Complex Number Calculator
Add, subtract, multiply, or divide two complex numbers in the form a + bi.
What complex numbers represent
A complex number has two parts: a real part and an imaginary part (a multiple of i, where i is defined as the square root of -1). They're written in the form a + bi, and while 'imaginary' sounds abstract, complex numbers are essential tools in engineering, physics, and signal processing — anywhere oscillating or rotational behavior needs to be modeled mathematically.
Multiplying and dividing complex numbers is less intuitive than adding or subtracting, since it requires distributing terms and using the fact that i² = -1 — this calculator handles that algebra automatically.
Understanding magnitude
The magnitude (or modulus) of a complex number is its distance from the origin when plotted on the complex plane (real part on the x-axis, imaginary part on the y-axis) — calculated the same way as ordinary 2D distance, using the Pythagorean theorem on the real and imaginary parts.
Magnitude is useful in applications like electrical engineering, where complex numbers represent AC circuit quantities and their magnitude corresponds to a physically meaningful amplitude.
Frequently asked questions
Why does multiplying complex numbers involve four separate products?
Multiplying (a+bi)(c+di) requires distributing each term, similar to multiplying two binomials, and then simplifying using the identity i² = -1 — this produces the standard formula this calculator uses.
What is the magnitude of a complex number used for?
Magnitude represents the complex number's distance from the origin on the complex plane, which has physical meaning in fields like electrical engineering (representing amplitude) and signal processing.