For quite some time I had the wrong impression that classical Greek mathematicians didn't use fractions. (I don't remember where I had this from.) But I recently looked into Heath's book about Archimedes and apparently Archimedes used fractions pretty much like we use them today - except for utilizing a different notation.

My question now is: How did Archimedes and his contemporaries view fractions?

To be a bit more clear about what I mean: In our "modern" view, fractions are simply numbers. We can add or multiply them to get other fractions. Natural numbers are just special fractions in the sense that $7$ is identified with $7/1$. It makes sense to compare natural numbers and fractions as in $1 < 6/5 < 2$, you can "mix" natural numbers and fractions in computations, and the sum or the product of two fractions can be a natural number.

Is this essentially also how Archimedes saw it or were fractions completely different from natural numbers in the sense that natural numbers were the only "real" numbers while fractions were just a convenient device to express ratios between geometric magnitudes? (If the latter were the case, it wouldn't make much sense to add fractions, for example.)


In searching for the source of my misconception I found this sentence in Dantzig's "Number - The Language of Science": "Diophantus was the first Greek mathematician who frankly recognized fractions as numbers."

  • $\begingroup$ This was not a misconception, the statement is more or less accurate. We do not really see rationals treated "as numbers" before Diophantus. $\endgroup$ – Conifold Aug 27 '20 at 10:35
  • $\begingroup$ Philolaos 'even-odd' numbers were discussed here hsm.stackexchange.com/questions/6721/… $\endgroup$ – sand1 Aug 27 '20 at 18:42

No, Archimedes, and ancient Greeks generally, did not see fractions as numbers, and they did not use fractions as we use them today, they did not use them at all. What they used was ratios of magnitudes. Despite some superficial similarities, ratios were not fractions, and they were not single entities, numbers or otherwise, they were relations between magnitudes, numbers, lines, areas, volumes, etc. Both relata were preserved in the ratio, 7:1 was not identified with 7, inscribed sphere:cylinder was not identified with 2:3, even though Archimedes proved them equal. Equality was not identity, areas and volumes were not numbers attached to geometric objects, they were the objects.

Ratios could be compared using Eudoxus's trick, multiplied only when it made sense geometrically, the product of line ratios was an area ratio, but not added or subtracted. See What was the aftermath of the proof of irrationality of $\sqrt{2}$ for the Greeks? for more details and references, and Did Eudoxus really set out to present irrationals as Dedekind cuts? on how Eudoxian ratio theory compares to modern real numbers. Here is from Ratio and Proportion in Euclid by Madden:

"We think of a ratio as a number obtained from other numbers by division. A proportion, for us, is a statement of equality between two “ratio‐numbers”. When we write a proportion such as a/b=c/d, the letters refer to numbers, the slashes are operations on numbers and the expressions on either side of the equals sign are numbers (or at least become numbers when the numerical values of the letters are fixed). This was not the thought pattern of the ancient Greeks. When Euclid states that the ratio of A to B is the same as the ratio of C to D, the letters A, B, C and D do not refer to numbers at all, but to segments or polygonal regions or some such magnitudes. The ratio itself, according to Definition V.3, is just “a sort of relation in respect of size” between magnitudes.

If we wish to compare two magnitudes, the first thing about them that we observe is their relative size. They may be the same size, or one may be smaller than the other. If one is smaller, we acquire more information by finding out how many copies of the smaller we can fit inside the larger. We can get even more information if we look at various multiples of the larger, and for each multiple, determine how many copies of the smaller fit inside. So, a ratio is implicitly a comparison of all the potential multiples of one magnitude to all the potential multiples of the other. (Two magnitudes are incommensurable exactly when no multiple of one is ever exactly equal to any multiple of the other.) To compare two ratios, A:B and C:D, then, we ought to be prepared to compare the array of all possible (whole‐number) multiples of the first pair with the array of all possible (whole‐number) multiples of the second."

It is this sort of comparison that Archimedes uses in his Measurement of the Circle, and which is today reinterpreted as producing fractional "estimates" of $\pi$, see Who was the first to calculate $\pi$?

  • 1
    $\begingroup$ Thanks. So, if Heath says that Archimedes wrote something like $\frac{14688}{4673\frac12} = 3+\frac{667\frac12}{4673\frac12} < 3\frac17$ this is Heath's interpretation (of some secondary source) and not to be read literally? $\endgroup$ – Frunobulax Aug 27 '20 at 10:43
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    $\begingroup$ @Frunobulax Heath is translating Archimedes's geometric constructions into modern notation to make them more accessible to the readers. $\endgroup$ – Conifold Aug 27 '20 at 11:59

Whilst Archimedes may not have viewed fractions as we view them, this doesn't mean that they didn't have the concept of a fraction. This is in part, because there is more than one way to view the concept of a number.

One of the viewpoints put forward by category theory is that the natural integers (the strictly positive integers, that is not including zero) are a decategorification of FinSet, the category of all finite sets. This is different from von Neumanns definition of a natural integer, which is to take the quotient by the natural equivalence relation that identifies all sets of the same cardinality.

To reassert FinSet, in place of the natural integers is to say that we categorify the positive integers. That this is a natural thing to do is shown by the fact that in certain places where we would use the natural integers, it is more natural to parametrise by finite sets.

Likewise, we can categorify fractions as we use them today, and find that we have the fractions that Archimedes used. In that language, whilst 8/1 is not 8, they are most assuredly isomorphic - and this means here, equivalent.

We might then say that Archimedes is being more modern than the most modern conceptions of number ...


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