Irrational number Quotes
A curated collection of the best quotes about Irrational number.
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“Square root of 2 is irrational*ut-the-Knot.org”
“Weisstein, Eric W., "Irrational Number" from MathWorld, A Wolfram Web Resource”
“Michael Stifel... the greatest German algebraist of the sixteenth century... studied German and Italian works, and published in 1544, in Latin, a book entitled Arithmetica Integra. Melanchthon wrote a preface to it. Its three parts treat respectively of rational numbers, irrational numbers, and algebra.”
“Another Pythagorean of about the same date as Archytas was Theodorus of Cyrene, who is said to have proved geometrically that and are incommensurable with unity. Theaetetus was one of his pupils.”
“Theaetetus developed the theory of incommensurable magnitudes. The only theorem that we can now definitely ascribe to the latter is given by Euclid in the ninth proposition of the tenth book of the Elements, namely, that the squares on two commensurable right lines have one to the other a ratio which a square number has to a square number (and conversely); but the squares on two incommensurable right lines have one to the other a ratio which cannot be expressed as that of a square number to a square number (and conversely). This theorem includes the results given by Theodorus.”
“If we construct a square whose side measures one unit, there is no way we can measure its diagonal exactly using just the rational numbers. No matter how we argue that by dividing that one unit indefinitely into ever smaller and smaller fractions we ought in the end to be able to measure the diagonal exactly, any reasonably competent mathematician can prove that our argument is false. ...in practice, however, ...we can still measure the diagonal as accurately as is necessary. Mathematicians respond to this situation, known since the time of the Pythagoreans, by classifying the square root of two as an irrational number. Yet, even if we add to the rational numbers all of those of the same kind as the square root of two, we still do not have enough numbers to measure continuous quantities exactly, even in theory.”
“We are unable to reproduce the whole body of Pythagorean teaching on this subject, but we gather from the notes of Proclus on Euclid, and from a few stray remarks in other writers, that it included the following propositions...
(viii) From their phraseology in the science of numbers and from other occasional remarks, it would seem that they were acquainted with the methods used in the second and fifth books of Euclid, and knew something of irrational magnitudes. In particular, there is reason to believe that Pythagoras proved that the side and the diagonal of a square were incommensurable, and that it was this discovery that led the early Greeks to banish the conception of number and measurement from their geometry. A proof of this proposition which may be that of Pythagoras is given below [page 60].”
“The subject of similar figures was studied and partly developed by Hippocrates. This involved the theory of proportion. Proportion had, thus far, been used by the Greeks only in numbers. They never succeeded in uniting the notions of numbers and magnitudes. The term "number" was used by them in a restricted sense. What we call irrational numbers was not included under this notion. Not even rational fractions were called numbers. They used the word in the same sense as we use "integers." Hence numbers were conceived as discontinuous, while magnitudes were continuous. The two notions appeared, therefore, entirely distinct. The chasm between them is exposed to full view in the statement of Euclid that "incommensurable magnitudes do not have the same ratio as numbers." In Euclid's Elements we find the theory of proportion of magnitudes developed and treated independent of that of numbers. The transfer of the theory of proportion from numbers to magnitudes (and to lengths in particular) was a difficult and important step.”
“[An] important generalisation, says Hankel, was this that the Hindoos never confined their arithmetical operations to rational numbers. For instance, Bhaskara showed how, by the formula the square root of the sum of rational and irrational numbers could be found. The Hindoos never discerned the dividing line between numbers and magnitudes, set up by the Greeks, which, though the product of a scientific spirit, greatly retarded the progress of mathematics. They passed from magnitudes to numbers and from numbers to magnitudes without anticipating that gap which to a sharply discriminating mind exists between the continuous and discontinuous. Yet by doing so the Indians greatly aided the general progress of mathematics. "Indeed, if one understands by algebra the application of arithmetical operations to complex magnitudes of all sorts, whether rational or irrational numbers or space-magnitudes, then the learned Brahmins of Hindostan are the real inventors of algebra.”
