Absolute Value of a Complex Number and Complex Conjugates

The Geometric Interpretation of Complex NumbersFunctions

Absolute Value of a Complex Number and Complex Conjugates

The absolute value (or modulus) of a complex number z = x + y i is its distance from the origin in the complex plane:

| z | = x 2 + y 2

The complex conjugate of z = a + b i is z ¯ = a − b i , obtained by reflecting z across the real axis.

Quick Reference

Concept Formula Example
Modulus $ z
Conjugate $\bar{z} = a - bi < / t d >< t d > \overline{3 + 4i} = 3 - 4i < / t d >< / t r >< t r >< t d > z \bar{z} < / t d >< t d > z
Product of moduli $ zw
Quotient of moduli $ z/w
Modulus of conjugate < / t d >< t d > z ¯ < / t d >< / t r >< / t b o d y >< / t a b l e >< p >< s p a n i d = " m o d u l u s − d e f i n i t i o n " >< / s p a n >< / p >< h 2 > T h e M o d u l u s ( A b s o l u t e V a l u e ) < / h 2 >< d i v c l a s s = " h i g h l i g h t " >< p >< s t r o n g > D e f i n i t i o n . < / s t r o n g > T h e < s t r o n g > m o d u l u s < / s t r o n g > ( o r < s t r o n g > a b s o l u t e v a l u e < / s t r o n g > ) o f t h e c o m p l e x n u m b e r z = x + yi$ is:

| z | = 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 $z . < / l i >< l i > |z| = 0$ if and only if $z = 0 ( i . e . , x = 0 a n d y = 0$).
  • For a real number $a + 0i$, $|a + 0i| = \sqrt{a^2} = |a|$, consistent with the usual absolute value.

Example 1. Find the modulus of $3 + 4i$, $-2 + 0i , a n d 0 + 5i$.

Solution.

| 3 + 4 i | = 9 + 16 = 25 = 5 | − 2 + 0 i | = 4 + 0 = 2 | 0 + 5 i | = 0 + 25 = 5

Properties of the Modulus

Multiplicative properties of the modulus. For any complex numbers $z a n d w$ (with $w \neq 0$):

| z w | = | z | | w | , | z w | = | z | | w |
Proof of $|zw| = |z||w|$ (click to expand)

Let $z = a + bi a n d w = c + di$. Then:

z w = ( a c − b d ) + ( a d + b c ) i | z w | 2 = ( a c − b d ) 2 + ( a d + b c ) 2

Expanding:

= a 2 c 2 − 2 a b c d + b 2 d 2 + a 2 d 2 + 2 a b c d + b 2 c 2 = a 2 c 2 + b 2 d 2 + a 2 d 2 + b 2 c 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:

z ¯ = a − b i

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.
  • $\overline{z + w} = \bar{z} + \bar{w} < / l i >< l i > \overline{zw} = \bar{z}\,\bar{w} < / l i >< l i > \overline{\bar{z}} = z$ (the conjugate of the conjugate is the original number)
  • $|\bar{z}| = |z|$

Example 2. Let $z = 2 - 3i$. Find $\bar{z}$, $z + \bar{z}$, $z - \bar{z} , a n d z\bar{z}$.

Solution.

$\bar{z} = 2 + 3i < / p >< p > z + \bar{z} = (2 - 3i) + (2 + 3i) = 4$ (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 = 1 z = z ¯ z z ¯ = 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.

1 1 + 2 i = 1 + 2 i ― | 1 + 2 i | 2 = 1 − 2 i 1 + 4 = 1 5 − 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.


Why is $z\bar{z}$ always a non-negative real number? Writing $z = a + bi , w e g e t z\bar{z} = (a+bi)(a-bi) = a^2 + b^2$. This is a sum of squares of real numbers, so it is always non-negative. It equals zero only when $a = b = 0 , i . e . , z = 0$.

What does the conjugate look like geometrically? In the complex plane, the conjugate $\bar{z}$ is the reflection of $z$ across the real (horizontal) axis. If $z = a + bi$ is plotted at $(a, b)$, then $\bar{z} = a - bi$ is plotted at $(a, -b)$.

How does the modulus relate to distance between two complex numbers? The distance between two complex numbers $z_1 a n d z_2$ in the complex plane equals $|z_1 - z_2|$. This generalizes the idea that $|z|$ is the distance from $z$ to the origin.

Why is the property $|zw| = |z||w|$ useful? It means that multiplication by a complex number $w$ scales distances by the factor $|w|$. This is fundamental to understanding multiplication geometrically (rotation and scaling), which is developed further in polar form and Euler's formula in more advanced courses.
The Geometric Interpretation of Complex NumbersFunctions
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