# Complex Number Calculator

> Free complex number calculator with step-by-step solutions. Add, subtract, multiply, divide complex numbers, find modulus, argument, conjugate, polar form.

**URL:** https://calculators.im/complex-number-calculator  
**Category:** math  
**Last updated:** 2026-07-03  

Perform all complex number operations with step-by-step solutions. Add, subtract, multiply, divide complex numbers, find modulus, argument, conjugate, polar form, De Moivre powers, and nth roots. Supports both rectangular (a+bi) and polar (r∠θ) forms.

Our Complex Number Calculator handles every standard complex arithmetic operation in one place. Enter numbers in rectangular form (a + bi) or polar form (r∠θ), then choose from addition, subtraction, multiplication, division, modulus, argument, conjugate, polar conversion, exponentiation using De Moivre's theorem, nth roots, and complex exponential e^z. Every operation shows detailed step-by-step working so you can follow the method, verify your work, or learn the underlying mathematics.

## Formula

```
z = a + bi,  |z| = √(a²+b²),  z^n = rⁿ(cos nθ + i sin nθ)
```

**Variables:**

- **a** — Real part of complex number
- **b** — Imaginary part of complex number
- **|z|** — Modulus (absolute value) — distance from origin
- **θ** — Argument — angle from positive real axis
- **z*** — Complex conjugate — z* = a − bi
- **r∠θ** — Polar form — r is modulus, θ is argument

A complex number z = a + bi has real part a and imaginary part b. The modulus is |z| = √(a²+b²). The argument θ = atan2(b, a). Multiplication is easiest in polar form: |z₁||z₂|∠(θ₁+θ₂). Division: (|z₁|/|z₂|)∠(θ₁−θ₂). Powers by De Moivre: z^n = rⁿ(cos nθ + i·sin nθ).

## Worked Example

### Example: Multiplying two complex numbers

Multiply using FOIL. The key identity is i² = −1, which converts the i² term back to a real number.

## How to Use the Complex Number Calculator

1. **Enter Z₁** — Enter the real and imaginary parts of the first complex number Z₁ = a + bi
2. **Enter Z₂ (if needed)** — For binary operations (add, subtract, multiply, divide), also enter Z₂ = c + di
3. **Choose Operation** — Select from 11 operations: arithmetic, modulus, argument, conjugate, polar form, power, roots, or e^z
4. **View Result & Steps** — See the result in both rectangular (a+bi) and polar (r∠θ) forms with step-by-step working

## Pro Tips

- Remember: i² = −1. This is the key identity for complex multiplication
- Modulus and argument: |z| = √(a²+b²), θ = atan2(b, a)
- Conjugate of (a+bi) is (a−bi). Multiplying z·z* = |z|² (always real)
- For powers and roots, De Moivre's theorem is far easier than expanding
- Polar multiplication: multiply moduli, add arguments
- The n-th roots of z are equally spaced around a circle of radius ⁿ√|z|

## Frequently Asked Questions

### How do you add and subtract complex numbers?

Add or subtract the real and imaginary parts separately: (a+bi) ± (c+di) = (a±c) + (b±d)i. For example: (3+2i) + (1−5i) = (3+1) + (2−5)i = 4 − 3i.

### How do you multiply complex numbers?

Use FOIL: (a+bi)(c+di) = ac + adi + bci + bdi². Since i² = −1: result = (ac−bd) + (ad+bc)i. Example: (2+3i)(1+4i) = (2−12) + (8+3)i = −10 + 11i.

### How do you divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = [(a+bi)(c−di)] / (c²+d²). Example: (1+2i)/(1+i) = [(1+2i)(1−i)] / 2 = (3+i) / 2 = 1.5 + 0.5i.

### What is the modulus of a complex number?

The modulus |z| is the distance from the origin in the complex plane: |a+bi| = √(a²+b²). Example: |3+4i| = √(9+16) = √25 = 5. It represents the magnitude of the complex number.

### What is polar form of a complex number?

Polar form is z = r(cos θ + i sin θ) = r∠θ, where r = |z| = √(a²+b²) is the modulus and θ = atan2(b,a) is the argument (angle). To convert back: a = r cos θ, b = r sin θ.

### How do you raise a complex number to a power?

Use De Moivre's theorem: z^n = r^n (cos nθ + i sin nθ). First convert to polar form (r, θ), then raise r to the power n and multiply the angle by n. Example: (1+i)² has r=√2, θ=45°, so (1+i)² = 2∠90° = 2i.