“The conception of "number" has been much extended in our time. With the Greeks it included only the ordinary positive whole numbers; Diophantus added rational fractions to the domain of numbers. Later negative numbers and imaginaries came gradually to be recognised. Descartes fully grasped the notion of the negative; Gauss that of the imaginary. With Euclid, a ratio, whether rational or irrational, was not a number. The recognition of ratios and irrationals as numbers took place in the sixteenth century, and found expression with Newton. By the ratio method, the continuity of the real number system has been based on the continuity of space, but in recent time three theories of irrationals have been advanced by Weierstrass, J. W. R. Dedekind, G. Cantor, and Heine, which prove the continuity of numbers without borrowing it from space. They are based on the definition of numbers by regular sequences, the use of series and limits, and some new mathematical conceptions.”
“Leonardo of Pisa is the man to whom we owe the first renaissance of mathematics on Christian soil. He is also called Fibonacci i.e. son of Bonaccio. ...
After the publication of the Liber Abaci Leonardo was presented by the astronomer Dominicus to Emperor Frederick II of Hohenstaufen. On that occasion, John of Palermo, an imperial notary, proposed several problems, which Leonardo solved promptly. ...
The second problem proposed to Leonardo at the famous scientific tournament which accompanied the presentation of this celebrated algebraist to that great patron of learning, Emperor Frederick II., was the solving of the equation . As yet cubic equations had not been solved algebraically. Instead of brooding stubbornly over this knotty problem, and after many failures still entertaining new hopes of success, he changed his method of inquiry and showed by clear and rigorous demonstration that the roots of this equation could not be represented by the Euclidean irrational quantities, or, in other words, that they could not be constructed with the ruler and compass only. He contented himself with finding a very close approximation to the required root. His work on this cubic is found in the Flos [1225]...”
“In the study of the right triangle there doubtless arose questions of puzzling subtlety. Thus, given a number equal to the side of an isosceles right triangle, to find the number which the hypotenuse is equal to. The side may have been taken equal to... any other number, yet in every instance all efforts to find a number exactly equal to the hypotenuse must have remained fruitless. The problem may have been attacked again and again, until finally "some rare genius, to whom it is granted, during some happy moments, to soar with eagle's flight above the level of human thinking," grasped the happy thought that this problem cannot be solved. In some such manner probably arose the theory of irrational quantities, which is attributed by Eudemus to the Pythagoreans. It was indeed a thought of extraordinary boldness, to assume that straight lines could exist, differing from one another not only in length,—that is, in quantity,—but also in a quality, which, though real, was absolutely invisible. Need we wonder that the Pythagoreans saw in irrationals a deep mystery, a symbol of the unspeakable? We are told that the one who first divulged the theory of irrationals, which the Pythagoreans kept secret, perished in consequence in a shipwreck. ...The first incommensurable ratio known seems to have been that of the side of a square to its diagonal, as .”
“We mentioned... the dictum of Proclus... that Pythagoras discovered the theory or study of irrationals. This subject was regarded by the Greeks as belonging to geometry rather than arithmetic. The irrationals in Euclid, Book X, are straight lines or areas, and Proclus mentions as special topics in geometry matters relating (1) to positions (for numbers have no position) (2) to contacts (for tangency is between continuous things), and (3) to irrational straight lines (for where there is division ad infinitum, there also is the irrational). ...it is certain that the incommensurability of the diagonal of a square with its side, that is, the irrationality of , was discovered in the school of Pythagoras... the traditional proof of the fact depends on the elementary theory of numbers, and... the Pythagoreans invented a method of obtaining an infinite series of arithmetical ratios approaching more and more closely to the value of .
The actual method by which the Pythagoreans proved the fact that is incommensurable with 1 was doubtless that indicated by Aristotle, a reductio ad absurdum showing that, if the diagonal of a square is commensurable with its side, it will follow that the same number is both odd and even. This is evidently the proof interpolated in the texts of Euclid as X. 117...”