ALG-02

AM–GM inequality

(x+y)/2 ≥ √(xy) for x,y ≥ 0, equality iff x = y.

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InequalitiesAM–GM

Governing equation

\dfrac{x+y}{2}\ge\sqrt{xy}

where

x: x (—)
y: y (—)
AM: Arithmetic mean (—)
GM: Geometric mean (—)

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-02 — AM–GM inequality) is the form associated with AM–GM. Working symbols: $x$ , $y$ $\rightarrow$ $AM$ , $GM$ . The arithmetic mean is at least the geometric mean — Jensen on ln, or (√x−√y)² ≥ 0.

Purpose

Purpose: compute

AM

GM

from

x

y

in Algebra via

\dfrac{x+y}{2}\ge\sqrt{xy}

(x+y)/2 ≥ √(xy) for x,y ≥ 0, equality iff x = 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 = 4.000\,\mathrm{—}

y = 9.000\,\mathrm{—}

, the governing relation

\dfrac{x+y}{2}\ge\sqrt{xy}

yields

AM = 6.500\,\mathrm{—}

GM = 6.000\,\mathrm{—}

. Two positive numbers, two means. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

x x4 —y y9 —

Outputs

Arithmetic mean AM6.500 —
Geometric mean GM6.000 —

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Narration of this film

Two positive numbers, two means.

The arithmetic mean is at least the geometric mean — Jensen on ln, or (√x−√y)² ≥ 0.

Reading speed

Watch on YouTube