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I am referring to the construction using pairs of natural numbers in 1858.

Since we use pretty much the same construction today in some analysis courses (Analysis 1, Terence Tao), except without the formal notion of "equivalence class", I'm assuming that his constructions must have become standard.

If not, then what was, and what happened to his works?

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    $\begingroup$ It is a stretch to attribute this construction to Dedekind, although he did present it more rigorously. Ratios of integers appear already in Euclid's Elements, although he does not add or subtract them. And the relation to common fractions, which are pairs of integers, together with the standard arithmetic rules was understood already in late antiquity. It was routine by the end of middle ages to identify ratios with common fractions and treat them as numbers, so it became standard long before Dedekind. $\endgroup$
    – Conifold
    Mar 3 at 5:20

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