a+bi Form (Complex Number) Calculator

Convert between rectangular (a + bi) and polar form. Calculate modulus, argument, conjugate, and Euler form.

Enter Complex Number

Real component of z = a + bi
Coefficient of i in z = a + bi

Results

Rectangular Form
3 + 4i
Polar Form
5 ∠ 53.13°
Modulus |z| 5
Argument θ (Degrees) 53.1301°
Argument θ (Radians) 0.9273 rad
Conjugate 3 - 4i
Euler's Form 5 · e^(0.9273i)
r = sqrt(3² + 4²) = sqrt(25) = 5

Understanding Complex Numbers & the a+bi Form

A complex number is a number of the form z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = -1. Complex numbers extend the real number system and are essential in mathematics, physics, and engineering.

The Imaginary Unit i

The imaginary unit i was introduced to solve equations like x² + 1 = 0, which has no real solution. The powers of i cycle every four steps: i, -1, -i, 1, i, ...

Converting Between Forms

Rectangular → Polar

Given z = a + bi, find r and θ.

r = √(a² + b²)
θ = atan2(b, a)

Polar → Rectangular

Given r and θ, find a and b.

a = r · cos(θ)
b = r · sin(θ)

Conjugate

The conjugate of z = a + bi is a - bi.

z · z̄ = |z|²

Euler's Formula

Connects exponential and trigonometric functions.

e^(iθ) = cosθ + i·sinθ

The Argand Diagram

The Argand diagram is a way to visualize complex numbers as points in a 2D plane. The horizontal axis represents the real part and the vertical axis represents the imaginary part. The modulus |z| is the distance from the origin, and the argument θ is the angle from the positive real axis.

Euler's Formula and Its Significance

Euler's formula states that e^(iθ) = cosθ + i·sinθ. Setting θ = π gives Euler's identity: e^(iπ) + 1 = 0, which connects five fundamental constants: e, i, π, 1, and 0.

Applications of Complex Numbers

  • Electrical Engineering: AC circuit analysis uses complex impedance and phasors.
  • Signal Processing: Fourier transforms decompose signals into complex exponentials.
  • Quantum Mechanics: Wave functions are complex-valued.
  • Fluid Dynamics: Conformal mappings solve 2D flow problems.
  • Control Theory: Transfer functions and stability analysis use the complex plane.
  • Fractals: The Mandelbrot set is defined by iterating z → z² + c.

Key Properties

  • |z| ≥ 0, and |z| = 0 only if z = 0
  • |z · w| = |z| · |w|
  • |z + w| ≤ |z| + |w| (triangle inequality)
  • z · z̄ = |z|²
  • arg(z · w) = arg(z) + arg(w)