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.