Multiply and divide complex numbers

Learning Objectives Multiply complex numbers in rectangular form (a + bi) using the distributive property. Identify and use the complex conjugate to divide complex numbers in rectangular form. Convert complex numbers between rectangular and polar forms. Multiply complex numbers in polar form by multiplying their moduli and adding their arguments. Divide complex numbers in polar form by dividing their moduli and subtracting their arguments. Solve multi-step problems involving both multiplication and division of complex numbers. How can multiplying two numbers cause a rotation in a 2D plane? 🤯 Complex numbers hold the key to this geometric magic! This tutorial will guide you through the two primary methods for multiplying and dividing complex numbers: the algebraic approach...

TermDefinitionExample Complex Number (Rectangular Form)A number written in the form z = a + bi, where 'a' is the real part and 'b' is the imaginary part. It can be plotted on a 2D plane called the complex plane.z = 3 + 4i, where the real part is 3 and the imaginary part is 4. Imaginary Unit (i)The imaginary unit 'i' is defined as the principal square root of -1. The most critical property to remember for multiplication is that i² = -1.i = \sqrt{-1}, so i² = -1, i³ = -i, and i⁴ = 1. Complex ConjugateThe complex conjugate of a complex number a + bi is a - bi. It is found by changing the sign of the imaginary part. We denote the conjugate of z as z̅.The conjugate of z = 5 - 2i is z̅ = 5 + 2i. ModulusThe modulus of a complex number z = a + bi, denoted |z|, is its...

Multiplication in Rectangular Form (a + bi)(c + di) = (ac - bd) + (ad + bc)i Use the FOIL (First, Outer, Inner, Last) method as you would with binomials. The key is to replace any instance of i² with -1 and then combine like terms (real parts with real parts, imaginary parts with imaginary parts). Division using the Complex Conjugate \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c² + d²} To divide, you must eliminate the imaginary unit from the denominator. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. This process is similar to 'rationalizing the denominator' with square roots. Multiplication in Polar Form Let z₁ = r₁(cos(θ₁) + i sin(θ₁)) and z₂...