🔢 Complex Number Calculator – Arithmetic, Polar Form & Argand Diagram
A complex number is a number of the form a + bi, where a is the real part, b is the imaginary part, and i = √(−1). Complex numbers extend the real number line into a two-dimensional plane, making them essential in electrical engineering, quantum mechanics, signal processing, control theory, and advanced mathematics.
📐 Rectangular vs. Polar Form
Every complex number can be expressed in two equivalent ways. Rectangular form a + bi is ideal for addition and subtraction. Polar form r∠θ (or exponential form r·e^(iθ)) is more natural for multiplication, division, powers, and roots.
| Property | Rectangular | Polar |
|---|---|---|
| Notation | a + bi | r∠θ |
| Best for | Addition, subtraction | Multiplication, powers, roots |
| Magnitude | r = √(a² + b²) | r (given directly) |
| Angle | θ = atan2(b, a) | θ (given directly) |
➕ Basic Arithmetic Operations
The four arithmetic operations on complex numbers follow straightforward formulas:
- Addition:
(a + bi) + (c + di) = (a+c) + (b+d)i— add real and imaginary parts separately. - Subtraction:
(a + bi) − (c + di) = (a−c) + (b−d)i— subtract components independently. - Multiplication:
(a + bi)(c + di) = (ac − bd) + (ad + bc)i— expand with FOIL, usingi² = −1. - Division:
(a + bi) / (c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²)— multiply numerator and denominator by the conjugate of the denominator.
🌀 Properties: Modulus, Argument, Conjugate & Inverse
The modulus (absolute value) of z = a + bi is |z| = √(a² + b²), representing the distance from the origin in the Argand plane. The argument is the angle θ = atan2(b, a), measured counterclockwise from the positive real axis. The conjugate z̄ = a − bi reflects the point across the real axis, and the inverse (reciprocal) 1/z = z̄ / |z|² is defined for any non-zero complex number.
🔋 Powers: De Moivre's Theorem
Raising a complex number to the n-th power is most efficient in polar form. De Moivre's Theorem states:
z^n = r^n · [cos(nθ) + i·sin(nθ)]For example, (1 + i)⁸ = (√2)⁸ · [cos(8 × 45°) + i·sin(360°)] = 16. This works for any real exponent, including fractions and negatives.
🌿 n-th Roots
Every non-zero complex number z = r∠θ has exactly n distinct n-th roots, given by:
z_k = r^(1/n) · [cos((θ + 2πk)/n) + i·sin((θ + 2πk)/n)]for k = 0, 1, …, n−1. The roots are equally spaced at 360°/n intervals on a circle of radius r^(1/n) in the Argand plane. For instance, the three cube roots of 8 lie on a circle of radius 2, separated by 120°.
📊 Argand Diagram Visualization
The Argand plane (also called the complex plane) plots complex numbers as points or vectors: the horizontal axis is the real part, the vertical axis is the imaginary part. This calculator renders all input numbers and the computed result as vector arrows from the origin, with projection guide lines showing the real and imaginary components. When computing n-th roots, all roots are plotted simultaneously, clearly showing their equal angular spacing on the root circle.
💡 Tips for Using This Calculator
- Input flexibility: Rectangular inputs accept forms like
3 + 4i,-2i,5, or1 - 2i. Spaces around the+or−sign are optional. - Polar input: Enter the magnitude (r ≥ 0) and angle separately, then choose your preferred angle unit.
- Step-by-step mode: Enable the "Show step-by-step" toggle to see all intermediate calculations, which is helpful for studying or verifying exam work.
- Precision: Adjust decimal places from 0 to 10. Higher precision is useful when intermediate rounding could accumulate errors in chained calculations.
⚙️ Common Applications
Complex numbers are used across many fields. In electrical engineering, impedance in AC circuits is expressed as Z = R + jX (using j instead of i). In signal processing, Fourier transforms represent frequency components as complex amplitudes. In control theory, poles and zeros of transfer functions are complex numbers whose positions determine system stability. This calculator handles all the numeric heavy-lifting so you can focus on the analysis.