ALG-22

Harmonic mean of two

H = 2xy / (x+y). Reciprocal of the mean of reciprocals.

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MeansHarmonic mean

Governing equation

H=\dfrac{2xy}{x+y}

$x \in \left[0.1, 40\right]$

where

x: x (—)
y: y (—)
H: Harmonic mean (—)

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

H=\dfrac{2xy}{x+y}

$x \in \left[0.1, 40\right]$

$x = 0.1$ $H = 0.198$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.1,\quad x_{\max} = 40

$x \in \left[0.1, 40\right]$

$x = 0.1$ $x = 0.1$

Along the curve

③

Response curve — green

H=\dfrac{2xy}{x+y}

$x \in \left[0.1, 40\right]$

$x = 0.1$ $H = 0.198$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}H}{\mathrm{d}x}

$x \in \left[0.1, 40\right]$

$x = 0.351$ $H^{\prime} = 1.889$

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-22 — Harmonic mean of two) is the form associated with Harmonic mean. Working symbols: $x$ , $y$ $\rightarrow$ $H$ . AM ≥ GM ≥ HM. The harmonic mean is the right average for rates (parallel resistors, average speed).

Purpose

Purpose: compute

H

from

x

y

in Algebra via

H=\dfrac{2xy}{x+y}

H = 2xy / (x+y). Reciprocal of the mean of reciprocals. 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 = 6.000\,\mathrm{—}

y = 12.000\,\mathrm{—}

, the governing relation

H=\dfrac{2xy}{x+y}

yields

H = 8.0000\,\mathrm{—}

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

Calculator

Inputs

x x6 —y y12 —

Outputs

Harmonic mean H8.0000 —

Reading speed

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Narration of this film

Two positive numbers.

AM ≥ GM ≥ HM. The harmonic mean is the right average for rates (parallel resistors, average speed).

Reading speed

Watch on YouTube