Complex Number Calculator

Complex Number Calculator

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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

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