Complex Number Calculator
About Complex Numbers
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, which satisfies the equation i² = −1. In this expression, a is the real part and b is the imaginary part of the complex number.
Basic Operations
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Division: (a + bi)/(c + di) = [(ac + bd)/(c² + d²)] + [(bc - ad)/(c² + d²)]i
Conjugate: (a + bi)* = a - bi
Modulus: |a + bi| = √(a² + b²)
Argument: arg(a + bi) = tan⁻¹(b/a)
Forms of Complex Numbers
Rectangular form: z = a + bi
Polar form: z = r(cos θ + i sin θ), where r = |z| and θ = arg(z)
Exponential form: z = re^(iθ), where r = |z| and θ = arg(z)
Examples
Example 1: Add (3 + 2i) and (1 - 4i)
(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i
Example 2: Multiply (2 + 3i) and (1 + i)
(2 + 3i)(1 + i) = 2(1) + 2(i) + 3i(1) + 3i(i)
= 2 + 2i + 3i + 3i²
= 2 + 5i + 3(-1)
= 2 - 3 + 5i
= -1 + 5i