How do you find the conjugate of a complex number?

To find the conjugate, keep the real part unchanged and negate the imaginary part. For example, the conjugate of 3 + 4i is 3 − 4i, and the conjugate of −2 − 5i is −2 + 5i. Purely real numbers are their own conjugates.

What is the modulus of a complex number?

The modulus |z| of a complex number z = a + bi is its distance from the origin, calculated as |z| = √(a² + b²). This comes from the Pythagorean theorem applied to the right triangle formed by the real and imaginary components.

Why does z times its conjugate equal the modulus squared?

When you multiply z = a + bi by z̄ = a − bi, the cross terms cancel: (a + bi)(a − bi) = a² − (bi)² = a² + b². This equals |z|² = (√(a² + b²))² = a² + b². The result is always a non-negative real number.

What is the complex conjugate used for?

Conjugates are essential for dividing complex numbers and rationalizing complex denominators. Multiplying by the conjugate eliminates imaginary parts from denominators. They also appear in finding polynomial roots, signal processing, and quantum mechanics.

Complex Conjugate Visualizer

Complex Conjugate & Modulus

Drag the point to explore z, z̄, and |z|

Try these

Set z manually

Re:Im:limit: ±10

Values

z3 + 2i

z̄3 − 2i

|z|3.6

|z|²13

z · z̄(3 + 2i)(3 − 2i)

z · z̄ = 3² + 2² = 9 + 4 = 13

|z|² = (3.6)² = 13

z · z̄ = |z|² ✓

Key Ideas

The conjugate z̄ reflects z across the real axis — same real part, negated imaginary part.

The modulus |z| is the distance from the origin. Both z and z̄ share the same modulus.

Multiplying z · z̄ always gives a real number equal to |z|². This is why we multiply by the conjugate to rationalize complex denominators.

How to Use the Visualizer

Understanding the Display

Using the Values Panel

The Proof Box Explained

Exploring Presets and Special Cases

Special Cases to Investigate

What is a Complex Conjugate?

The Modulus of a Complex Number

Why z · z̄ = |z|²

Related Concepts and Tools

How to Use the Visualizer

This tool lets you explore the relationship between a complex number, its conjugate, and its modulus through direct manipulation on the Argand plane.

Getting started:

z anywhere on the complex plane
z̄ (the conjugate) mirror your movements across the real axis

Manual input method:

Enter specific values using the Re and Im number inputs on the right panel. Values are constrained to the range ±10. If you enter a value outside this range, the tool automatically clamps it and displays a warning message.

Quick exploration:

Click any preset button (like 3+2i, −1+4i, or 3i) to jump to interesting examples, or use the Random button to generate unexpected combinations.

Understanding the Display

The complex plane visualization shows several elements that update as you move z:

Points and vectors:

Geometric elements:

Axis labels:

The horizontal axis shows the real part (Re), while the vertical axis shows the imaginary part (Im). Tick marks appear at every 2 units for readability.

Using the Values Panel

The Values panel on the right displays all computed quantities in real time as you manipulate z:

Basic values:

z: The complex number you're exploring, shown in standard form a + bi
z̄: The conjugate, with the same real part but negated imaginary part
|z|: The modulus (distance from origin), calculated as $\sqrt{a^2 + b^2}$
|z|²: The modulus squared, equal to $a^2 + b^2$

Product display:

z · z̄: Shows the multiplication setup as (a + bi)(a − bi)

Each row highlights on hover, making it easy to track specific values. The color coding matches the diagram: navy for z, orange for z̄, blue for modulus-related quantities.

The Proof Box Explained

The green proof box demonstrates a fundamental identity: z · z̄ = |z|². This proof updates live with your current values.

First line — computing z · z̄:

The tool calculates

a^2 + b^2

directly from your real and imaginary parts. For example, with z = 3 + 2i:

3^2 + 2^2 = 9 + 4 = 13

.

Second line — computing |z|²:

Starting from the modulus

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

, squaring gives

|z|^2 = a^2 + b^2

.

Conclusion:

Both calculations yield the same result, confirming z · z̄ = |z|². The checkmark indicates the identity holds for your current z value. Try different numbers — the identity always works!

Exploring Presets and Special Cases

The preset buttons offer carefully chosen examples that highlight different behaviors:

3+2i: A standard complex number in the first quadrant — good starting point for understanding the basics.

