ALG-17

Complex conjugate product

z z̄ = x² + y² = |z|². Conjugation flips the sign of y.

Reading speed

ComplexConjugate

Governing equation

z\bar z=x^2+y^2

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

where

x: Real (—)
y: Imag (—)
z\bar z: |z|² (—)
\bar y: Conjugate imag (—)

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.

00:0 / 00:08

①

Governing law — red

z\bar z=x^2+y^2

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

$y = -20$ $z\bar z = 409$

Along the curve

②

Domain of the variable — blue

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

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

$y = -20$ $y = -20$

Along the curve

③

Response curve — green

z\bar z=x^2+y^2

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

$y = -20$ $z\bar z = 409$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}z\bar z}{\mathrm{d}y}

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

$y = -19.75$ $z\bar z^{\prime} = -39.5$

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-17 — Complex conjugate product) is the form associated with Conjugate. Working symbols: $x$ , $y$ $\rightarrow$ $z\bar z$ , $\bar y$ . Conjugation is the Galois action of ℂ/ℝ. Real numbers are the fixed points z = z̄.

Purpose

Purpose: compute

z\bar z

\bar y

from

x

y

in Algebra via

z\bar z=x^2+y^2

z z̄ = x² + y² = |z|². Conjugation flips the sign of y. 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\bar z=x^2+y^2

yields

z\bar z = 25.0000\,\mathrm{—}

\bar y = -4.0000\,\mathrm{—}

. Cartesian z, output |z|² and the conjugate's imaginary part −y. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Real x3 —Imag y4 —

Outputs

|z|² z\bar z25.0000 —
Conjugate imag \bar y-4.0000 —

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Cartesian z, output |z|² and the conjugate's imaginary part −y.

Conjugation is the Galois action of ℂ/ℝ. Real numbers are the fixed points z = z̄.

Reading speed

Watch on YouTube