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Proposition: 5.03: Multiplication of Numbers is Associative

(Proposition 3 from Book 5 of Euclid's “Elements”)

If a first (magnitude) and a third are equal multiples of a second and a fourth (respectively), and equal multiples are taken of the first and the third, then, via equality, the (magnitudes) taken will also be equal multiples of the second (magnitude) and the fourth, respectively.

• For let a first (magnitude) $A$ and a third $C$ be equal multiples of a second $B$ and a fourth $D$ (respectively), and let the equal multiples $EF$ and $GH$ have been taken of $A$ and $C$ (respectively).
• I say that $EF$ and $GH$ are equal multiples of $B$ and $D$ (respectively).

Modern Formulation

If we are given two¹ positive real numbers \(\alpha\), \(\beta\), and the following multiples of aliquot parts \(m\ge 1\), \(n\ge 1\): \[m\alpha=m\beta,\quad n(m\alpha)=n(m\beta)\quad\quad( * )\] then it follows \[(nm)\alpha=(nm)\beta.\]

General Modern Formulation

See multiplication of real numbers is associative.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1
Sections: 2

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016

Footnotes