Modulus and conjugate

Two of the most important operations on complex numbers are the modulus (size) and the conjugate (reflection in the real axis). They satisfy a dozen clean identities, and together they let us divide complex numbers and prove inequalities.

Definitions

Let $z = a + bi$ with $a, b \in \mathbb{R}$.

The modulus (or absolute value) of $z$ is $|z| = \sqrt{a^2 + b^2}.$ It is the distance of $z$ from the origin in the Argand plane. For real $z$, this agrees with the ordinary $|x|$ , so the notation is unambiguous.

The conjugate of $z$ is $\bar{z} = a - bi.$ It is the reflection of $z$ across the real axis.

The master identity

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

This single identity is the engine behind division of complex numbers, and behind the inequality theory.

Properties of conjugate

For all $z, w \in \mathbb{C}$:

1. $\overline{(\bar{z})} = z$ (involution).
2. $\overline{z + w} = \bar{z} + \bar{w}$.
3. $\overline{zw} = \bar{z}\,\bar{w}$.
4. $\overline{z/w} = \bar{z}/\bar{w}$ (when $w \ne 0$).
5. $z + \bar{z} = 2\,\text{Re}(z)$.
6. $z - \bar{z} = 2i\,\text{Im}(z)$.
7. $z = \bar{z} \iff z$ is real. $z = -\bar{z} \iff z$ is purely imaginary.

Each is proved by direct expansion.

Properties of modulus

For all $z, w \in \mathbb{C}$:

1. $|z| \ge 0$, with equality iff $z = 0$.
2. $|z| = |\bar{z}|$.
3. $|zw| = |z| \cdot |w|$ (multiplicative).
4. $\left|\dfrac{z}{w}\right| = \dfrac{|z|}{|w|}$.
5. $|z^n| = |z|^n$.
6. Triangle inequality: $|z + w| \le |z| + |w|$.
7. Reverse triangle: $\big||z| - |w|\big| \le |z - w|$.

The multiplicative property is striking: complex multiplication multiplies sizes. The triangle inequality is a direct generalisation of the inequality $|x + y| \le |x| + |y|$ for real numbers.

Proof of $|zw| = |z||w|$

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

Taking positive square roots, $|zw| = |z||w|$. $\qed$

Computing $1/z$

For $z \ne 0$: $\frac{1}{z} = \frac{\bar{z}}{z \bar{z}} = \frac{\bar{z}}{|z|^2} = \frac{a - bi}{a^2 + b^2}.$

This is the practical method for dividing complex numbers.

Worked examples

Example 1. Find the modulus and conjugate of $z = 3 - 4i$.

$|z| = \sqrt{9 + 16} = 5$. $\bar{z} = 3 + 4i$.

Example 2. Verify $|zw| = |z||w|$ for $z = 1 + i$, $w = 2 - i$.

$zw = 2 - i + 2i - i^2 = 3 + i$. $|zw| = \sqrt{9 + 1} = \sqrt{10}$. $|z| = \sqrt{2}, |w| = \sqrt{5}$, product $\sqrt{10}$. Verified.

Example 3. Find $\dfrac{3 + 2i}{1 - i}$ using the conjugate technique.

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

Example 4. Find all $z$ with $z + \bar{z} = 4$ and $z \bar{z} = 13$.

$z + \bar{z} = 2\text{Re}(z) = 4 \Rightarrow \text{Re}(z) = 2$. $z \bar{z} = |z|^2 = 13$. If $z = 2 + bi$, then $|z|^2 = 4 + b^2 = 13 \Rightarrow b = \pm 3$. So $z = 2 + 3i$ or $z = 2 - 3i$.

Example 5 (harder). Show that if $|z| = 1$, then $z + \dfrac{1}{z}$ is real.

If $|z| = 1$, then $z\bar{z} = 1$, so $\bar{z} = 1/z$. Hence $\dfrac{1}{z} = \bar{z}$, and $z + \dfrac{1}{z} = z + \bar{z} = 2\text{Re}(z) \in \mathbb{R}$. $\qed$

Try it yourself

1. Find $|z|$ and $\bar{z}$ for $z = -1 + 2i$.
2. Show $|3 + 4i| = 5$, $|5 - 12i| = 13$.
3. Find $z$ such that $z \bar{z} = 25$ and $z + \bar{z} = 6$.
4. Verify the triangle inequality for $z = 3 + 4i, w = 5 - 12i$.
5. Show $\overline{(z + w)} = \bar{z} + \bar{w}$ by direct computation.
6. If $z = 2 + i$, compute $\dfrac{1}{z}$.
7. Find all $z$ with $|z - 1| = 1$ and $z \bar{z} = 1$.
8. Prove $|z|^2 + |w|^2 = \tfrac{1}{2}(|z + w|^2 + |z - w|^2)$ (parallelogram law).
9. If $|z + 1| = |z - 1|$, prove $z$ is purely imaginary.
10. Find all $z$ with $z = \bar{z}^2$.
11. Show $|z + w|^2 = |z|^2 + |w|^2 + 2\text{Re}(z \bar{w})$.
12. If $|z_1| = |z_2| = |z_3| = 1$ and $z_1 + z_2 + z_3 = 0$, prove $|z_1 + z_2| = 1$.

Pitfalls / Tricks

• $|z|$ is a real number, never complex. So $|z| \ge 0$ and $|z|^2 \ne z^2$.
• $\overline{z^2} = \bar{z}^2$, but $z^2 \ne |z|^2$ unless $z$ is real.
• The triangle inequality is not always tight: equality holds iff $z, w$ lie on the same ray from the origin.
• Insight. $z\bar{z} = |z|^2$ is the key identity. Whenever you see $z$ in a denominator or under a modulus squared, multiply by its conjugate.