How do you divide complex numbers?

Multiply both numerator and denominator by the conjugate of the denominator. This makes the denominator real (equal to |z₂|²), reducing the problem to dividing each component by a real number. Alternatively, in polar form, divide the moduli and subtract the arguments.

What does complex division look like geometrically?

Dividing z₁ by z₂ shrinks or stretches z₁ by a factor of 1/|z₂| and rotates it by −θ₂. The quotient modulus is |z₁|/|z₂| and the quotient angle is θ₁ − θ₂. It is the inverse of multiplication's scale-and-rotate operation.

What is 1 divided by i?

1/i = −i. Since i has modulus 1 and angle 90°, dividing by i rotates 90° clockwise without changing the length. Algebraically, multiply top and bottom by −i: 1/i × (−i)/(−i) = −i/1 = −i.

Why do you multiply by the conjugate when dividing complex numbers?

Multiplying the denominator by its conjugate gives z₂ · z̄₂ = |z₂|², which is a real number. This eliminates the imaginary part from the denominator, turning complex division into simple real division of the numerator components.

Can you divide by zero in complex numbers?

No. Division by zero is undefined in complex arithmetic, just as in real arithmetic. The denominator z₂ must be nonzero, meaning at least one of its real or imaginary components must be nonzero.

Complex Numbers Division

Complex Division

Divide magnitudes, subtract angles

Try these

z₁ — Numerator

Re:Im:

4 + 2i|z₁| = 4.47, θ₁ = 26.57°

z₂ — Denominator

Re:Im:

1 − i|z₂| = 1.41, θ₂ = -45°

z₁ / z₂ — Quotient

1 + 3i

|z₁/z₂| = 3.16, θ = 71.57°

Algebraic Method (Multiply by Conjugate)

4 + 2i1 − i × 1 + i1 + i

Multiply top and bottom by the conjugate of z₂:

Denominator: (1 − i)(1 + i) = |z₂|² = 1² + 1² = 2

Numerator: (4 + 2i)(1 + i) = 2 + 6i

Result: 2 + 6i2 = 1 + 3i

Geometric Method (Polar)

|z₁| ÷ |z₂| = 4.47 ÷ 1.41 = 3.16

θ₁ − θ₂ = 26.57° − -45° = 71.57°

Divide the lengths, subtract the angles. The quotient vector is 3.16 units long at 71.57° from the real axis.

Key Ideas

To divide complex numbers algebraically, multiply the numerator and denominator by the conjugate of the denominator. This makes the denominator real (since z·z̄ = |z|²), turning the problem into simple real division.

Geometrically, dividing z₁ by z₂ shrinks z₁ by a factor of |z₂| and rotates it by −θ₂. The result: |z₁/z₂| = |z₁|/|z₂| and arg(z₁/z₂) = θ₁ − θ₂.

Dividing by i rotates by −90°. Try "1/i" — i has angle 90°, so dividing subtracts 90°, giving angle −90° = −i. This confirms that 1/i = −i.

Dividing conjugates always gives a result on the unit circle. Try "(3+4i)/(3−4i)" — since |z| = |z̄|, the magnitudes cancel to 1, and only the angle doubles.

Getting Started — Set Numerator and Denominator

The Three Angle Arcs — Subtraction in Action

Division of Pure Real Numbers

Division of Pure Imaginary Numbers

Dividing by i — The −90° Rotation

Conjugate Pair Division

The Divide-by-Zero State

Off-Screen Quotients and the Zoom Inset

The Conjugate Multiplication Method

The Polar (Geometric) Method

Related Concepts and Tools

Getting Started — Set Numerator and Denominator

Two draggable points represent

z_1

(navy, the numerator) and

z_2

(orange, the denominator). Drag either point on the complex plane and the green quotient vector

z_1 / z_2

updates instantly, along with both step-by-step solution panels.

You can also type values directly into the input fields — each has separate real and imaginary inputs accepting

-10

10

. Six presets are available below the plane:

(4+2i)/(1-i)

6/3

4i/2i

1/i

(3+4i)/(3-4i)

, and

(-2+6i)/(1+2i)

. Click Random to generate a random pair (the denominator is guaranteed non-zero).

Each input panel displays the complex number in rectangular form alongside its modulus and argument, so you can follow both the algebraic and geometric methods simultaneously.

