Complex Numbers: A Checklist for Complex Number Properties

The conjugate of a complex number a + bi is a - bi. Its useful for dividing complex numbers and finding the modulus.

The modulus of a complex number z = a + bi is |z| = √(a² + b²), representing its distance from the origin in the complex plane.

Polar form: z = r(cos θ + i sin θ), where r is the modulus and θ is the argument. Exponential form: z = re^(iθ).

Add or subtract the real and imaginary parts separately: (a + bi) ± (c + di) = (a ± c) + (b ± d)i.

De Moivres Theorem states (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). Its useful for finding powers and roots of complex numbers.

Multiply like binomials, remembering that i² = -1: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

Multiply the numerator and denominator by the conjugate of the denominator.

When multiplying complex numbers, their arguments add. When dividing, the argument of the divisor is subtracted from the argument of the dividend.