### How do you find the nth roots of a complex number?

The n nth-roots of z = r∠θ are: z_k = ⁿ√r ∠(θ/n + 2πk/n) for k = 0, 1, ..., n−1. They are equally spaced on a circle of radius ⁿ√r. Example: the 3 cube roots of 8 are 2∠0°, 2∠120°, 2∠240°.

## Deep Dive

Complex numbers extend the real number system by introducing the imaginary unit i, where i-squared equals negative one. Far from being a purely abstract concept, complex numbers are essential tools in electrical engineering for analyzing AC circuits using phasor notation, in signal processing for representing frequency components via the Fourier transform, in control systems for stability analysis, and in quantum mechanics for describing wave functions. A complex number z = a + bi has a real part a and imaginary part b, and can equivalently be expressed in polar form as r times e-to-the-i-theta, where r is the magnitude and theta is the argument. Converting between rectangular and polar forms, computing products and quotients, finding powers and roots — these operations follow specific rules that are straightforward but tedious to perform by hand, especially when precision matters. This complex number calculator handles all standard operations including addition, subtraction, multiplication, division, powers, nth roots, conjugates, and magnitude calculations, showing detailed step-by-step solutions for each operation.

### What is a Complex Number Calculator?

A **complex number calculator** performs arithmetic and geometric operations on numbers of the form a + bi, where i is the imaginary unit (i² = −1).

Our calculator supports all standard operations:

- addition
- subtraction
- multiplication
- division
- modulus
- argument
- conjugate
- polar form conversion
- De Moivre powers
- nth roots
- complex exponential

### Rectangular vs Polar Form

Complex numbers can be expressed in **rectangular form** (a + bi) or **polar form** (r∠θ).

The modulus r = √(a²+b²) is the distance from the origin, and the argument θ = atan2(b,a) is the angle.

Polar form makes multiplication, division, and powers much simpler: multiply moduli and add arguments.

### What Is a Complex Number and Why Does i Exist?

A **complex number** is a number of the form a + bi, where a and b are real numbers and i is the **imaginary unit** defined by i² = −1. The real part is a and the imaginary part is b.

Complex numbers arose because equations like x² + 1 = 0 have no real solution, so mathematicians extended the number system to include √−1.

According to Wolfram MathWorld, the complex numbers form an **algebraically closed field**, meaning every non-constant polynomial has a root among them.

Geometrically, a + bi is a point (a, b) in the complex plane, with the horizontal axis for real values and the vertical axis for imaginary values.

### How Do You Add, Subtract, Multiply, and Divide Complex Numbers?

- **To add or subtract** complex numbers, combine the real and imaginary parts separately: (a+bi) ± (c+di) = (a±c) + (b±d)i.
- **Multiplication** uses the FOIL method and the identity i² = −1: (a+bi)(c+di) = (ac−bd) + (ad+bc)i. For example, (3+4i)(1−2i) = 11 − 2i.
- **Division** multiplies numerator and denominator by the conjugate of the denominator, giving (a+bi)/(c+di) = [(ac+bd) + (bc−ad)i]/(c²+d²).

Khan Academy notes that multiplying by the conjugate clears the imaginary part from the denominator, since (c+di)(c−di) = c²+d² is always real. These four operations preserve the a + bi structure.

### What Is the Modulus and Argument of a Complex Number?

The **modulus** of a complex number z = a + bi is its distance from the origin in the complex plane, given by |z| = √(a²+b²). For instance, |3+4i| = √(9+16) = √25 = 5.

The **argument** is the angle θ from the positive real axis, computed as θ = atan2(b, a), typically reported between −π and π.

As NIST's Digital Library of Mathematical Functions describes, the modulus and argument together give the polar coordinates of z. The modulus is always non-negative, and multiplying a complex number by its conjugate yields the modulus squared: z·z* = |z|², which is always a real number.

### How to Convert Between Rectangular and Polar Form

To convert a complex number from **rectangular form** a + bi to **polar form** r∠θ, compute the modulus r = √(a²+b²) and the argument θ = atan2(b, a).

To go back from polar to rectangular, use a = r·cos θ and b = r·sin θ.

For example, 1 + i has r = √2 ≈ 1.414 and θ = 45°, so its polar form is √2 ∠45°.