The Three Angle Arcs — Subtraction in Action

Three concentric arcs near the origin show the arguments of the numerator, denominator, and quotient. The inner navy arc is

\theta_1

, the middle orange arc is

\theta_2

, and the outer green arc is

\theta_1 - \theta_2

.

Division subtracts angles — the opposite of multiplication, which adds them. Click

(4+2i)/(1-i)

to see:

\theta_1 \approx 26.6°

\theta_2 \approx -45°

, and the quotient angle is

26.6° - (-45°) = 71.6°

. Subtracting a negative angle effectively adds, so dividing by a number with a negative argument rotates the result upward.

Click

1/i

for the most striking case. The numerator

z_1 = 1

has angle

0°

and the denominator

z_2 = i

has angle

90°

. The quotient angle is

0° - 90° = -90°

, placing the result at

-i

on the negative imaginary axis. Dividing by

i

rotates

90°

clockwise — the reverse of multiplying by

i

Division of Pure Real Numbers

Click

6/3

to see the simplest case: both numbers are real. The navy vector for

z_1 = 6

and the orange vector for

z_2 = 3

both lie on the positive real axis with angles

0°

. The quotient is

6/3 = 2

, also real and on the positive axis.

The green quotient vector is shorter than

z_1

— the modulus is divided:

|6|/|3| = 2

. The angle is

0° - 0° = 0°

, confirming no rotation. This is ordinary real division rendered on the complex plane.

Now change

z_2

-3

. The denominator angle becomes

180°

, so the quotient angle is

0° - 180° = -180°

, which wraps to

180°

. The result is

-2

— dividing by a negative real number flips the direction. The green vector points left along the negative real axis.

These real-only cases produce the cleanest SVG snapshots: all vectors collinear along the real axis, with the quotient either shorter or longer than the numerator depending on whether

|z_2| > 1

|z_2| < 1

Division of Pure Imaginary Numbers

Click

4i / 2i

to divide two purely imaginary numbers. Both vectors point along the imaginary axis —

z_1 = 4i

90°

and

z_2 = 2i

90°

. The quotient angle is

90° - 90° = 0°

, so the result lands on the positive real axis.

The result is

4i / 2i = 2

— a real number. The imaginary units cancel, leaving a real quotient. The modulus is

4/2 = 2

. In the visualizer, both input vectors are vertical but the green quotient vector is horizontal.

This is a general rule: dividing two pure imaginary numbers always yields a real result because both arguments are

\pm 90°

and their difference is

0°

180°

. Try

4i / (-2i)

: the angles are

90°

and

-90°

, giving a quotient angle of

180°

, so the result is

-2

— a negative real number.

Dividing by i — The −90° Rotation

Click

1/i

to see the reciprocal of the imaginary unit. Since

i

has modulus

1

and angle

90°

, dividing by

i

scales by

1/1 = 1

(no change in length) and rotates by

-90°

(clockwise quarter-turn).

The result is

1/i = -i

, which sits at angle

-90°

on the negative imaginary axis. The green vector points straight down. This is confirmed algebraically by multiplying top and bottom by

-i

\frac{1}{i} \cdot \frac{-i}{-i} = \frac{-i}{1} = -i

.

Dividing by

i

is the inverse of multiplying by

i

. Multiplication by

i

rotates

+90°

(counterclockwise); division by

i

rotates

-90°

(clockwise). Together they cancel —

(z \cdot i) / i = z

.

Try changing

z_1

to other values while keeping

z_2 = i

. Every quotient is the numerator rotated

90°

clockwise. Setting

z_1 = 1 + i

gives

(1+i)/i = 1 - i

, confirmed by the green vector appearing in Quadrant IV.

Conjugate Pair Division

Click

(3+4i)/(3-4i)

to divide a complex number by its own conjugate. Conjugates share the same modulus (

|3+4i| = |3-4i| = 5

) but have opposite angles. The quotient modulus is

5/5 = 1

, so the result lies exactly on the unit circle.

The quotient angle is

\theta_1 - \theta_2

. Since

\theta_2 = -\theta_1

, the difference is

2\theta_1

. The result is a pure rotation with unit modulus — no scaling at all.

In the visualizer, the green quotient point sits at distance

1

from the origin. The angle is double the numerator's angle. This is a distinctive SVG state: one point above the real axis, one below (mirror images), and the quotient on the unit circle.

