Complex Number Calculator

Add, subtract, multiply, and divide complex numbers (a + bi) with step-by-step solutions.

Enter Complex Numbers

First Complex Number (z1)

Second Complex Number (z2)

Result

Result
4 + 2i
complex number
z1 3 + 4i
z2 1 - 2i
Operation Addition
Real Part 4
Imaginary Part 2
Magnitude |z| 4.4721

Step-by-Step Solution

(3 + 4i) + (1 - 2i) = 4 + 2i

What Are Complex Numbers?

A complex number is a number that has both a real part and an imaginary part, written in the form 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 many areas of mathematics, physics, and engineering.

The set of all complex numbers is denoted by the symbol C. Every real number is also a complex number (with b = 0), and every purely imaginary number has a = 0. The complex number 0 + 0i is simply 0.

Operations on Complex Numbers

Addition

Add the real parts together and the imaginary parts together.

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

Subtraction

Subtract the real parts and the imaginary parts separately.

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

Multiplication

Use FOIL method, remembering that i² = -1.

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

Division

Multiply numerator and denominator by the conjugate of the denominator.

(a+bi)/(c+di) = [(ac+bd) + (bc-ad)i] / (c²+d²)

The Complex Plane

Complex numbers can be visualized on a two-dimensional plane called the complex plane (or Argand diagram). The horizontal axis represents the real part and the vertical axis represents the imaginary part. Each complex number a + bi corresponds to the point (a, b) on this plane.

Key Properties

  • Conjugate: The conjugate of a + bi is a - bi. Multiplying a complex number by its conjugate always gives a real number: (a+bi)(a-bi) = a² + b².
  • Magnitude (Modulus): |a + bi| = sqrt(a² + b²). This is the distance from the origin to the point (a, b) on the complex plane.
  • Argument: The angle the complex number makes with the positive real axis, measured counterclockwise.
  • Commutativity: Both addition and multiplication of complex numbers are commutative.
  • Associativity: Both addition and multiplication are associative.

Applications of Complex Numbers

  • Electrical Engineering: AC circuit analysis uses complex impedance to represent resistance and reactance.
  • Quantum Mechanics: Wave functions are complex-valued, and probability amplitudes involve complex numbers.
  • Signal Processing: The Fourier transform converts signals between time and frequency domains using complex exponentials.
  • Control Theory: Transfer functions and stability analysis rely on complex number mathematics.
  • Fluid Dynamics: Complex potential functions describe two-dimensional fluid flow.
  • Fractals: The Mandelbrot set and Julia sets are defined using complex number iterations.

Euler's Formula

One of the most beautiful relationships in mathematics connects complex numbers to trigonometry through Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta). When theta = pi, this gives Euler's identity: e^(i*pi) + 1 = 0, which remarkably connects five fundamental mathematical constants.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has at least one complex root. This means that the complex numbers are algebraically closed -- you never need to invent new types of numbers to solve polynomial equations.