Friday 21 October 2022

#25 Ratio ... from old Math Terms notes


The ancient root of ratio comes from the same early Indo-European root that gave us arithmetic. It is sometimes given as ar and sometimes ree. In its earliest incarnations the word may have related to "fitting together", but quickly took on a meaning related to counting (putting all the items together into one group, perhaps). By the Latin reri it had taken on the ideas of "reason", from which comes rational, and ratio for a comparison of two magnitudes. Rate is a synonym for ratio and comes from the same source. The word rational is used in common language to mean a method of thinking based on logic and reason, and in mathematics to describe a comparison of two magnitudes. A rational number is a number that may be expressed as a ratio of two integers. The letter Q is generally used to represent the set of rational numbers. At Jeff Miller's web site on the earliest use of some math symbols, I found the statement, "Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. (N. Bourbaki was a group of mostly French mathematicians which began meeting in the 1930s, aiming to write a thorough unified account of all mathematics.) The letters stand for the German Quotient and Zahlen. These notations occur in Bourbaki's Algébre, Chapter 1. "

The real numbers may be divided into two sets by separating numbers into the rational numbers, and the irrational numbers. Rational numbers are numbers that may be expressed as the ratio of two integers. All common fractions would be in this category, 2/3 or 5/4, as well as the integers themselves since 3 can be expressed as the ratio of 3/1. Any decimal expression that terminates after some time can be expressed as a rational number also. As an example, .35 can be written as 35/100 or 7/20. Decimal numbers that repeat the same string of digits forever are also rational numbers. Expressions like .444….. can also be represented as 4/9, and in general it is easy to express any decimal fraction that repeats right from the decimal point by writing the repeating string in the numerator and as many nines as there are digits in the repeat string in the denominator. For example the three digit repeat sequence .453453453…. can be written as 453/999. A little algebra allows us to show that if the number repeats after some initial non-repeating sequence, it can still be rewritten as a rational. For example .23453453…. can be written as a rational with the numerator equal to 23453-23= 23430; and the denominator equal to 99900 (note that three digits repeated, hence three nines, and two did not, hence two zeros).

Irrational numbers are real numbers that can NOT be expressed as a ratio of two rational numbers. The story of the irrationals probably starts with the Pythagorean discovery that the diagonal of a square could not be expressed as a ratio of the sides in any way. If the sides of the square are 1 unit in length, the diagonal will have a length that is the square root of two, so  is irrational. The  is approximately equal to 1.41423156… but the decimal expansion never reaches a point where some cycle repeats itself forever. In fact all square roots of integers that are not perfect squares (numbers like 1, 4, 9, 16, etc) are irrational. Other famous numbers that are irrational include Pi, which is appx 3.14159265… and e, which is appx 2.7182818284590… and the golden ratio which is appx 1.6180339… .

A recent discussion on the Historia Matematica list explains the origin and development of irrational. I have clipped parts of several documents.

In the eminent website Earliest Known Uses of Some of the Words of Mathematics I read about the history of the word irrational:
Cajori (1919, page 68) writes, "It is worthy of note that Cassiodorius was the first writer to use the terms 'rational' and 'irrational' in the sense now current in arithmetic and algebra."
Irrational is used in English by Robert Recorde in 1551 in The Pathwaie to Knowledge: "Numbres and quantitees surde or irrationall."
Heath mentions (Vol 1 p. 92) that Magnus Aurelius Cassiodorius presented Greek geometry in his encyclopaedia "De artibus ac disciplinis liberalium literarum" (about 475 A.D.). I suppose it is there Cajori has found the terms 'rational' and 'irrational'. Perhaps in his writing about proposition 47, book I? But did Cassiodorius really mention irrational numbers? Does 'now current' mean that he in some sense had a numberline?
I know that Jacques Peletier in 1563 uses the word irrational number and he means that the sum of a rational and a irrational number is always irrational according to, what he calls a philosophical axiom. [Staffan Rodhe]

Let me intervene in this learned discussion with the following remarkable observation made by Johan Kepler in the first pages of his Harmoniae Mundi. The Greek words translated by the Latin "rational" and "irrational" (segments, i.e., numbers) are "logos" and "alogos", resp. When in Greek mathematics one mentions "logoi" or "alogoi" in connection with segments (such as the side and diagonal of a quadrilateral) it means "expressible" or "un-expressible", resp., Hence the Latin translation "rational" and "irrational" is a mis-translation, and it should better be translated as "expressible" or "inexpressible", resp., when appearing in the mathematical context. But now is too late for such a reformation of terminology. Yaakov S. Kupitz

There are actually three Greek words having similar meaning. In Plato one finds occasionally "arrhetos" (unspeakable, inexpressible, related to "rhetoric") and "alogos" (irrational, "illogical"). The word in Euclid is "a-sym-metra" (plural) referring to two in-com-mensurables (a piece-for-piece translation of "asymmetra", and somewhat distinct from the English cognate "asymmetric"). I find it interesting that the English word "unspeakable" carries a heavy emotional connotation of being "too horrible for words," but that connotation is not in the Greek "arrhetos". [Roger Cooke]


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