Complex Number Calculator — Step-by-Step Solutions
The Complex Number Calculator handles all operations with complex numbers in the form a + bi. Perform arithmetic, find the modulus and conjugate, convert between rectangular and polar forms, and compute powers using De Moivre's theorem.
Enter the real and imaginary parts of your complex numbers to get instant results with detailed step-by-step working.
Your calculations will appear here
How to Use the Complex Number Calculator
- Select the operation mode: Arithmetic, Modulus & Conjugate, Polar Form, Powers & Roots, or Euler Form.
- Enter the real and imaginary parts of your complex number(s).
- For arithmetic, select the operation (add, subtract, multiply, or divide).
- View the result with step-by-step KaTeX-rendered working.
Formula and Theory
A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i^2 = -1.
The modulus of z = a + bi is |z| = sqrt(a^2 + b^2). The conjugate is z-bar = a - bi.
Every complex number can be written in polar form as r(cos(theta) + i*sin(theta)), where r is the modulus and theta is the argument.
De Moivre's theorem: [r*cis(theta)]^n = r^n * cis(n*theta).
Worked Examples
Example 1: Complex Multiplication
Problem: (3 + 2i)(1 - 4i)
Solution: 3 - 12i + 2i - 8i^2 = 3 - 10i + 8
Answer: 11 - 10i
Example 2: Modulus
Problem: Find |3 + 4i|
Solution: sqrt(9 + 16) = sqrt(25)
Answer: 5
Example 3: Polar Form
Problem: Convert 1 + i to polar
Solution: r = sqrt(2), theta = 45 degrees
Answer: sqrt(2) * cis(45 degrees)
Frequently Asked Questions
What is a complex number?
How do you divide complex numbers?
What is Euler's formula?
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