Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
To add or subtract complex numbers, combine the real parts and the imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.
Multiply complex numbers using the distributive property (FOIL method), remembering that i² = -1. For example, (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i.
The complex conjugate of a + bi is a - bi. Its used to rationalize denominators in complex fractions and find the modulus of a complex number.
To divide complex numbers, multiply both the numerator and the denominator by the complex conjugate of the denominator. This eliminates the imaginary part from the denominator.
The modulus (or absolute value) of a complex number z = a + bi is denoted as |z| and is calculated as √(a² + b²). It represents the distance from the origin to the point (a, b) in the complex plane.
The argument of a complex number z = a + bi is the angle θ that the vector from the origin to the point (a, b) makes with the positive real axis. It can be found using tan⁻¹(b/a), considering the quadrant of (a, b).
To convert z = a + bi to polar form, find the modulus r = √(a² + b²) and the argument θ = tan⁻¹(b/a). Then, z = r(cos θ + i sin θ).
De Moivres Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). Its used to find powers and roots of complex numbers.