Encyclopaedia Britannica explains that polar representation makes multiplication and division intuitive, since you multiply or divide the moduli and add or subtract the arguments. Rectangular form is best for addition and subtraction, while polar form simplifies products, quotients, powers, and roots.

### How Do You Raise a Complex Number to a Power Using De Moivre's Theorem?

**De Moivre's theorem** states that for z = r(cos θ + i sin θ), the nth power is z^n = rⁿ(cos nθ + i sin nθ). Raise the modulus to the power n and multiply the argument by n.

For example, (1+i) has r = √2 and θ = 45°, so (1+i)² = (√2)²(cos 90° + i sin 90°) = 2(0 + i) = 2i, which matches direct expansion.

Wolfram MathWorld presents De Moivre's formula as a direct consequence of Euler's formula e^(iθ) = cos θ + i sin θ. This method avoids the tedious repeated multiplication needed to expand high powers in rectangular form.

### How to Find the Nth Roots of a Complex Number

Every non-zero complex number has exactly n distinct **nth roots**. For z = r∠θ, the roots are z_k = ⁿ√r ∠(θ/n + 360°k/n) for k = 0, 1, …, n−1. They lie equally spaced on a circle of radius ⁿ√r in the complex plane.

For example, the three cube roots of 8 are 2∠0°, 2∠120°, and 2∠240°.

The special case of the **nth roots of unity** — the solutions of zⁿ = 1 — are catalogued in the OEIS and form the vertices of a regular n-gon on the unit circle.

As NIST DLMF notes, this equal spacing follows directly from adding 2π/n to the angle for each successive root.

### What Is the Complex Conjugate and How Is It Used?

The **complex conjugate** of z = a + bi is z* = a − bi, obtained by reversing the sign of the imaginary part. Geometrically it reflects the number across the real axis.

The conjugate has two key properties:

- z + z* = 2a is always real
- z·z* = a² + b² = |z|² is always a non-negative real number

Khan Academy highlights that conjugates are the standard tool for dividing complex numbers, because multiplying the denominator by its conjugate removes the imaginary part.

Conjugates also appear in physics and signal processing, where quantities like power and probability depend on multiplying a complex value by its conjugate to obtain a real magnitude.

### Where Are Complex Numbers Used in the Real World?

**Complex numbers** are indispensable across engineering and science:

- In **electrical engineering**, alternating-current circuits are analyzed with phasors, where impedance is a complex quantity combining resistance and reactance.
- In **signal processing**, the Fourier transform represents signals as sums of complex exponentials e^(iωt), separating them into frequency components.
- **Control systems** use the complex plane to assess stability by locating poles and zeros.
- **Quantum mechanics** is fundamentally built on complex-valued wave functions.

Encyclopaedia Britannica notes that Euler's identity e^(iπ) + 1 = 0 elegantly links the constants e, i, π, 1, and 0. Fluid dynamics and fractal geometry, such as the Mandelbrot set, also rely on complex arithmetic.

### Common Mistakes When Working With Complex Numbers

- The most frequent error is forgetting that i² = −1, which changes the sign of the product term during multiplication; writing i² = 1 gives a wrong result.
- Another mistake is applying √(ab) = √a·√b to negative numbers — this rule fails, since √−1·√−1 = −1, not √1 = 1.
- When finding the argument, many students use plain arctan(b/a) and land in the wrong quadrant; use atan2(b, a) instead.
- People also drop roots when solving zⁿ = w, forgetting there are exactly n of them.
- Finally, mixing degrees and radians in polar calculations produces incorrect angles.

Always confirm which unit your formula expects before computing sines and cosines.

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## Authoritative Sources

- [Wolfram MathWorld - Complex Number](https://mathworld.wolfram.com/ComplexNumber.html) — Rigorous reference on complex numbers, De Moivre's theorem, and the complex plane
- [NIST Digital Library of Mathematical Functions - Complex Arithmetic](https://dlmf.nist.gov/1.9) — Authoritative definitions of modulus, argument, roots, and complex analysis
- [Khan Academy - Complex Numbers](https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex) — Free lessons on complex arithmetic, conjugates, and polar form
- [Encyclopaedia Britannica - Complex Number](https://www.britannica.com/science/complex-number) — Overview of complex numbers, Euler's identity, and real-world applications
- [OEIS - Roots of Unity and Integer Sequences](https://oeis.org/) — On-Line Encyclopedia of Integer Sequences, including sequences related to roots of unity

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