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The Greeks knew that numbers of the form $\sqrt{n}$ for nonsquare integers $n$ are not rational. Much later, Lambert (1768) proved that the values of $e^x$ and $\tan x$ are irrational for nonzero rational numbers $x$ (and conjectured that these values are transcendental, hence cannot be constructed using ruler and compass). My question is; what happened in between?

Here's what little I have found:

  • Fibonacci (Flos) showed that the real root of $x^3 + 2x^2 + 10x = 20$ is neither rational, nor the square root of a rational, nor equal to one of the other irrational
    numbers occurring in Euclid X.

  • M. Stifel (Arithmetica integra, 1544) at least claimed that e.g. cube roots of noncube integers are not rational.

  • Fermat claimed to have a proof that if $a$ and $b$ are positive rational numbers such that $a^2 + b^2 = 2(a+b)x + x^2$, then neither $x$ nor $x^2$ are rational.

Apart from occasional claims that Euler proved the irrationality of $e$ there seem to be no results in this direction between Euclid and Lambert.

Are there any irrationality proofs going beyond the square roots of integers and known before Euler and Lambert?

Edit Following up on Michael Hardy's suggestion, I haven't found anything predating Euler. On the other hand, Euler, in his Introductio in analysin infinitorum, claims that logarithms $\log_a b$ are "neither rational nor irrational" for integers $a, b > 1$. He does not prove that the logarithms are irrational (probably because he regarded it as trivial), and claims that they are not irrational (meaning it is not the square root of a nonsquare rational) since otherwise we would have $a^{\sqrt{m}} = b$, "which is impossible" (again, no proof, but this time it is not at all obvious but a very special case of Gelfond-Schneider).

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If $a+b$ is to $a$ as $a$ is to $b$, and $a/b$ is in lowest terms, then $b/(a-b)$ is the same thing in still lower terms. Contradiction. That's not the square root of an integer, although it's close: it's a rational number plus the square root of a rational number. BUT: This proof is much simpler than if you'd first proved that it's $(1+\sqrt{5})/2$ and that $\sqrt{5}$ is irrational and that the simple operations done preserve irrationality. –  Michael Hardy Dec 16 '10 at 18:26
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Yes, but transcendence implies non-constructibility. –  Ben Webster Dec 16 '10 at 19:06
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That's the Dark Ages for you. :-( –  Nikita Sidorov Dec 16 '10 at 19:50
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My history of math professor always took great pains to point out that the Greeks didn't know that \sqrt{n} was irrational for all nonsquare n, because they didn't have a concept of a rational or irrational number. Instead, they knew that certain geometric lengths were incommensurable: for example you can't find A, B so that A diagonals of a square is equivalent to B sides of the same square. Perhaps this is just semantics... –  Chris Taylor Dec 23 '10 at 14:03
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@chris: this is indeed true: the only objects that Euclid accepted as numbers were 2, 3, 4, 5, ..., the proper multiples of the unit. Arabic mathematicians at the end of the first millenium started accepting ratios of magnitudes as numbers. –  Franz Lemmermeyer Dec 23 '10 at 14:15

2 Answers 2

I don't know when things like $\log_2 3$ were first proved irrational, but the proof is simpler than the proofs involving square roots of integers: If $2^n = 3^m$, then an even number equals an odd number.

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The only such evidence I have so far is that people were working with logarithms and the proof is really simple, so one should look at whether it may have happened. –  Michael Hardy Dec 16 '10 at 19:12
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I find Michael's suggestion intriguing and worth following up on. One can easily imagine a 17th century mathematician, knowing about logarithms and something about the history of tuning and temperament in Western music, being mindful of the irrationality of $\log_2 3$. –  Todd Trimble Dec 16 '10 at 19:44
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A propos of the theory of music, Stevin around 1585 gave a spirited defence of irrational numbers in the equal-tempered scale. See The Principal Works of Simon Stevin, vol. V, p.441. He was speaking of $2^{7/12}$, I think, but $\log_2 3$ is certainly lurking in the same neighborhood. –  John Stillwell Dec 16 '10 at 20:07
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@John: Indeed, if you follow the standard "circle of fifths", you multiply the frequency 12 times by approximately 3/2, and in the process you go up 7 octaves. Thus $(3/2)^{12} \approx 2^7$, so $3^{12} \approx 2^{19}$, which says $\log_2 3 \approx 19/12 = 1.583333\dots$. –  Michael Hardy Dec 17 '10 at 1:00
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@Michael: And to restate your irrationality argument in these terms: the irrationality of $\log_2 3$ is equivalent to the incommensurability of the fifth and the octave. –  John Stillwell Dec 17 '10 at 2:46

