Complex Numbers
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
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