ALG-16

Complex modulus

|z| = √(x² + y²) for z = x + iy.

Reading speed

ComplexComplex modulus

Governing equation

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

$x \in \left[-20, 20\right]$

where

x: Real (—)
y: Imag (—)
|z|: Modulus (—)

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}

$x \in \left[-20, 20\right]$

$x = -20$ $|z| = 20.4$

Along the curve

②

Domain of the variable — blue

x_{\min} = -20,\quad x_{\max} = 20

$x \in \left[-20, 20\right]$

$x = -20$ $x = -20$

Along the curve

③

Response curve — green

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

$x \in \left[-20, 20\right]$

$x = -20$ $|z| = 20.4$

Along the curve

④

Rate of change — amber

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

$x \in \left[-20, 20\right]$

$x = -19.75$ $|z|^{\prime} = -0.98$

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-16 — Complex modulus) is the form associated with Complex modulus. Working symbols: $x$ , $y$ $\rightarrow$ $|z|$ . |z|² = z z̄. The modulus is the Euclidean length in the Argand plane.

Purpose

Purpose: compute

|z|

from

x

y

in Algebra via

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

|z| = √(x² + y²) for z = x + iy. 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| = 5.0000\,\mathrm{—}

. Cartesian components. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Real x3 —Imag y4 —

Outputs

Modulus |z|5.0000 —

Reading speed

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Narration of this film

Cartesian components.

|z|² = z z̄. The modulus is the Euclidean length in the Argand plane.

Reading speed

Watch on YouTube