Understanding Modulus and Conjugate of a Complex Number

The modulus and conjugate of a complex number are fundamental concepts describing the magnitude and a key symmetry transformation of any complex number $z = a + ib$, where $a, b \in \mathbb{R}$ and $i$ is the imaginary unit with $i^2 = -1$.

Mathematical Definitions of Modulus and Conjugate for Complex Numbers

The modulus of a complex number $z = a + ib$ is the non-negative real number defined as $|z| = \sqrt{a^2 + b^2}$.

The conjugate of a complex number $z = a + ib$ is the complex number $\overline{z} = a - ib$.

For $z = a + ib$, $a$ is called the real part and $b$ is called the imaginary part of $z$. Both $|z|$ and $\overline{z}$ are functions that assign, respectively, a non-negative real number and another complex number to $z$.

Algebraic Derivation of the Modulus Formula

Consider $z = a + ib$, with $a, b \in \mathbb{R}$. The modulus $|z|$ is derived from the product of $z$ and its conjugate $\overline{z}$.

The conjugate is given by $\overline{z} = a - ib$.

Form the product $z \cdot \overline{z}$:

$z \cdot \overline{z} = (a + ib)(a - ib)$

Expand the product:

$= a(a) + a(-ib) + ib(a) + ib(-ib)$

$= a^2 - a ib + a ib - i^2 b^2$

$= a^2 - a ib + a ib + b^2$ (since $i^2 = -1$ so $-i^2 = +1$)

$= a^2 + b^2$

By definition of modulus, $|z| = \sqrt{z \cdot \overline{z}} = \sqrt{a^2 + b^2}$.

Key Properties of the Modulus and Conjugate of Complex Numbers

For any $z = a + ib$, $|z| \geq 0$, and $|z| = 0$ if and only if $a = 0$ and $b = 0$, i.e., $z = 0$.

For any complex number $z$, $|\overline{z}| = |z|$.

Let $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$. Then $|z_1 z_2| = |z_1| \, |z_2|$, and $|z_1 / z_2| = |z_1| / |z_2|$ (provided $z_2 \neq 0$).

The modulus satisfies the triangle inequality: $|z_1 + z_2| \leq |z_1| + |z_2|$.

For addition, $z + \overline{z} = 2a$ is a real number; for multiplication, $z \cdot \overline{z} = a^2 + b^2$ is also a real number.

Complex Numbers Overview

Algebraic Properties of the Conjugate Operation

For any complex number $z = a + ib$ and any other $w = c + id$, the following hold:

$\overline{\overline{z}} = z$

$\overline{z+w} = \overline{z} + \overline{w}$

$\overline{z \cdot w} = \overline{z} \cdot \overline{w}$

$\overline{z / w} = \overline{z} / \overline{w}$ (when $w \neq 0$)

For any real $k$, $\overline{k z} = k \, \overline{z}$

Practice Paper on Modulus and Conjugate

Proof: Equality of Modulus for a Number and its Conjugate

Let $z = a + ib$, then $\overline{z} = a - ib$.

Compute $|\overline{z}|$:

$|\overline{z}| = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2}$

So $|\overline{z}| = |z|$.

Worked Example: Find the Modulus and Conjugate of $z = 3 - 4i$

Example

Given $z = 3 - 4i$

Here $a = 3$ and $b = -4$

Substitute in modulus formula:

$|z| = \sqrt{3^2 + (-4)^2}$

$= \sqrt{9 + 16}$

$= \sqrt{25}$

$= 5$

The conjugate is $\overline{z} = 3 + 4i$.

Complex Numbers and Quadratic Equations

Relation between Modulus and Conjugate of Complex Number

The modulus of any complex number $z$ is equal to the positive square root of the product of $z$ and its conjugate: $|z| = \sqrt{z \cdot \overline{z}}$.

For $z = a + ib$, $z \cdot \overline{z} = (a + ib)(a - ib) = a^2 + b^2$ (see expanded derivation above).

Therefore, $|z| = \sqrt{a^2 + b^2} = \sqrt{z \cdot \overline{z}}$.

Geometry of Complex Numbers

Special Cases and Additional Results

If $a = 0$ and $b \neq 0$, then $z = ib$ and $|z| = |b|$; its conjugate is $-ib$.

If $b = 0$ and $a \neq 0$, then $z = a$ is real; the conjugate is $a$ itself; $|z| = |a|$.

If $z$ is real, then $z = \overline{z}$ and $|z|$ becomes the absolute value of $z$.

Detailed Discussion on Modulus

Exam Notes and Error Prevention for Modulus and Conjugate Questions

Common Error: Do not change the sign of the real part when calculating the conjugate; only the sign of the imaginary part must be changed.

Exam Tip: When proving identities involving modulus and conjugate, always explicitly show expansion of products and simplification, especially justifying $i^2 = -1$.