Absolute Value of a Complex Number and Complex Conjugates
The absolute value (or modulus) of a complex number $z = x + yi$ is its distance from the origin in the complex plane:
$|z| = \sqrt{x^2 + y^2}$The complex conjugate of $z = a + bi$ is $\bar{z} = a - bi$, obtained by reflecting $z$ across the real axis.
Quick Reference
| Concept | Formula | Example |
|---|---|---|
| Modulus | $ | z |
| Conjugate | $\bar{z} = a - bi$ | $\overline{3 + 4i} = 3 - 4i$ |
| $z \bar{z}$ | $ | z |
| Product of moduli | $ | zw |
| Quotient of moduli | $ | z/w |
| Modulus of conjugate | $ | \bar{z} |
The Modulus (Absolute Value)
Definition. The modulus (or absolute value) of the complex number $z = x + yi$ is:
$|z| = \sqrt{x^2 + y^2}$Geometrically, $|z|$ is the distance from the point $z$ to the origin in the complex plane.
Key facts about the modulus:
- $|z| \geq 0$ for all complex numbers $z$.
- $|z| = 0$ if and only if $z = 0$ (i.e., $x = 0$ and $y = 0$).
- For a real number $a + 0i$, $|a + 0i| = \sqrt{a^2} = |a|$, consistent with the usual absolute value.
Example 1. Find the modulus of $3 + 4i$, $-2 + 0i$, and $0 + 5i$.
Solution.
$ |3 + 4i| = \sqrt{9 + 16} = \sqrt{25} = 5 $$ |-2 + 0i| = \sqrt{4 + 0} = 2 $$ |0 + 5i| = \sqrt{0 + 25} = 5 $
Properties of the Modulus
Multiplicative properties of the modulus. For any complex numbers $z$ and $w$ (with $w \neq 0$):
$|zw| = |z|\,|w|, \qquad \left|\frac{z}{w}\right| = \frac{|z|}{|w|}$Proof of $|zw| = |z||w|$ (click to expand)
Let $z = a + bi$ and $w = c + di$. Then:
$zw = (ac - bd) + (ad + bc)i$$|zw|^2 = (ac-bd)^2 + (ad+bc)^2$Expanding:
$= a^2c^2 - 2abcd + b^2d^2 + a^2d^2 + 2abcd + b^2c^2 = a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2$$= (a^2 + b^2)(c^2 + d^2) = |z|^2 |w|^2$Taking square roots (both sides are non-negative): $|zw| = |z||w|$.
The Complex Conjugate
Definition. The complex conjugate of $z = a + bi$ is:
$\bar{z} = a - bi$Geometrically, $\bar{z}$ is the reflection of $z$ across the real axis in the complex plane.
Key properties of conjugates:
- $z + \bar{z} = 2a = 2\operatorname{Re}(z)$, which is a real number.
- $z - \bar{z} = 2bi = 2i\operatorname{Im}(z)$, which is purely imaginary.
- $z\bar{z} = a^2 + b^2 = |z|^2$, which is a non-negative real number.
- $\overline{z + w} = \bar{z} + \bar{w}$
- $\overline{zw} = \bar{z}\,\bar{w}$
- $\overline{\bar{z}} = z$ (the conjugate of the conjugate is the original number)
- $|\bar{z}| = |z|$
Example 2. Let $z = 2 - 3i$. Find $\bar{z}$, $z + \bar{z}$, $z - \bar{z}$, and $z\bar{z}$.
Solution.
$\bar{z} = 2 + 3i$
$z + \bar{z} = (2 - 3i) + (2 + 3i) = 4$ (real)
$z - \bar{z} = (2 - 3i) - (2 + 3i) = -6i$ (purely imaginary)
$z\bar{z} = (2 - 3i)(2 + 3i) = 4 + 9 = 13 = |z|^2$
Using the Conjugate to Find the Reciprocal
Since $z \bar{z} = |z|^2$ is a positive real number (for $z \neq 0$), the reciprocal of $z$ can be written as:
$ z^{-1} = \frac{1}{z} = \frac{\bar{z}}{z\bar{z}} = \frac{\bar{z}}{|z|^2} $This confirms the formula from Section 4.3 and shows clearly that the conjugate is the key tool for inverting a complex number.
Example 3. Find the reciprocal of $1 + 2i$.
Solution.
$ \frac{1}{1 + 2i} = \frac{\overline{1+2i}}{|1+2i|^2} = \frac{1 - 2i}{1 + 4} = \frac{1}{5} - \frac{2}{5}\,i $Frequently Asked Questions
What is the difference between the absolute value and the modulus?
They are the same thing for complex numbers. The term absolute value extends from the real number concept $|a| = \sqrt{a^2}$, while modulus is the standard term for complex numbers. For a real number $a$, the modulus $|a + 0i| = \sqrt{a^2} = |a|$ matches the usual absolute value.
