# Does rational come from ratio or ratio come from rational?

Going through law school we often used the Latin phrase ratio decidendi, meaning the reasoning of a decision. In this context we took the latin word ratio to mean thinking process.

Recently I saw an author make this observation:

Whoa. Term "Rational" number comes from ratio, since all rational numbers are the result of a ratio. How did I never learn that in school?

The author's argument seems to be that the Latin word ratio has the same meaning as the English word ratio, and then that mathematical reasoning preceded philosophical reasoning, leading to the word rational having a mathematical origin. I believe this is based on an incorrect assumption.

Now I'm inclined to think that the term rational number has a slightly different origin.

My question is: Does rational come from ratio, or ratio come from rational?

• Why does the word "rational" in the sense of "rational number" need to have exactly the same etymology as the word "rational" in the sense of "according to reason"? Couldn't they have arisen from different senses of the Latin word "ratio"? Commented Jan 1, 2015 at 22:41
• According to the Mathematics Exchange, “rational” meaning a mathematical ratio and meaning ‘according to reason’ indeed have different etymologies, as Peter Shor suggests. Commented Jan 1, 2015 at 22:58
• Actually, it says that ratio comes from rational (or more properly from the Latin word ratio meaning '(something) rational; reason', since it was a legal term. Translating Euclid into Latin, the word the Greeks used for mathematical ratio (logos) got assigned to Latin ratio. In the 15th or 16th century; but the 'rational' part was much older and continued its own way in Latin, being stamped on the cannon of Louis XIV: ULTIMA RATIO REGUM 'the final argument of kings'. Commented Jan 1, 2015 at 23:21
• The word ratio is the Latin noun corresponding to the adjective rata as in pro rata (meaning in proportion, i.e., according to the proper ratio; rata is also the source, via French, of the word rate). So the English mathematical meaning is not too distant from Latin. So both rational number and rational thought came from different meanings of the word ratio in Latin. Commented Jan 1, 2015 at 23:58
• Come on lads, you know better than to answer in comments. Commented Jan 2, 2015 at 0:12

The mathematical meaning of ratio comes from the mathematical meaning of rational, which in turn comes from the mathematical meaning of irrational.

The OED says that we get the word irrational from the Latin word irrationalis, which was used in Latin for both the mathematical and the non-mathematical sense of irrational. It appears that the mathematical senses of both ratio and rational are backformations from irrational.

Euclid called irrational numbers ἄλογος (alogos). Since the word logos in Greek means either word or reason, this would have meant either unsayable numbers or unreasonable numbers. I expect the name originally was intended to mean unsayable numbers; the ancient Greeks used rationals to identify numbers, so an irrational number would have been a number without a name. Furthermore, the word for the mathematical sense of irrational in modern Greek is άρρητος (arretos), which also means unsayable.

Anyway, the word ἄλογος was translated into Latin as irrationalis, and according to the OED, the mathematical sense of irrational is first attested in English in 1551, and the mathematical sense of rational in 1570. The mathematical sense of ratio is first attested in English in 1660, so it seems that the mathematical meaning of ratio was a backformation from rational. The word ratio means both reason and calculation in Latin, but I haven't found any evidence it was used to mean ratio, i.e., one quantity divided by another. One of the meanings of the related adjective rata, as in pro rata, does seem to have been in ratio, so the choice of ratio to mean this was not too unreasonable.

The backformation from rational to ratio presumably happened in English, because the French word ratio (same meaning as in English) was borrowed from English and not Latin (see this link to le trésor de la langue française informatisé).

The backformation from irrationalis to rationalis happened in Latin.

I read in a philosophy book (citation not kept) that the connection between ratio and rationality comes through Euclid. That for centuries, Euclidean geometry was held as the paradigm for reasoning, so much so that Descartes modelled his Cartesian method (an early milestone in the philosophy of science) on Euclidean geometry. So much so that when consistent non-Euclidean geometries were formulated in the 19th century, a moral and intellectual tremor shook Europe.

Euclid's method was to proceed by ratios. Euclid's arguments often take the form of if A:B as C:D. Two examples:

Proposition VI.1 asserts: • Suppose a and b to be two line segments, and suppose that we erect on each a rectangle of the same height h. Then the ratio of the areas A and B of these rectangles is that same as the ratio of the lengths of the segments a and b.

Proposition VI.4 asserts: • Suppose that two triangles hve the same angles. Then the ratios of corresponding sides are equal.

Both of these examples are from this document, "Euclid's Theory of Ratios." http://www.math.ubc.ca/~cass/courses/m446-03/ratios.pdf

Euclid's method of reasoning came to be called "to proceed by the method of ratios."

Hence, ratio as the proportion of one number compared to another. Hence too, ratio as rationality.

• Do you mean “consistent non- Euclidean geometries”?  Please do not respond in comments; edit your answer if necessary. Commented Jul 23, 2018 at 3:20
• Indeed I did. Duly edited. Commented Jul 24, 2018 at 7:48

From the perspective of mathematics, at first we believed everything can be compared. Thus we got the ratio. And the number to represent such comparion is called rational. Then math crisis arose and we continued to discover more and more types of numbers that cannot be reprsented as ratio. So we collectively call them irrational, which merely means they are not rational. It's just a way of classification. Please avoid the overtune of the word in daily conversation.

With rational and irrational, we got the real number, which is closed under arithmetic and limit process. We call it continuum.

But unfortunately/interestingly, there are still more types of numbers to go...

## ADD 1 - 5:56 PM 7/11/2023

What I said above can be found in the book Number: The Language of Science. Unfortunately, I don't remember the exact locations to cite.