−1+4i: Second quadrant example where the real part is negative. Notice z̄ appears in the third quadrant.

3i: A purely imaginary number lying on the imaginary axis. Here z and z̄ are symmetric about the origin, and both have the same distance from it.

4: A purely real number. The conjugate equals the original:

\bar{4} = 4

. Both points overlap on the real axis.

−2−3i: Third quadrant example. The conjugate z̄ appears in the second quadrant.

Random: Generates arbitrary values to test that the z · z̄ = |z|² identity holds universally.

Special Cases to Investigate

Certain values reveal important properties of conjugates and modulus:

Purely real numbers (Im = 0):

Set z = 4 + 0i. The conjugate equals the original number. Both points overlap on the real axis. Real numbers are their own conjugates.

Purely imaginary numbers (Re = 0):

Set z = 0 + 3i. The conjugate is z̄ = −3i. Points appear on opposite sides of the origin along the imaginary axis.

Origin (z = 0):

Both z and z̄ collapse to the origin. Modulus is zero, and z · z̄ = 0.

Unit circle:

Try values where

a^2 + b^2 = 1

, such as 0.6 + 0.8i. The modulus equals 1, so z · z̄ = 1.

Equal real and imaginary parts:

Set z = 3 + 3i. The point lies on a 45° diagonal, and z̄ reflects to z = 3 − 3i at −45°.

What is a Complex Conjugate?

The complex conjugate of a number z = a + bi is denoted z̄ (or sometimes z*) and defined as:

\bar{z} = a - bi

Geometrically, the conjugate is the reflection of z across the real axis. The real part stays the same; only the imaginary part changes sign.

Key properties:

$\overline{\bar{z}} = z$ — conjugating twice returns the original
$\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$ — conjugate of a sum is the sum of conjugates
$\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}$ — conjugate of a product is the product of conjugates
$z + \bar{z} = 2a$ — sum gives twice the real part
$z - \bar{z} = 2bi$ — difference gives twice the imaginary part

For deeper theory on complex number fundamentals, see complex numbers introduction.

The Modulus of a Complex Number

The modulus (or absolute value) of z = a + bi measures its distance from the origin:

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

This formula comes directly from the Pythagorean theorem. On the complex plane, z forms a right triangle with legs of length |a| and |b|, and the modulus is the hypotenuse.

Key properties:

$|z| \geq 0$ and $|z| = 0$ only when z = 0
$|\bar{z}| = |z|$ — a number and its conjugate have the same modulus
$|z_1 \cdot z_2| = |z_1| \cdot |z_2|$ — modulus of a product is the product of moduli
$|z_1 / z_2| = |z_1| / |z_2|$ — modulus of a quotient is the quotient of moduli

The dashed circle in the visualizer shows all complex numbers sharing the same modulus as your current z. For more on distance calculations, see complex plane geometry.

Why z · z̄ = |z|²

This identity is one of the most useful in complex number algebra. Here's the algebraic proof:

z \cdot \bar{z} = (a + bi)(a - bi)

Expanding using FOIL:

= a^2 - abi + abi - b^2i^2

= a^2 - b^2(-1)

= a^2 + b^2

Since

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

, we have:

z \cdot \bar{z} = |z|^2

Why this matters:

This identity is essential for rationalizing complex denominators. To divide by a complex number, multiply numerator and denominator by the conjugate:

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

The denominator becomes a real number, eliminating the imaginary part.

Related Concepts and Tools

Explore more complex number topics with these resources:

Related visualizers:

Complex Number Explorer — visualize complex numbers with modulus, quadrants, and coordinate display
Complex Operations Visualizer — see addition, subtraction, and multiplication geometrically
Polar Form Visualizer — explore the relationship between rectangular and polar representations

Theoretical foundations:

Complex Numbers — comprehensive introduction to a + bi form
Complex Plane — understanding the Argand diagram coordinate system
Polar Form — representing complex numbers as r(cos θ + i sin θ)

Calculators:

Complex Number Calculator — perform arithmetic operations
Polar to Rectangular Converter — convert between coordinate systems

The conjugate and modulus concepts connect to advanced topics like complex division, roots of unity, and Euler's formula.