This property is used in signal processing to extract phase information. Dividing a complex number by its conjugate strips away the magnitude and doubles the phase angle.

The Divide-by-Zero State

Drag

z_2

to the origin (set both components to

0

). The quotient becomes undefined — you cannot divide by zero in complex arithmetic, just as in real arithmetic.

The visualizer handles this gracefully: the green quotient vector disappears, both step-by-step panels hide, and a red warning message appears in the result box: "Cannot divide by zero — move z₂ away from the origin."

The Random button deliberately prevents this by ensuring

z_2

has at least one component with

|\text{value}| \geq 0.5

. But you can still reach the zero state manually by typing

0

into both inputs.

This is the only configuration where the tool cannot display a result. Every other combination of

z_1

and

z_2

(including

z_1 = 0

, which gives quotient

0

) produces a valid output.

Off-Screen Quotients and the Zoom Inset

When the denominator is much smaller than the numerator, the quotient modulus can exceed the

\pm 10

grid. In this case the green vector becomes a dashed ray pointing toward the edge with an arrow and the label "

z_1/z_2 \to

". The exact numerical result still appears in the result panel.

Try

z_1 = 10

and

z_2 = 0.5

: the quotient is

20

, well beyond the visible range. The dashed ray points right along the real axis.

Conversely, when the denominator is much larger than the numerator, the quotient is tiny. If the quotient modulus falls below

1

, a zoom inset appears in the upper-right corner of the plane. This magnified window shows a small grid around the origin with the quotient point plotted at readable scale, complete with its own labeled coordinates.

Try

z_1 = 0.5

and

z_2 = 5 + 5i

: the quotient modulus is approximately

0.07

, invisible at normal scale but clearly shown in the inset.

The Conjugate Multiplication Method

The standard algebraic method for dividing complex numbers is to multiply both numerator and denominator by the conjugate of the denominator:

\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{z_2 \cdot \overline{z_2}} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}

The denominator becomes real because

z_2 \cdot \overline{z_2} = |z_2|^2 = a^2 + b^2

for

z_2 = a + bi

. Once the denominator is real, the division reduces to dividing each component of the numerator by a single real number.

The Algebraic Method panel shows this process with fraction notation. It displays the original fraction, the conjugate multiplier, the expanded denominator

|z_2|^2

, the expanded numerator, and the final simplified quotient.

This technique is sometimes called rationalizing the denominator — it eliminates the imaginary part from the denominator, converting the problem from complex division to real division.

The Polar (Geometric) Method

In polar form, division is the inverse of multiplication:

\frac{z_1}{z_2} = \frac{r_1}{r_2} \cdot e^{i(\theta_1 - \theta_2)}

Moduli divide and arguments subtract. The Geometric Method panel shows these two calculations side by side:

|z_1| \div |z_2|

for the modulus and

\theta_1 - \theta_2

for the angle, followed by a summary sentence.

This is dramatically simpler than the conjugate multiplication method. Where the algebraic approach requires expanding a product, computing

|z_2|^2

, and dividing two components, the polar approach is just one division and one subtraction.

The trade-off is that you need the moduli and arguments first. If

z_1

and

z_2

are given in rectangular form, you must convert to polar — which involves square roots and arctangent. If they are already in polar or exponential form, division is trivial.

The three angle arcs on the plane make the subtraction visible: the green arc is always the angular difference between the navy and orange arcs.

Related Concepts and Tools

Complex division is the inverse of complex multiplication. Explore these related pages for the complete picture of complex arithmetic.

Complex Multiplication Visualizer — the counterpart to this tool. Multiplication adds angles and multiplies moduli; division subtracts angles and divides moduli.

Complex Addition & Subtraction Visualizer — addition and subtraction work component-wise with no cross-terms, unlike multiplication and division which involve the

i^2 = -1

rule.

Polar-Rectangular Converter — convert between

a + bi

and

re^{i\theta}

forms. The polar form makes division simple: just divide moduli and subtract arguments.

Euler's Formula Explorer — the identity

e^{i\theta} = \cos\theta + i\sin\theta

provides the theoretical basis for the polar division rule.

Powers of i Calculator — since

1/i = -i

and

1/i^2 = -1

, reciprocals of powers of

i

follow the same mod 4 cycle.

Complex Numbers — foundational theory covering algebraic operations, conjugates, and the modulus-argument representation.