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

How to use

Drag the navy point z anywhere on the plane; the orange point z̄ follows as its mirror image across the real axis. Learn more about using the visualizer
The plane draws $z$ as a solid navy vector, $\bar{z}$ as a dashed orange vector, the dashed modulus circle through both, light right triangles on the real part, and a purple dashed segment from $z$ to $\bar{z}$ . Learn more about the display
The Try these row loads $3 + 2i$ , $-1 + 4i$ , $3i$ , $4$ and $-2 - 3i$ ; Random picks any value inside the plane. Learn more about the presets
Set z manually takes exact values in the Re and Im boxes; anything beyond $\pm 10$ is clamped to $\pm 10$ with a short warning. Learn more about manual input
The Values panel lists $z$ , $\bar{z}$ , $|z|$ , $|z|^2$ and the product $z \cdot \bar{z}$ written as two factors. Learn more about the values panel
The proof box under the values computes $z \cdot \bar{z} = a^2 + b^2$ and $|z|^2$ from the current $z$ and closes with a check mark. Learn more about the proof box
The Key Ideas panel keeps three standing notes and adds a fourth when $z$ sits exactly on a preset value or the origin. Learn more about the Key Ideas panel
Type 0 into both boxes for the origin, or zero out one part to put $z$ on an axis. Learn more about the special cases

Complex Conjugate & Modulus

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

Try these

Set z manually

Re:Im:limit: ±10

Values

z3 + 2i

z̅3 − 2i

|z|3.6

|z|²13

z · z̅(3 + 2i)(3 − 2i)

z · z̅ = 3² + 2² = 9 + 4 = 13

|z|² = (3.6)² = 13

z · z̅ = |z|² ✓

Key Ideas

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

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

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

This starting value is the tool’s generic example — nothing vanishes, so every element is visible. Learn more about 3 + 2i · All presets

How to Use the Visualizer

Understanding the Display

Using the Values Panel

The Proof Box Explained

The Key Ideas Panel

Exploring Presets and Special Cases

The Starting Point: 3 + 2i

Negative Real Part: −1 + 4i

Purely Imaginary: 3i

Purely Real: 4

Third Quadrant: −2 − 3i

Special Cases to Investigate

The Origin: z = 0

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

DemoType, clamp, drag

Step 0 of 4

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:

Drag the navy blue point labeled z anywhere on the complex plane
Watch the orange point z̄ (the conjugate) mirror your movements across the real axis
Observe how the dashed circle representing the modulus updates in real time

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:

Navy blue point and vector: your complex number z
Orange point and dashed vector: the conjugate z̄
Small gray dot at center: the origin (0, 0)

Geometric elements:

Dashed blue circle: shows all points with the same modulus as z — notice both z and z̄ lie on this circle
Light right triangles: connect origin → real part → z (and similarly for z̄), showing how modulus relates to the Pythagorean theorem
Purple dashed vertical line: connects z to z̄, emphasizing reflection symmetry across the real axis

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

DemoValues and proof

Step 0 of 5

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!

The Key Ideas Panel

The Key Ideas panel sits at the bottom of the right column, under the proof box. It is a reading surface: nothing in it can be clicked except its links.

Three standing notes:

Navy dot: the conjugate $\bar{z}$ reflects $z$ across the real axis, keeping the real part and negating the imaginary part.
Blue dot: the modulus $|z|$ is the distance from the origin, and $z$ and $\bar{z}$ share it.
Teal dot: the product $z \cdot \bar{z}$ is always a real number equal to $|z|^2$ , the reason a conjugate is used to clear a complex denominator.

A fourth note for the named values:

When

z

lands exactly on one of the preset values (

3 + 2i

-1 + 4i

3i

4

-2 - 3i

) or on the origin, an orange-dot note appears below the three. It says what is special about that value and ends with two links: one to the section for that value further down this page, and one back to the presets overview or the special cases.

The note appears for a preset button, a typed value or a drag that stops exactly on the value, and it disappears as soon as

z

moves off it. Values that match no preset, including most Random results, show only the three standing notes.

Exploring Presets and Special Cases

DemoFive presets

Step 0 of 5

The preset buttons offer carefully chosen examples that highlight different behaviors — each preset has a dedicated section below with the tool frozen on it:

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.

The Starting Point: 3 + 2i

The tool opens at

z = 3 + 2i

— a first-quadrant number with nothing special about it, which is exactly the point: every element of the visualizer is visible at once from an ordinary example.

z = 3 + 2i, frozen

Solid navy vector to z, dashed orange vector to z̅ = 3 − 2i below the axis, and both points pinned to one dashed modulus circle of radius √13.

Read the picture: the navy vector reaches

z = 3 + 2i

, the dashed orange vector reaches

\bar{z} = 3 - 2i

, and the purple dashed line between them crosses the real axis at

3

— their shared real part. The two vectors are mirror images across the highlighted real axis.

