ALG-17

Complex conjugate product

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

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ComplexConjugate

Governing equation

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

where

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

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 —

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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