In mathematics, the irrational numbers are all the real numbers that are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers. Among irrational numbers are the ratio Π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.
- Let us suppose... ... This is the identical problem which... caused the existential crisis in ancient Greek mathematics. The exists (as the diagonal of the unit square); yet it does not exist (as a fraction).
Solution I. Notice that... it would follow that . Now, if... is... underestimated, the will be overestimated. ...halfway between [these estimates]... should be a better estimate than either or . Formalizing...
Solution II. Consider the graph of... . ...a parabola ...When . When . As moves continuously from to , moves continuously from from a negative to a positive... Hence, there must be a value for where , or, equivalently, where .
Solution I is algorithmic mathematics. Solution II is... dialectic [mathematics]. In a certain sense, neither solution... is a solution at all. Solution I gives a better and better approximation... Solution II tells us that an exact solution "exists."
- 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.
- Graham Flegg, Numbers: Their History and Meaning (1983)
- The Eleatic school attacked the numerical atomism of the Pythagoreans. They used what was to become the Socratic method, that is, by assuming their opponents' tenets they followed through arguments based on these, which led to absurd conclusions. The best known of these arguments are those of Zeno. ...
The effect of such arguments was to reinforce the incommensurable difficulties... often described as the great crisis in the evolution of Greek mathematics... particular attention was paid to finding a solution to the problems raised by incommensurability and the dialectical attacks of the Eleatics.
The solution... is usually attributed to Eudoxes... This was to make a distinction between numbers and magnitudes, and to provide definitions and theorems about ratios of magnitudes which replaced those only of numbers. The diagonal of a unit square was now regarded as a magnitude instead of as a length equal to the ratio of two numbers. ...Since lengths and area were magnitudes of different kinds, they could not be compared. This was less flexible a situation than that of Babylonian mathematics. It avoided the problems raised by irrational numbers, but at the cost of separating number theory and geometry for many centuries to come.
- Graham Flegg, Numbers: Their History and Meaning (1983)
- 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...
- We have first the passage of the Theaetetus recording that Theodoras proved the incommensurability of , after which Theaetetus generalized the theory of such 'roots.'... The subject of incommensurables comes up again in the Laws, where Plato inveighs against the ignorance prevailing among the Greeks of his time of the fact that lengths, breadths, and depths may be incommensurable as well as commensurable with one another, and appears to imply that he himself had not learnt the fact till late, so that he was ashamed for himself as well as for his countrymen in general. But the irrationals known to Plato included more than mere 'surds' or the sides of non-squares; in one place he says that, just as an even number may be the sum of either two odd or two even numbers, the sum of two irrationals may be either rational or irrational. An obvious illustration of the former case is afforded by a rational straight line divided 'in extreme and mean ratio'. (Euclid XIII. 6) proves that each of the segments is a particular kind of irrational straight line called by him in Book X an apotome; and to suppose that the irrationality of the two segments was already known to Plato is natural enough if we are correct in supposing that 'the theorems which' (in the words of Proclus) 'Plato originated regarding the section' were theorems about what came to be called the 'golden section', namely the division of a straight line in extreme and mean ratio as in Eucl. II. 11 and VI. 30. The appearance of the latter problem in Book II, the content of which is probably all Pythagorean, suggests that the incommensurability of the segments with the whole line was discovered before Plato's time, if not as early as the irrationality of .
- 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 Ionian and Pythagorean Schools," pp. 23-24.
- 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.
- "The Ionian and Pythagorean Schools," p. 30.
- 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.
- "The Schools of Athens and Cyzicus," p. 48.
- In the tenth book Euclid deals with certain irrational magnitudes; and since the Greeks possessed no symbolism for surds, he was forced to adopt a geometrical representation. Propositions 1 to 21 deal generally with incommensurable magnitudes. The rest of the book, namely, propostions 22 to 117, is devoted to the discussion of every possible variety of lines which can be represented by , where a and b denote commensurable lines. There are twenty-five species of such lines, and that Euclid could detect and classify them all is in the opinion of so competent an authority as Nesselmann the most striking illustration of his genius. No further advance in the theory of incommensurable magnitudes was made until the subject was taken up by Leonardo and Cardan after the interval of more than a thousand years.
In the last proposition of the tenth book [prop. 117] the side and diagonal of a square are proved to be incommensurable. The proof is so short and easy that I may quote it. If possible let the side be to the diagonal in a commensurable ratio, namely, that of two integers, a and b. Suppose this ratio reduced to its lowest terms so that a and b have no common divisor other than unity, that is, they are prime to one another. Then (by Euc. I, 47) ; therefore is an even number; therefore b is an even number; hence, since a is prime to b, a must be an odd number. Again, since it has been shown that b is an even number, b may be represented by 2n; therefore ; therefore ; therefore is an even number; therefore a is an even number. Thus the same number a must be both odd and even, which is absurd; therefore the side and the diagonal are incommensurable. Hankel believes that this proof is due to Pythagoras, and this is not unlikely. This proposition is also proved in another way in Euc. X, 9, and for this and other reasons it is now usually believed to be an interpolation by some commentator on the Elements.
- "The First Alexandrian School," p. 60.
A History of Mathematics (1893)Edit
- 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.
- "The Greeks," p. 26.
- 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 .
- "The Greeks," pp. 69-70.
- [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.
- "Middle Ages," pp. 93-94.
- 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 ...
- "Middle Ages," pp. 128-132.
- 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.
- "Modern Europe," p. 151
- 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.
- p. 372