Complex Number Calculator

What is a Complex Number?

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i² = -1. The real part is a, and the imaginary part is b.

Complex numbers extend the concept of one-dimensional number line to a two-dimensional complex plane by introducing the imaginary unit i. They are essential in mathematics, physics, engineering, signal processing, and many other fields.

Forms of Complex Numbers

Rectangular Form (a + bi)

The rectangular (or Cartesian) form expresses a complex number as a sum of its real and imaginary parts: z = a + bi, where a is the real part and b is the imaginary part.

Example: 3 + 4i has real part 3 and imaginary part 4

Polar Form (r∠θ)

The polar form expresses a complex number using its magnitude (modulus) and angle (argument): z = r∠θ or z = r·e^(iθ), where r is the modulus and θ is the argument.

The modulus r = √(a² + b²) represents the distance from the origin, and the argument θ = arctan(b/a) represents the angle from the positive real axis.

Example: 5∠53.13° is equivalent to 3 + 4i

Converting Between Forms

Rectangular to Polar: r = √(a² + b²), θ = arctan(b/a)

Polar to Rectangular: a = r·cos(θ), b = r·sin(θ)

Complex Number Operations

Addition and Subtraction

To add or subtract complex numbers in rectangular form, add or subtract the real parts and imaginary parts separately:

(a + bi) + (c + di) = (a + c) + (b + d)i

Multiplication

In rectangular form: (a + bi)(c + di) = (ac - bd) + (ad + bc)i

In polar form: (r₁∠θ₁)(r₂∠θ₂) = r₁r₂∠(θ₁ + θ₂)

Division

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

In polar form: (r₁∠θ₁) ÷ (r₂∠θ₂) = (r₁/r₂)∠(θ₁ - θ₂)

Complex Conjugate

The conjugate of a + bi is a - bi. Geometrically, it's the reflection across the real axis.

Important: z × z̄ = a² + b² (always a real number)

Powers and Roots

De Moivre's Theorem: (r∠θ)ⁿ = rⁿ∠(nθ)

The nth root of a complex number has n distinct values distributed evenly around a circle.

Complex Functions

Many mathematical functions extend to complex numbers:

• Exponential: e^(a+bi) = e^a(cos b + i·sin b)
• Natural logarithm: ln(r∠θ) = ln(r) + iθ
• Trigonometric functions: Defined using Euler's formula

Applications of Complex Numbers

Complex numbers have wide-ranging applications:

• Electrical Engineering: AC circuit analysis and impedance calculations
• Signal Processing: Fourier transforms and frequency analysis
• Quantum Mechanics: Wave functions and probability amplitudes
• Control Theory: Stability analysis and transfer functions
• Fluid Dynamics: Potential flow and conformal mapping
• Computer Graphics: Transformations and fractals (Mandelbrot set)
• Aerodynamics: Wing design and airflow analysis

Tips for Using the Calculator

• Choose the appropriate input format (rectangular or polar) based on your problem
• Multiplication and division are often easier in polar form
• Addition and subtraction are easier in rectangular form
• Toggle between degrees and radians for the argument as needed
• The conjugate operation is useful for division and finding magnitudes