Complex Number Calculator

Complex numbers extend the real number system by including imaginary numbers. They are essential in many areas of mathematics, physics, and engineering.

Basic Form

A complex number is written as a + bi where:

• a is the real part
• b is the imaginary part
• i is the imaginary unit (√-1)

Operations

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
• Conjugate: The conjugate of (a + bi) is (a - bi)
• Magnitude: |a + bi| = √(a² + b²)

Polar Form

Complex numbers can be represented in polar form as r(cos θ + i sin θ) or re^(iθ)

Applications

• Electrical engineering (AC circuits)
• Signal processing
• Quantum mechanics
• Computer graphics

Related Tools

Explore other mathematical calculators:

• Quaternion Calculator
• Vector Calculator
• Polar Coordinate Converter