Complex Number Calculator
Add, subtract, multiply, divide, raise to powers and find roots of complex numbers. Full step-by-step solutions with magnitude, argument and polar form conversion.
(a+bi) + (c+di)
Result
Enter complex numbers to calculate
Key Formulas
Multiplication:
(a+bi)(c+di) = (ac-bd) + (ad+bc)i
Division:
(a+bi)/(c+di) = multiply by conj(c+di)/(c^2+d^2)
Magnitude:
|z| = sqrt(a^2 + b^2)
De Moivre:
z^n = r^n * (cos(n*t) + i*sin(n*t))
8 Operations
Add, subtract, multiply, divide, power (De Moivre), nth root, conjugate and reciprocal with instant results.
Step-by-Step
Every calculation shows the complete solution process so you can follow along or check your homework.
Polar Form
Automatic conversion to polar form with magnitude and argument in both radians and degrees for every input and result.
Complex Number Operations Reference
| Operation | Formula | Example |
|---|---|---|
| Addition | (a+bi) + (c+di) = (a+c) + (b+d)i | (3+2i) + (1+4i) = 4+6i |
| Subtraction | (a+bi) - (c+di) = (a-c) + (b-d)i | (3+2i) - (1+4i) = 2-2i |
| Multiplication | (a+bi)(c+di) = (ac-bd) + (ad+bc)i | (3+2i)(1+4i) = -5+14i |
| Division | (a+bi)/(c+di) = [(ac+bd)+(bc-ad)i]/(c^2+d^2) | (3+2i)/(1+4i) = 11/17 - 10/17 i |
| Conjugate | conj(a+bi) = a-bi | conj(3+2i) = 3-2i |
| Magnitude | |a+bi| = sqrt(a^2+b^2) | |3+4i| = 5 |
| Power | z^n = r^n(cos(nt)+i*sin(nt)) | (1+i)^4 = -4 |
| Nth Root | z^(1/n): n roots at angles (t+2pk)/n | sqrt(i) = (1+i)/sqrt(2) or (-1-i)/sqrt(2) |
How to Use the Complex Calculator
Complex Numbers: A Complete Guide
What Are Complex Numbers?
A complex number has the form a + bi, where a is the real part and b is the imaginary part. The symbol i represents the square root of negative one. Complex numbers extend the real number line into a two-dimensional plane, letting us solve equations like x^2 + 1 = 0 that have no real solutions.
Every real number is also a complex number with b = 0. Every purely imaginary number has a = 0. The complex plane plots the real part on the horizontal axis and the imaginary part on the vertical axis, giving each complex number a unique position.
Arithmetic with Complex Numbers
Addition and subtraction work component by component: add the real parts together and the imaginary parts together. This is identical to vector addition in two dimensions.
Multiplication uses the distributive property (FOIL) combined with the rule that i^2 = -1. The product of (a+bi) and (c+di) is (ac - bd) + (ad + bc)i. In polar form, you simply multiply the magnitudes and add the angles, which is often the easier approach for repeated multiplication.
Division requires eliminating i from the denominator. Multiply both numerator and denominator by the conjugate of the denominator. This converts the denominator to a real number (c^2 + d^2), making the division straightforward.
Polar Form and Euler's Formula
Any complex number can be written as z = r(cos theta + i sin theta), where r = |z| is the magnitude and theta is the argument (angle from the positive real axis). This is the polar form. Euler's formula gives the compact version: z = r * e^(i * theta).
Polar form makes multiplication and division intuitive. To multiply two complex numbers in polar form, multiply their magnitudes and add their angles. To divide, divide magnitudes and subtract angles. This is much faster than FOIL for repeated operations.
Powers and Roots with De Moivre
De Moivre's theorem says z^n = r^n(cos(n*theta) + i*sin(n*theta)). Raise the magnitude to the power and multiply the angle by the power. This makes computing (1+i)^10 or (2+3i)^5 practical without expanding everything out.
For nth roots, the magnitude becomes r^(1/n) and the angle splits into n equally spaced values: (theta + 2*pi*k)/n for k = 0 through n-1. Every nonzero complex number has exactly n distinct nth roots, arranged symmetrically around a circle on the complex plane.
Where Complex Numbers Are Used
- Electrical engineering: AC circuit analysis uses complex impedance. Voltage and current are represented as phasors (complex numbers in polar form) which simplify the math for circuits with resistors, capacitors and inductors.
- Signal processing: The Fourier transform converts signals between time and frequency domains using complex exponentials. Every audio codec, image compressor and radio receiver relies on this.
- Quantum mechanics: Wave functions are complex-valued. The probability of a particle being at a location is the squared magnitude of the complex wave function at that point.
- Control systems: Transfer functions and stability analysis (poles and zeros) use the complex plane. The Nyquist plot maps a system's frequency response as a curve in the complex plane.
- Fractals: The Mandelbrot set and Julia sets are defined by iterating complex number operations. The stunning fractal images come from plotting how quickly complex sequences diverge.
Common Mistakes to Avoid
- Forgetting that i^2 = -1 during multiplication. Every i*i term flips sign.
- Trying to divide by just writing the fraction. You must multiply by the conjugate first.
- Mixing up rectangular and polar forms. Make sure you convert before applying De Moivre.
- Using degrees in formulas that expect radians (or vice versa). Most math functions in code use radians.
- Missing roots. The nth root always produces n distinct answers, not just one.