The real numbers: Stevin to Hilbert
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:-
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 $x^{2} = 2$, 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
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
When progress came in giving a rigorous definition of a real number, there was a sudden flood of contributions. Dedekind worked out his theory of Dedekind cuts in 1858 but it remained unpublished until 1872. Weierstrass gave his own theory of real numbers in his Berlin lectures beginning in 1865 but this work was not published. The first published contribution regarding this new approach came in 1867 from Hankel who was a student of Weierstrass. Hankel, for the first time, suggests a total change in out point of view regarding the concept of a real number:-
Two years after the publication of Hankel's monograph, Méray published Remarques sur la nature des quantités Ⓣ in which he considered Cauchy sequences of rational numbers which, if they did not converge to a rational limit, had what he called a "fictitious limit". He then considered the real numbers to consist of the rational numbers and his fictitious limits. Three years later Heine published a similar notion in his book Elemente der Functionenlehre Ⓣ although it was done independently of Méray. It was similar in nature with the ideas which Weierstrass had discussed in his lectures. Heine's system has become one of the two standard ways of defining the real numbers today. Essentially Heine looks at Cauchy sequences of rational numbers. He defines an equivalence relation on such sequences by defining
Cantor also published his version of the real numbers in 1872 which followed a similar method to that of Heine. His numbers were Cauchy sequences of rational numbers and he used the term "determinate limit". It was clear to Hankel (see the quote above) that the new ideas of number had suddenly totally changed a concept which had been motivated by measurement and quantity. Similarly Cantor realised that if he wants the line to represent the real numbers then he has to introduce an axiom to recover the connection between the way the real numbers are now being defined and the old concept of measurement. He writes about a distance of a point from the origin on the line:-
Another definition, similar in style to that of Heine and Cantor, appeared in a book by Thomae in 1880. Thomae had been a colleague of Heine and Cantor around the time they had been writing up their ideas. He claimed that the real numbers defined in this way had a right to exist because:-
By the beginning of the 20th century, then, the concept of a real number had moved away completely from the concept of a number which had existed from the most ancient times to the beginning of the 19th century, namely its connection with measurement and quantity.
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's Reine 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 $x^{2} = 2$, 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 $x \mapsto x^{2}$. Hamilton writes:-
$b' < a < b''$
$c' < a < c''$
$d' < a < d''$
...
If x undergoes a continuous and constant increase from zero, then will pass successively through every state of positive ration b, and therefore that every determined positive ration b has one determined square root √b which will be commensurable or incommensurable according as b can or cannot be expressed as the square of a fraction. When b cannot be so expressed, it is still possible to approximate in fractions to the incommensurable square root √b by choosing successively larger and larger positive denominators ...One can see what Hamilton is getting at, but much here is without justification - can a quantity undergo a continuous and constant increase. Even if one got round this problem he is only defining numbers given by a law. It is unclear whether he thought that all real numbers would arise in this way.
When progress came in giving a rigorous definition of a real number, there was a sudden flood of contributions. Dedekind worked out his theory of Dedekind cuts in 1858 but it remained unpublished until 1872. Weierstrass gave his own theory of real numbers in his Berlin lectures beginning in 1865 but this work was not published. The first published contribution regarding this new approach came in 1867 from Hankel who was a student of Weierstrass. Hankel, for the first time, suggests a total change in out point of view regarding the concept of a real number:-
Today number is no longer an object, a substance which exists outside the thinking subject and the objects giving rise to this substance, an independent principle, as it was for instance for the Pythagoreans. Therefore, the question of the existence of numbers can only refer to the thinking subject or to those objects of thought whose relations are represented by numbers. Strictly speaking, only that which is logically impossible (i.e. which contradicts itself) counts as impossible for the mathematician.In his 1867 monograph Hankel addressed the question of whether there were other "number systems" which had essentially the same rules as the real numbers.
Two years after the publication of Hankel's monograph, Méray published Remarques sur la nature des quantités Ⓣ in which he considered Cauchy sequences of rational numbers which, if they did not converge to a rational limit, had what he called a "fictitious limit". He then considered the real numbers to consist of the rational numbers and his fictitious limits. Three years later Heine published a similar notion in his book Elemente der Functionenlehre Ⓣ although it was done independently of Méray. It was similar in nature with the ideas which Weierstrass had discussed in his lectures. Heine's system has become one of the two standard ways of defining the real numbers today. Essentially Heine looks at Cauchy sequences of rational numbers. He defines an equivalence relation on such sequences by defining
$a_{1} , a_{2} , a_{3} , a_{4} , ...$ and $b_{1}, b_{2} , b_{3} , b_{4} , ...$
to be equivalent if the sequence of rational numbers $a_{1} - b_{1}, a_{2} - b_{2} , a_{3} - b_{3} , a_{4} - b_{4} , ...$ converges to 0. Heine then introduced arithmetic operations on his sequences and an order relation. Particular care is needed to handle division since sequences with a non-zero limit might still have terms equal to 0.
