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, defined as the square root of -1. a is the real part, and b is the imaginary part.
To add or subtract complex numbers, simply add or subtract 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.
To divide complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator.
The complex conjugate of a + bi is a - bi. Its important because multiplying a complex number by its conjugate results in a real number, which is useful for division and finding the modulus.
The modulus (or absolute value) of a complex number a + bi is given by |a + bi| = √(a² + b²). It represents the distance of the complex number from the origin in the complex plane.
The argument of a complex number a + bi is the angle θ that the vector representing the complex number makes with the positive real axis in the complex plane. It can be found using tan⁻¹(b/a), considering the quadrant of the complex number.
To convert a complex number a + bi to polar form (r, θ), find the modulus r = √(a² + b²) and the argument θ = tan⁻¹(b/a), considering the quadrant of the complex number. The polar form is then r(cos θ + i sin θ).
De Moivres Theorem states that (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for any integer n. Its used to find powers and roots of complex numbers in polar form.
Common mistakes include forgetting that i² = -1, incorrectly applying the distributive property, not using the conjugate correctly when dividing, and not considering the correct quadrant when finding the argument.