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Date: 11-10-2015
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Date: 11-10-2015
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Date: 11-10-2015
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By the time Stevin proposed the use of decimal fractions in 1585, the concept of a number had developed little from that of Euclid's Elements. Details of the earlier contributions are examined in some detail in our article: The real numbers: Pythagoras to Stevin
If we move forward almost exactly 100 years to the publication of A treatise of Algebra by Wallis in 1684 we find that he accepts, without any great enthusiasm, the use of Stevin's decimals. He still only considers finite decimal expansions and realises that with these one can approximate numbers (which for him are constructed from positive integers by addition, subtraction, multiplication, division and taking nth roots) as closely as one wishes. However, Wallis understood that there were proportions which did not fall within this definition of number, such as those associated with the area and circumference of a circle:-
... such proportion is not to be expressed in the commonly received ways of notation: particularly that for the circles quadrature. ... Now, as for other incommensurable quantities, though this proportion cannot be accurately expressed in absolute numbers, yet by continued approximation it may; so as to approach nearer to it than any difference assignable.
For Wallis there were a variety of ways that one might achieve this approximation, so coming as close as one pleased. He considered approximations by continued fractions, and also approximations by taking successive square roots. This leads into the study of infinite series but without the necessary machinery to prove that these infinite series converged to a limit, he was never going to be able to progress much further in studying real numbers. Real numbers became very much associated with magnitudes. No definition was really thought necessary, and in fact the mathematics was considered the science of magnitudes. Euler, in Complete introduction to algebra (1771) wrote in the introduction:-
Mathematics, in general, is the science of quantity; or, the science which investigates the means of measuring quantity.
He also defined the notion of quantity as that which can be continuously increased or diminished and thought of length, area, volume, mass, velocity, time, etc. to be different examples of quantity. All could be measured by real numbers. However, Euler's mathematics itself led to a more abstract idea of quantity, a variable x which need not necessarily take real values. Symbolic mathematics took the notion of quantity too far, and a reassessment of the concept of a real number became more necessary. By the beginning of the nineteenth century a more rigorous approach to mathematics, principally by Cauchy and Bolzano, began to provide the machinery to put the real numbers on a firmer footing. Grabiner writes [2]:-
... though Cauchy implicitly assumed several forms of the completeness axiom for the real numbers, he did not fully understand the nature of completeness or the related topological properties of sets of real numbers or of points in space. ... Cauchy did not have explicit formulations for the completeness of the real numbers. Among the forms of the completeness property he implicitly assumed are that a bounded monotone sequence converges to a limit and that the Cauchy criterion is a sufficient condition for the convergence of a series. Though Cauchy understood that a real number could be obtained as the limit of rationals, he did not develop his insight into a definition of real numbers or a detailed description of the properties of real numbers.
Cauchy, in Cours d'analyse (1821), did not worry too much about the definition of the real numbers. He does say that a real number is the limit of a sequence of rational numbers but he is assuming here that the real numbers are known. Certainly this is not considered by Cauchy to be a definition of a real number, rather it is simply a statement of what he considers an "obvious" property. He says nothing about the need for the sequence to be what we call today a Cauchy sequence and this is necessary if one is to define convergence of a sequence without assuming the existence of its limit. He does define the product of a rational number A and an irrational number B as follows:-
Let b, b', b'', ... be a sequence of rationals approaching B closer and closer. Then the product AB will be the limit of the sequence of rational numbers Ab, Ab', Ab'', ...
Bolzano, on the other hand, showed that bounded Cauchy sequence of real numbers had a least upper bound in 1817. He later worked out his own theory of real numbers which he did not publish. This was a quite remarkable achievement and it is only comparatively recently that we have understood exactly what he did achieve. His definition of a real number was made in terms of convergent sequences of rational numbers and is explained in [22] where Rychlik describes it as "not quite correct". In [28] van Rootselaar disagrees saying that "Bolzano's elaboration is quite incorrect". However in J Berg's edition of Bolzano'sReine Zahlenlehre which was published in 1976, Berg points out that Bolzano had discovered the difficulties himself and Berg found notes by Bolzano which proposed amendments to his theory which make it completely correct. As Bolzano's contributions were unpublished they had little influence in the development of the theory of the real numbers.
Cauchy himself does not seem to have understood the significance of his own "Cauchy sequence" criterion for defining the real numbers. Nor did his immediate successors. It was Weierstrass, Heine, Méray, Cantor and Dedekind who, after convergence and uniform convergence were better understood, were able to give rigorous definitions of the real numbers.
Up to this time there was no proof that numbers existed that were not the roots of polynomial equations with rational coefficients. Clearly √2 is the root of a polynomial equation with rational coefficients, namely x2 = 2, and it is easy to see that all roots of rational numbers arise as solutions of such equations. A number is called transcendental if it is not the root of a polynomial equation with rational coefficients. The word transcendental is used as such number transcend the usual operations of arithmetic. Although mathematicians had guessed for a long time that π and e were transcendental, this had not been proved up to the middle of the 19th century. Liouville's interest in transcendental numbers stemmed from reading a correspondence between Goldbach and Daniel Bernoulli. Liouville certainly aimed to prove that e is transcendental but he did not succeed. However his contributions led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. These were the first numbers to be proved transcendental. In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number
0.1100010000000000000000010000...
where there is a 1 in place n! and 0 elsewhere.
One of the first people to attempt to give a rigorous definition of the real numbers was Hamilton. Perhaps, if one thinks about it, it is logical that he would be interested in this since his introduction of the quaternions had shown that there were new previously unstudied number systems. In fact came close to the idea of a Dedekind cut, as Euclid had done in the Elements, but failed to make the idea into a definition (again Euclid had spotted the property but never thought to use it as a definition). For a number a he noted that there are rationals a', a", b', b", c', c", d', d", ... with
a' < a < a"
b' < a < b"
c' < a < c"
d' < a < d"
...
but he never thought to define a number by the sets {a', b', c', d', ... } and {a", b", c", d", ... }. He tried another approach of defining numbers given by some law, say x