Cantor also published his version of the real numbers in 1872 which followed a similar method to that of Heine. His numbers were Cauchy sequences of rational numbers and he used the term "determinate limit". It was clear to Hankel (see the quote above) that the new ideas of number had suddenly totally changed a concept which had been motivated by measurement and quantity. Similarly Cantor realised that if he wants the line to represent the real numbers then he has to introduce an axiom to recover the connection between the way the real numbers are now being defined and the old concept of measurement. He writes about a distance of a point from the origin on the line:-
If this distance has a rational relation to the unit of measure, then it is expressed by a rational quantity in the domain of rational numbers; otherwise, if the point is one known through a construction, it is always possible to give a sequence of rationals $a_{1} , a_{2} , a_{3} , ..., a_{n} , ...$which has the properties indicated and relates to the distance in question in such a way that the points on the straight line to which the distances $a_{1} , a_{2} , a_{3} , ..., a_{n} , ...$ are assigned approach in infinity the point to be determined with increasing n. ... In order to complete the connection presented in this section of the domains of the quantities defined [his determinate limits] with the geometry of the straight line, one must add an axiom which simple says that every numerical quantity also has a determined point on the straight line whose coordinate is equal to that quantity, indeed, equal in the sense in which this is explained in this section.As we mentioned above, Dedekind had worked out his idea of Dedekind cuts in 1858. When he realised that others like Heine and Cantor were about to publish their versions of a rigorous definition of the real numbers he decided that he too should publish his ideas. This resulted in yet another 1872 publication giving a definition of the real numbers. Dedekind considered all decompositions of the rational numbers into two sets $A_{1} , A_{2}$ so that $a_{1} < a_{2}$ for all $a_{1}$ in $A_{1}$ and $a_{2}$ in $A_{2}$. He called $(A_{1}, A_{2})$ a cut. If the rational $a$ is either the maximum element of $A_{1}$ or the minimum element of $A_{2}$ then Dedekind said the cut was produced by $a$. However not all cuts were produced by a rational. He wrote:-
In every case in which a cut $(A_{1}, A_{2})$ is given that is not produced by a rational number, we create a new number, an irrational number a, which we consider to be completely defined by this cut; we will say that the number a corresponds to this cut or that it produces the cut.He defined the usual arithmetic operations and ordering and showed that the usual laws apply.
Another definition, similar in style to that of Heine and Cantor, appeared in a book by Thomae in 1880. Thomae had been a colleague of Heine and Cantor around the time they had been writing up their ideas. He claimed that the real numbers defined in this way had a right to exist because:-
... the rules of combination abstracted from calculations with integers may be applied to them without contradiction.Frege, however, attacked these ideas of Thomae. He wanted to develop a theory of real numbers based on a purely logical base and attacked the philosophy behind the constructions which had been published. Thomae added further explanation to his idea of "formal arithmetic" in the second edition of his text which appeared in 1898:-
The formal conception of numbers requires of itself more modest limitations than does the logical conception. It does not ask, what are and what shall the numbers be, but it asks, what does one require of numbers in arithmetic.Frege was still unhappy with the constructions of Weierstrass, Heine, Cantor, Thomae and Dedekind. How did one know. he asked, that the constructions led to systems which would not produced contradictions? He wrote in 1903:-
This task has never been approached seriously, let alone been solved.Frege, however, never completed his own version of a logical framework. His hopes were shattered when he learnt of Russell's paradox. Hilbert had taken a totally different approach to defining the real numbers in 1900. He defined the real numbers to be a system with eighteen axioms. Sixteen of these axioms define what today we call an ordered field, while the other two were the Archimedean axiom and the completeness axiom. The Archimedean axiom stated that given positive numbers $a$ and $b$ then it is possible to add $a$ to itself a finite number of times so that the sum exceed $b$. The completeness property says that one cannot extend the system and maintain the validity of all the other axioms. This was totally new since all other methods built the real numbers from the known rational numbers. Hilbert's numbers were unconnected with any known system. It was impossible to say whether a given mathematical object was a real number. Most seriously, there was no proof that any such system actually existed. If it did it was still subject to the same questions concerning its consistency as Frege had pointed out.
By the beginning of the 20th century, then, the concept of a real number had moved away completely from the concept of a number which had existed from the most ancient times to the beginning of the 19th century, namely its connection with measurement and quantity.
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Written by
J J O'Connor and E F Robertson
Last Update October 2005
Last Update October 2005