How do you multiply complex numbers?

Multiply using distribution and replace i squared with negative one.

Advanced Math Calculator

Complex Number Calculator

Calculate complex-number operations, magnitudes, conjugates, arguments, polar coordinates and exponential forms with instant results.

Instant calculations Rectangular and polar forms Mobile friendly

Complex Number Calculator

Enter the real and imaginary parts of two complex numbers

Ready

First Complex Number

z₁ = a + bi

Real Part

Imaginary Coefficient

z₁ = a + bi

Second Complex Number

z₂ = c + di

Real Part

Imaginary Coefficient

z₂ = c + di

Arithmetic Operations

Addition

z₁ + z₂

Subtraction

z₁ − z₂

Multiplication

z₁ × z₂

Division

z₁ ÷ z₂

First Number Properties

z₁

Magnitude 0

Conjugate 0

Argument 0 rad

Degrees 0°

Polar Form 0 ∠ 0°

Exponential 0

Second Number Properties

z₂

Magnitude 0

Conjugate 0

Argument 0 rad

Degrees 0°

Polar Form 0 ∠ 0°

Exponential 0

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What Is a Complex Number Calculator?

A complex number calculator performs calculations with numbers containing both real and imaginary parts. A complex number is normally written as a + bi, where a is the real component and b is the imaginary coefficient.

This calculator adds, subtracts, multiplies and divides two complex numbers. It also calculates their magnitudes, conjugates, arguments, polar coordinates and exponential forms.

Four arithmetic operations Calculate addition, subtraction, multiplication and division together.

Polar coordinates Convert rectangular complex numbers into magnitude and angle form.

Advanced properties Find conjugates, radians, degrees and exponential notation.

Simple Process

How the Complex Number Calculator Works

Calculate two complex numbers in three straightforward steps.

Enter the First Number

Add the real part and imaginary coefficient of the first complex number.

Enter the Second Number

Add the real and imaginary values of the second complex number.

View All Results

Calculate arithmetic operations, magnitudes, conjugates and polar forms instantly.

Formula Guide

Complex Number Formulas Explained

These formulas are used for complex-number arithmetic and coordinate conversion.

Addition and Subtraction

Combine the real components separately from the imaginary components.

z₁ ± z₂ = (a ± c) + (b ± d)i

Multiplication

Apply distribution and replace i² with −1.

z₁z₂ = (ac − bd) + (ad + bc)i

Division

Multiply by the conjugate of the denominator to remove the imaginary denominator.

z₁/z₂ = [(ac + bd) + (bc − ad)i]/(c² + d²)

Polar Form

Polar form uses the magnitude and argument instead of real and imaginary coordinates.

z = r(cos θ + i sin θ)

Worked Examples

Complex Number Examples

Compare common complex-number operations and their results.

First Number	Second Number	Sum	Difference	Product
3 + 4i	2 − i	5 + 3i	1 + 5i	10 + 5i
1 + 2i	3 + 4i	4 + 6i	−2 − 2i	−5 + 10i
5 − 3i	2 + i	7 − 2i	3 − 4i	13 − i
−2 + 5i	4 + 3i	2 + 8i	−6 + 2i	−23 + 14i

Helpful Answers

Complex Number Calculator FAQs

Find answers to common questions about imaginary numbers, conjugates and polar notation.

What is a complex number?

A complex number contains a real component and an imaginary component. It is commonly written as a + bi, where i is the square root of −1.

What does the imaginary coefficient mean?

The imaginary coefficient is the number multiplied by i. In 3 + 4i, the imaginary coefficient is 4.

How are complex numbers added?

Add the real parts together and add the imaginary coefficients together. For example, 2 + 3i plus 4 + 5i equals 6 + 8i.

What is the conjugate of a complex number?

The conjugate changes the sign of the imaginary component. The conjugate of 3 + 4i is 3 − 4i.

How is the magnitude calculated?

The magnitude is calculated using √(a² + b²). It represents the distance of the complex number from the origin on the complex plane.

What is the argument of a complex number?

The argument is the angle between the positive real axis and the line connecting the origin to the complex number.

Why is division by 0 + 0i undefined?

Dividing by 0 + 0i creates a zero denominator. Division by zero is undefined in both real and complex arithmetic.

Calculate Complex Numbers Instantly

Enter two complex numbers above to calculate operations, magnitudes, arguments, conjugates and polar forms.

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