One of the Pythogoreans, probably Hippasos of Metapont, showed the irrationality of $\sqrt 2$ about 500 BC. According to Platon, Theodorus of Cyrene knew that the square roots of all integers up to 17 are integers or irrationals, and Theaetetus proved this for all integers. Much of what Euclid writes about incommensurable magnitudes, distinguishing different kinds of irrationalities, stems from these scholars and Eudoxus of Knidos who also invented the theory of exhaustion. (Note that Euclid's understanding of irrational numbers is not the same as today, but this would deviate too much from answering this question.). Appolonius of Perga extended this theory, as can be obtained from an Arabic translation of a commentary of Pappus on Euclid's book X. No details, however, are known.

The Indian mathematician Bhāskarāchārya, a contemporary of Fibonacci, calculated square roots of sums of rational and irrational numbers and, like Fibonacci, treated polynomial equations of higher than second degree. Also in the 13th century Johannes de Sacrobosco (Johannes Campanus) proved the irrationality of the golden ratio (this result had been known to the old Greek but lost in the pass of times) by the method of descente infinie.

Simon Stevin, the inventor of the decimal system, knows that there are two cases which do not allow an exact decimal representation, fractions like 5/6 and irrationals. Marin Mersenne in a critique of a paper of Alfons Anton de Sarasa, mentions the difficulty to obtain by geometrical means the logarithm of a certain quantity, if two other logarithms are given, may they be rational or irrational. The correspondence between Leibniz and Newton is abundant with irrationalities and the praise of the own method to handle them better. But proofs of irrationalities are not contained. May it be that enough irrational numbers were already available, or that the proof of irrationality in case of logarithms is so easy (Euler mentiones it en passant).

The French algebraist de Lagny showed that a certain kind of polynomial equations have irrational roots (Histoire de l'Académie de Sciences, 1705, p. 294).

It is controversial whether Euler has implicitly proved the irrationality of $e$ and $\pi$ by means of continued fractions. Anyhow, his introductio in analysin infinitorum is full of irrationalities. Vol. 1, contains, in chapter 6, the assertion that with exception of the powers of the base, the logarithm of a number $h$ is not rational (and cannot be irrational, hence must be transcendental). In § 508 of vol. 2, he explains: The algebraic equations are either rational and do not contain other than integer exponents or irrational with broken exponents. But in the latter case, they can be made rational. If an equation of a graph neither is rational nor can be made rational, it is transcendental. If an equation contains powers, the exponents of which are neither integers nor fractions, it cannot be made rational. The graphs of these equations are the first and, so to say, simplest kind of transcendental graphs, namely such resulting from equations with irrational exponents. § 509 starts with the example $y = x^{\sqrt 2}$.

And then came Lambert.

Etymology

In Earliest Known Uses of Some of the Words of Mathematics we read: 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." The first citation of rational in the OED2 is by John Wallis in 1685. Irrational is used in English by Robert Recorde in 1551 in The Pathwaie to Knowledge: "Numbres and quantitees surde or irrationall."

M. Cantor in his Vorlesungen über die Geschichte der Mathematik, vol 2, credits the Italien (living is Spain) Gerard of Cremona (c. 1114 - 1187), the English mathematician and bishop Thomas Bradwardine (c. 1290 - 1349), the French mathematician and bishop Nicole d'Oresme (c. 1320 – 1382), and the German mathematician and bishop Albert of Saxony (c. 1320 - 1390) with early use of the word "irrational" in mathematical context, arguing how fast this notion spread in the world of mathematics in the 14th century. Nevertheless Vieta, Fermat, Newton and others used the word "asymetriae" or "quantitates surdae".

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