The proof box computes

z \cdot \bar{z} = 3^2 + 2^2 = 13

for this value, and

|z|^2 = (\sqrt{13})^2 = 13

— the identity in its most concrete form.

The other presets each break one thing at a time: −1 + 4i makes the real part negative, 4 and 3i drop the number onto an axis, and −2 − 3i negates both parts.

Negative Real Part: −1 + 4i

The preset

z = -1 + 4i

moves the number into the second quadrant: real part negative, imaginary part positive.

z = −1 + 4i, frozen

z sits upper-left, z̅ = −1 − 4i lower-left: conjugation reflects across the real axis, so a second-quadrant number always has a third-quadrant conjugate.

Conjugation never touches the real part, so

\bar{z} = -1 - 4i

keeps the same horizontal position and simply drops below the axis — from the second quadrant into the third. Compare this with the starting preset, whose conjugate stays on the right half of the plane.

The modulus ignores both signs:

|-1 + 4i| = \sqrt{(-1)^2 + 4^2} = \sqrt{17}

. The proof box still lands on a positive real product,

z \cdot \bar{z} = 1 + 16 = 17

— a negative real part cannot break the identity, because every term gets squared.

Notice the right-triangle guides: the vertical leg now hangs left of the imaginary axis, but its length

4

and the horizontal leg's length

1

still assemble the same Pythagorean picture that defines the modulus.

Purely Imaginary: 3i

The preset

z = 3i

places the number directly on the imaginary axis: real part zero.

z = 3i, frozen

z at 3i and z̅ at −3i sit symmetric about the origin: for a purely imaginary number, the conjugate coincides with the negative.

With no real part, reflection across the real axis is the same as reflection through the origin:

\overline{3i} = -3i = -(3i)

. The purple symmetry line runs straight down the imaginary axis, and the two vectors point in exactly opposite directions.

The product is still real and positive:

z \cdot \bar{z} = (3i)(-3i) = -9i^2 = 9 = |z|^2

. This is

i^2 = -1

doing its signature work — squaring a purely imaginary number gives a negative real number, and the conjugate's second sign flip makes it positive.

The mirror case is the purely real preset, where conjugation does nothing at all; between the two axes live the generic examples like 3 + 2i.

Purely Real: 4

The preset

z = 4

is an ordinary real number viewed inside the complex plane — imaginary part zero.

z = 4, frozen

The orange conjugate point hides exactly underneath the navy z: a real number is its own conjugate, and both vectors lie flat on the real axis.

Here

\bar{z} = z

: reflecting a point that already lies on the mirror leaves it fixed. The visualizer draws both points and both vectors, but they coincide — drag z slightly off the axis and watch them split apart.

The identity collapses to ordinary squaring:

z \cdot \bar{z} = 4 \cdot 4 = 16 = |4|^2

. For real numbers the modulus is just the absolute value, so nothing new happens — which is precisely the sense in which complex conjugation extends real arithmetic without disturbing it.

This is one of the two axis cases; the other is the purely imaginary preset, where conjugation flips the number to its negative instead of fixing it.

Third Quadrant: −2 − 3i

The preset

z = -2 - 3i

has both parts negative, putting the number in the third quadrant — and its conjugate above the axis in the second.

z = −2 − 3i, frozen

The reflection now goes upward: z̅ = −2 + 3i floats above the real axis while z hangs below. Same modulus circle, radius √13.

Conjugation is direction-blind: whichever side of the real axis

z

occupies,

\bar{z}

takes the other. Here that sends the third quadrant to the second — the mirror image of what happens with −1 + 4i.

By a small coincidence of arithmetic, this preset shares its modulus with the starting preset:

|-2 - 3i| = \sqrt{4 + 9} = \sqrt{13}

. Load the two presets in turn and the dashed circle does not move — several different numbers live on that one circle, and the proof box computes the same product

13

for each.

The identity

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

is quadrant-proof:

(-2)^2 + (-3)^2 = 4 + 9

, every sign erased by squaring.

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°.

The Origin: z = 0

Set both inputs to zero — or drag the point home — and the whole construction collapses:

z = 0

is its own conjugate and its own negative at once.

z = 0, frozen

Both points, both vectors and the modulus circle have collapsed into the origin dot: at z = 0 there is no direction and no distance left to draw.

Zero is the degenerate case of everything this tool shows:

\bar{0} = 0

|0| = 0

, and

z \cdot \bar{z} = 0 = |0|^2

— the identity holds with every quantity vanishing. It is the only complex number with modulus zero.

The proof box's checkmark still appears, and it should: the identity is universal, boundary case included. What disappears is the geometry — no vectors, no circle, no triangles — because all of it was built from a nonzero distance.

The special cases section lists the other boundary configurations worth loading: the axes, the unit circle, and the 45° diagonal.

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.