Complex Numbers Calculator

Perform arithmetic with complex numbers: addition, subtraction, multiplication, division, and modulus. Convert between rectangular (a+bi) and polar (r∠θ) forms.

∑ Algebra📐 z = a + bi where i = √(-1)🔢 Math
Operation
Real part a
Imaginary part b
Real part c
Imaginary part d
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Complex Numbers Calculator—z = a + bi where i = √(-1)complex number

Step-by-Step Examples

Example 1
Multiplication

(3+4i)(1−2i).

  • Real: 3×1−4×(−2)=3+8=11
  • Imag: 3×(−2)+4×1=−6+4=−2
  • Result: 11−2i
✓ 11 − 2i
Example 2
Modulus

z = 3+4i.

  • r = √(3²+4²) = √25 = 5
  • θ = arctan(4/3) = 53.13°
  • Polar: 5∠53.13°
✓ r=5, θ=53.13°
Example 3
Division

(3+4i)/(1+2i).

  • Multiply by conjugate (1−2i)
  • Numerator: (3+4i)(1−2i)=11−2i
  • Denominator: 1+4=5
  • Result: 11/5 − 2i/5 = 2.2−0.4i
✓ 2.2 − 0.4i

Real-World Applications

🧪
Physics
AC circuits analyzed using complex impedance; wave functions in quantum mechanics.
💻
Signal Processing
Fourier transforms use complex exponentials; complex arithmetic is fundamental.
📐
Algebra
Allows solving all polynomial equations, not just those with real roots.
📊
Control Theory
Transfer functions in Laplace domain use complex numbers.

Common Mistakes to Avoid

⚠️
i² = −1, not 1

The whole point of complex numbers is i = √(−1), so i² = −1. Never forget the sign change.

⚠️
Division requires conjugate

Multiply numerator and denominator by the conjugate (a−bi) of the denominator to get a real denominator.

⚠️
Modulus vs absolute value

For real numbers |a| = absolute value. For complex |a+bi| = √(a²+b²). This is the distance from origin in the complex plane.

Frequently Asked Questions

What is an imaginary number? ▾
i = √(−1). Imaginary numbers allow us to take square roots of negative numbers. i² = −1, i³ = −i, i⁴ = 1.
What is the complex conjugate? ▾
For z = a+bi, the conjugate is z* = a−bi. z×z* = a²+b² (real). Used to divide complex numbers.
What is Euler's formula? ▾
e^(iθ) = cos(θ) + i×sin(θ). Connects complex exponentials with trigonometry. Euler's identity: e^(iπ) + 1 = 0.
What is the complex plane? ▾
The 2D plane where real parts are on the x-axis and imaginary parts on the y-axis. Also called the Argand diagram.
Why do complex numbers matter? ▾
Every polynomial with real coefficients has roots (possibly complex) by the Fundamental Theorem of Algebra. Used everywhere in physics, engineering, and mathematics.

Related Math Calculators

∑ Quadratic📐 Trig Functions∑ Matrix∑ Vector∑ Inverse Function

Formula Explorer connections

Interpretation: This relationship converts, summarizes or checks numerical quantities using standard arithmetic and measurement rules. Assumption: Use consistent units, preserve enough significant digits, and round only the final result unless the method states otherwise.

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