# Complex Number Calculator

Add, subtract, multiply, and divide complex numbers. Shows results in rectangular (a+bi) and polar form with magnitude and angle.

## What this calculates

Perform arithmetic on complex numbers and see results in both rectangular and polar form. For example, (3 + 4i) x (1 - 2i) = 11 - 2i, which has magnitude 11.18 and angle -10.3 degrees.

## Inputs

- **Real part (a₁)** — Real part of the first complex number.
- **Imaginary part (b₁)** — Imaginary part of the first complex number (coefficient of i).
- **Operation** — options: Add (+), Subtract (−), Multiply (×), Divide (÷) — Choose the arithmetic operation.
- **Real part (a₂)** — Real part of the second complex number.
- **Imaginary part (b₂)** — Imaginary part of the second complex number.

## Outputs

- **Result (Rectangular)** — formatted as text — The result in a + bi form.
- **Result Magnitude (|z|)** — The distance from the origin: sqrt(a² + b²).
- **Result Angle (degrees)** — The angle in degrees from the positive real axis.
- **Result Angle (radians)** — The angle in radians from the positive real axis.
- **Result (Polar Form)** — formatted as text — The result in r(cos theta + i sin theta) form.

## Details

**Complex Number Arithmetic**

A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i = sqrt(-1).

**Addition and Subtraction**

Just combine the real and imaginary parts separately:
- (3 + 4i) + (1 - 2i) = 4 + 2i
- (3 + 4i) - (1 - 2i) = 2 + 6i

**Multiplication**

Use the distributive property (FOIL) and the fact that i² = -1:
- (3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i² = 3 - 2i + 8 = 11 - 2i

**Division**

Multiply numerator and denominator by the conjugate of the denominator:
- (3 + 4i) / (1 - 2i) = (3 + 4i)(1 + 2i) / ((1)² + (2)²) = (-5 + 10i) / 5 = -1 + 2i

**Polar Form**

Every complex number can also be written as r(cos theta + i sin theta), where:
- r = sqrt(a² + b²) is the magnitude (distance from origin)
- theta = atan2(b, a) is the angle from the positive real axis

Polar form is especially useful for multiplication and division, since you just multiply/divide magnitudes and add/subtract angles.

## Frequently Asked Questions

**Q: What is a complex number?**

A: A complex number is a number of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Complex numbers extend the real number line into a two-dimensional plane.

**Q: How do you multiply complex numbers?**

A: Use the distributive property (FOIL): (a + bi)(c + di) = ac + adi + bci + bdi². Since i² = -1, this simplifies to (ac - bd) + (ad + bc)i. For example, (2 + 3i)(1 + 4i) = (2 - 12) + (8 + 3)i = -10 + 11i.

**Q: What is polar form of a complex number?**

A: Polar form expresses a complex number using its distance from the origin (magnitude r) and angle from the positive real axis (theta): z = r(cos theta + i sin theta). It is also written as r * e^(i*theta). The conversion formulas are r = sqrt(a² + b²) and theta = atan2(b, a).

**Q: When are complex numbers used in real life?**

A: Complex numbers are essential in electrical engineering (AC circuit analysis uses impedance in complex form), quantum mechanics, signal processing, control theory, and fluid dynamics. They also appear in fractal geometry (the Mandelbrot set is defined using complex number iteration).

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Source: https://vastcalc.com/calculators/math/complex-number
Category: Math
Last updated: 2026-04-08