complex Conjugate

For complex numbers $z$ and $w$:

$\overline{\bar{z}} = z$
$\overline{z + w} = \bar{z} + \bar{w}$
$\overline{z - w} = \bar{z} - \bar{w}$
$\overline{z \cdot w} = \bar{z} \cdot \bar{w}$
$\overline{\left(\frac{z}{w}\right)} = \frac{\bar{z}}{\bar{w}}$ ($w \neq 0$)
$z + \bar{z} = 2\mathrm{Re}(z)$
$z - \bar{z} = 2i\mathrm{Im}(z)$
$z \cdot \bar{z} = |z|^2$ (see absolute value below)

Proof (selected)

Let $z = a + bi$ and $w = c + di$.

Proof of (2):

$$\overline{z + w} = \overline{(a+c) + (b+d)i} = (a+c) - (b+d)i = (a-bi) + (c-di) = \bar{z} + \bar{w}$$

Proof of (6):

$$z + \bar{z} = (a + bi) + (a - bi) = 2a = 2\mathrm{Re}(z)$$

Proof of (7):

$$z - \bar{z} = (a + bi) - (a - bi) = 2bi = 2i\mathrm{Im}(z)$$

Absolute Value

Definition: Absolute Value of a Complex Number

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

$$|z| = \sqrt{a^2 + b^2}$$

This corresponds to the distance from the origin in the complex plane.

Relationship Between Absolute Value and Conjugate

For $z = a + bi$:

$$z \cdot \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2 i^2 = a^2 + b^2 = |z|^2$$

Therefore, $|z| = \sqrt{z \cdot \bar{z}}$.

Properties of the Absolute Value

For complex numbers $z$ and $w$:

$|z| \geq 0$, with equality if and only if $z = 0$
$|\bar{z}| = |z|$
$|zw| = |z||w|$
$\displaystyle\left|\frac{z}{w}\right| = \frac{|z|}{|w|}$ ($w \neq 0$)
$|z + w| \leq |z| + |w|$ (triangle inequality)
$||z| - |w|| \leq |z - w|$

Proof (multiplicativity)

Let $z = a + bi$ and $w = c + di$.

$$|zw|^2 = (zw)\overline{(zw)} = zw \cdot \bar{z}\bar{w} = (z\bar{z})(w\bar{w}) = |z|^2|w|^2$$

Since $|zw| \geq 0$ and $|z||w| \geq 0$, it follows that $|zw| = |z||w|$.

Examples: Computing Absolute Values

$|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
$|1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$
$|5| = \sqrt{5^2 + 0^2} = 5$ (agrees with the real absolute value)
$|3i| = \sqrt{0^2 + 3^2} = 3$

Multiplicative Inverse

Inverse of a Complex Number

For $z \neq 0$, the multiplicative inverse $z^{-1} = \displaystyle\frac{1}{z}$ is given by:

$$\frac{1}{z} = \frac{\bar{z}}{|z|^2}$$

Derivation

$$\frac{1}{z} = \frac{1}{z} \cdot \frac{\bar{z}}{\bar{z}} = \frac{\bar{z}}{z\bar{z}} = \frac{\bar{z}}{|z|^2}$$

Example: Computing an Inverse

Find the inverse of $z = 3 + 4i$.

$|z|^2 = 3^2 + 4^2 = 25$, $\bar{z} = 3 - 4i$, so

$$\frac{1}{3 + 4i} = \frac{3 - 4i}{25} = \frac{3}{25} - \frac{4}{25}i$$

Summary

The imaginary unit $i$ satisfies $i^2 = -1$
A complex number $z = a + bi$ consists of a real part $a$ and an imaginary part $b$
Arithmetic uses the rule $i^2 = -1$
The complex conjugate $\bar{z} = a - bi$ reverses the sign of the imaginary part
The absolute value $|z| = \sqrt{a^2 + b^2}$ is the distance from the origin
$z\bar{z} = |z|^2$