12.4.1: Resources and Key Concepts — Read

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Key Concepts

Polar Form of a Complex Number: A way of writing a complex number \(z = x+yi\) using its modulus \(r\) and argument \(\theta\). The form is \(z = r(\cos(\theta) + i\sin(\theta))\), which is often abbreviated as \(z = r\text{cis}(\theta)\).
Modulus: The distance of the point \((x, y)\) from the origin in the complex plane, given by \(r = \sqrt{x^2+y^2}\). It is the same as the radius \(r\) in polar form.
Argument: The angle \(\theta\) in standard position whose terminal side passes through the point \((x, y)\) in the complex plane.

Cartesian Coordinates in Terms of Cosine and Sine: If a point \(P\) is located at a distance \(r\) from the origin in the direction specified by an angle \(\theta\) in standard position, then the coordinates of \(P\) are \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
Modulus in Polar Form: The modulus of the complex number \(z\), written in polar form \(z = r(\cos(\theta) + i\sin(\theta))\), is \(r\).
Products of Complex Numbers Written in Polar Form: If \(z_1 = r_1\text{cis}(\theta_1)\) and \(z_2 = r_2\text{cis}(\theta_2)\), then their product is \(z_1z_2 = r_1r_2\text{cis}(\theta_1 + \theta_2)\).
Quotients of Complex Numbers Written in Polar Form: If \(z_1 = r_1\text{cis}(\theta_1)\) and \(z_2 = r_2\text{cis}(\theta_2)\), then their quotient is \(\frac{z_1}{z_2} = \frac{r_1}{r_2}\text{cis}(\theta_1 - \theta_2)\).
De Moivre's Theorem: If \(z = r(\cos(\theta) + i\sin(\theta))\) is a complex number and \(n\) is a positive integer, then \(z^n = r^n(\cos(n\theta) + i\sin(n\theta))\), or \(z^n = r^n\text{cis}(n\theta)\).
nth Root Theorem: If \(z = r(\cos(\theta) + i\sin(\theta))\) is a complex number, then its \(n\) distinct nth roots are given by \(z^{\frac{1}{n}} = r^{\frac{1}{n}}\left[\cos\left(\frac{\theta+2k\pi}{n}\right) + i\sin\left(\frac{\theta+2k\pi}{n}\right)\right]\), where \(k = 0, 1, 2, \dots, n-1\).

Incorrectly Determining the Argument: When converting a complex number from rectangular form to polar form, the argument \(\theta\) is not simply \(\arctan(\frac{y}{x})\). You must consider the quadrant of the point \((x,y)\) to find the correct angle.
Evaluating Trigonometric Functions Prematurely: When converting a complex number *to* polar form, do not evaluate the trigonometric functions. The goal is to leave the number in the form \(r(\cos\theta + i\sin\theta)\). Evaluating them would convert the number back to rectangular form.
Forgetting to Find All Roots: A complex number has \(n\) distinct nth roots. A common mistake is to only find the principal root (for \(k=0\)) and forget to find the other roots by adding multiples of \(\frac{2\pi}{n}\) to the angle.

This page titled 12.4.1: Resources and Key Concepts was last modified on Tue, 08 Jul 2025 17:47:08 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.