complex number calculator online

What is a complex number calculator online?

A complex number calculator online helps you compute expressions that include both real and imaginary parts. Instead of manually expanding and simplifying expressions like (3 + 4i)(1 - 2i), the calculator handles the arithmetic instantly and accurately.

Complex numbers are written in the form a + bi, where:

• a is the real part
• b is the imaginary coefficient
• i is the imaginary unit, with i² = -1

Why use this calculator?

Complex numbers appear in electrical engineering, signal processing, control systems, quantum mechanics, and advanced algebra. A fast calculator saves time and reduces errors when working through homework, design calculations, and exam practice.

• Performs common operations with one click
• Shows answers in standard form (a + bi)
• Provides polar information where useful
• Handles decimal values and negative numbers

Supported operations

1) Addition and subtraction

Add or subtract corresponding parts:
(a + bi) ± (c + di) = (a ± c) + (b ± d)i

2) Multiplication

Multiply with distribution and use i² = -1:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i

3) Division

Division is done using the conjugate:
(a + bi)/(c + di) = [(a + bi)(c - di)] / (c² + d²)

4) Conjugate, magnitude, argument, reciprocal, and polar form

• Conjugate: if z = a + bi, then z̄ = a - bi
• Magnitude: |z| = √(a² + b²)
• Argument: arg(z) = atan2(b, a)
• Reciprocal: 1/z = (a - bi)/(a² + b²)
• Polar form: z = r(cosθ + i sinθ), with r = |z| and θ = arg(z)

How to use this complex number calculator

1. Enter the real and imaginary parts of z₁.
2. Enter z₂ only if your operation needs it (add, subtract, multiply, divide).
3. Choose an operation from the dropdown menu.
4. Click Calculate.
5. Read the output in standard form and, when relevant, in polar form.

Example problems

Example A: Multiplication

Let z₁ = 2 + 3i and z₂ = 4 - i.
(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i² = 11 + 10i

Example B: Magnitude

For z = 3 + 4i:
|z| = √(3² + 4²) = √25 = 5

Example C: Reciprocal

For z = 1 + 2i:
1/z = (1 - 2i)/(1² + 2²) = (1 - 2i)/5 = 0.2 - 0.4i

Common mistakes to avoid

• Forgetting that i² = -1, not +1
• Adding real and imaginary terms incorrectly
• Dividing without multiplying by the conjugate
• Using plain arctangent instead of atan2 for the argument

Final thoughts

If you regularly work with imaginary numbers, this complex number calculator online gives you a quick, dependable workflow. Use it to check your algebra, speed up assignments, and move confidently between rectangular and polar representations.