ALG-25

Complex modulus

Polar radius.

Reading speed

Algebracomplex

Governing equation

|z|=\\sqrt{x^2+y^2}

$x \in \left[-10, 10\right]$

where

x: x (—)
y: y (—)
|z|: |z| (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

|z|=\\sqrt{x^2+y^2}

$x \in \left[-10, 10\right]$

$x = -10$ $|z| = 10.77$

Along the curve

②

Domain of the variable — blue

x_{\min} = -10,\quad x_{\max} = 10

$x \in \left[-10, 10\right]$

$x = -10$ $x = -10$

Along the curve

③

Response curve — green

|z|=\\sqrt{x^2+y^2}

$x \in \left[-10, 10\right]$

$x = -10$ $|z| = 10.77$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}|z|}{\mathrm{d}x}

$x \in \left[-10, 10\right]$

$x = -9.874$ $|z|^{\prime} = -0.927$

Along the curve

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-25 — Complex modulus) is the form associated with complex. Working symbols: $x$ , $y$ $\rightarrow$ $|z|$ . Polar radius.

Purpose

Purpose: compute

|z|

from

x

y

in Algebra via

|z|=\\sqrt{x^2+y^2}

Polar radius. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

x = 3.000\,\mathrm{—}

y = 4.000\,\mathrm{—}

, the governing relation

|z|=\\sqrt{x^2+y^2}

yields

|z| = 5.000\,\mathrm{—}

. Polar radius. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

x x3 —y y4 —

Outputs

|z| |z|5.000 —

Reading speed

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Narration of this film

Polar radius.

Reading speed

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