Convert complex numbers from rectangular to polar form

Learning Objectives Identify the real (a) and imaginary (b) components of a complex number z = a + bi. Plot a complex number on the complex plane (Argand diagram) to determine its quadrant. Calculate the modulus (r) of a complex number using the formula r = sqrt(a^2 + b^2). Calculate the argument (θ) of a complex number using inverse tangent and quadrant analysis. Correctly adjust the argument based on the quadrant in which the complex number lies. Write a complex number in its final polar form, z = r(cos θ + i sin θ) or z = r cis θ. How can we describe a location with a distance and a direction instead of just left/right and up/down coordinates? 🤔 Let's apply this powerful idea to the world of complex numbers! This tutorial will guide you through the process of conve...

TermDefinitionExample Rectangular FormThe standard way of writing a complex number, z, using its real part (a) and imaginary part (b). It corresponds to Cartesian coordinates (a, b).z = 3 + 4i, where a = 3 and b = 4. Polar FormA way of writing a complex number, z, using its distance from the origin (modulus, r) and its angle relative to the positive real axis (argument, θ).z = 5(cos(0.927) + i sin(0.927)), where r = 5 and θ ≈ 0.927 radians. Complex Plane (Argand Diagram)A two-dimensional plane used to plot complex numbers. The horizontal axis represents the real part (Re), and the vertical axis represents the imaginary part (Im).The number z = -2 + i is plotted at the point (-2, 1) on the complex plane. Modulus (r)The magnitude or absolute value of a complex number, representing its dista...

Modulus Formula For z = a + bi, the modulus is r = |z| = sqrt(a^2 + b^2) Use this formula to find the distance from the origin to the point (a, b) on the complex plane. This is a direct application of the Pythagorean theorem. Argument Formula For z = a + bi, the argument is θ, found by using tan(θ) = b/a. First, calculate the reference angle α = arctan(|b/a|). Then, determine the quadrant from the signs of 'a' and 'b' to find the correct θ: Q1: θ = α, Q2: θ = π - α, Q3: θ = π + α (or α - π for principal), Q4: θ = 2π - α (or -α for principal). Polar Form Representation z = r(cos θ + i sin θ) or z = r cis θ Once you have calculated the modulus (r) and argument (θ), substitute them into this standard format to express the complex number in polar form...