Complex Number Calculator
Perform arithmetic on complex numbers in rectangular (a+bi) and polar form.
Add, subtract, multiply, divide, find modulus, argument, and conjugate.
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 (i² = −1). Complex numbers extend the real number line into a two-dimensional plane called the Argand diagram.
Basic operations:
Addition: (a + bi) + (c + di) = (a+c) + (b+d)i Example: (3 + 2i) + (1 + 5i) = 4 + 7i
Subtraction: (a + bi) − (c + di) = (a−c) + (b−d)i Example: (5 + 3i) − (2 + 7i) = 3 − 4i
Multiplication: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i (since i² = −1) Example: (2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i² = 2 + 11i − 12 = −10 + 11i
Division: multiply numerator and denominator by the conjugate (a − bi): (a + bi)/(c + di) = (a + bi)(c − di) / (c² + d²)
Modulus (magnitude): |a + bi| = √(a² + b²) Example: |3 + 4i| = √(9 + 16) = √25 = 5
Argument (angle): θ = arctan(b/a) Measured in radians from the positive real axis.
Polar form: r(cos θ + i sin θ) = r·eⁱᶿ (Euler’s formula)
Why complex numbers matter:
- Electrical engineering: AC circuit analysis (impedance = R + jX)
- Signal processing: Fourier transforms use e^(iωt)
- Quantum mechanics: wave functions are complex-valued
- Control systems: stability analysis via poles in the complex plane
- Computer graphics: fractals (Mandelbrot set uses complex iteration)
Euler’s identity e^(iπ) + 1 = 0 links five fundamental mathematical constants and is considered the most beautiful equation in mathematics.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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