The Garden of Archimedes  A museum for mathematics

The theory of irrational numbers

|    Cantor's irrational numbers    |    Dedekind's irrational numbers     |   
|    a passage from Heine.    |    a passage from Cantor.    |    a passage from Dedekind.    |    a passage from Meray.    |

Cantor's irrational numbers

In the article titled Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, which means "On the extension of a theorem of the theory of trigonometric series" appeared in 1872 in the "Matematische Annalen", Cantor finds himself considering infinite sets of points in relation to the problem of the convergence of series. In order to operate rigorously, he then proposes an arithmetic theory of irrational numbers, which he was to discuss in more detail later.
Irrational numbers are defined using sequences of rational numbers $a_1$, $a_2$,..., $a_n$, submitted to the condition that for every $\varepsilon>0$ all its terms except at most a finite number differ one from the other by less than $\varepsilon$, that is that a natural number $n_1$ exist, such that for any $n>n_1$ and for any $m$ one has $\vert a_{m+n} -a_n \vert<\varepsilon$. This is the condition today know as "Cauchy's condition" and that Cantor calls "fundamental".
He begins by affirming that if a sequence satisfies that condition, then it "has a determined limit $a$", or rather, in correcting the ambiguity of this expression, he later affirmed that the number a is "associated" to the sequence, that is the irrational numbers are identified with fundamental sequences. Two of those sequences, a_n and b_n, are the same irrational number if $\vert a_n - b_n\vert$ tends to zero. If, given any rational number, the members of the sequence - for a large enough n - are all smaller in absolute value than any given number, then $a=0$. If they are all greater than a certain positive rational then $a>0$. If they are all smaller than a certain negative rational then $a<0$.
Fundamental operations are extended to the new system, observing that if $a= a_n$ and $b=b_n$ are two fundamental sequences, also $a_n+b_n$ and $a_n \cdot b_n$ are, and they define $a+b$ and $ab$.
If $b_n$ is a fundamental sequence of irrational numbers then there exists only one irrational number a, determined by a sequence of rationals $a_n$ such that $b_n$ tends to $a$: fundamental sequences of irrational numbers does not create the necessity for new types of numbers. In other terms, irrational numbers constitute a complete system.

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Dedekind's irrational numbers

In 1872, Dedekind puts out the pamphlet Stetigkeit und irrationale Zahlen, that is "Continuity and irrational numbers", in which he presents a rigorous definition of the idea of the continuum. He starts from the study of rational numbers, and he points out three of their fundamental properties:

1. order: if $a>b$ and $b>c$ then $a>c$
2. density: if $a\neq b$ then there are infinite rationals between $a$ and $b$
3. section: if $a$ is a given rational, then all rationals can be divided into two classes $A_1$ and $A_2$ containing each an infinite number of elements, such that in the first are all the numbers smaller than $a$and in the second all the numbers larger than $a$, and $a$ can be in either the first or the second class.

If one fixes a segment as a measuring unit, then to each rational one can associate a point on a straight line, and the points on the line all respect similar properties of order, density and section. It is known, however, that the inverse correspondence is not true, because a line contains infinite points not corresponding to any rational. If one wants to create a number system that respects "the quality of being complete, without gaps, that is, continuous" of the line, then one must create new numbers, because rational numbers are not enough to describe arithmetically all the phenomena of the line.
The "essence of continuity" is recognised by Dedekind as the inverse of property (3) that is verified by every point of the line, that is in the fact - known as the "continuity" or "Dedekind" axiom - that if one creates a partition into two classes in a straight line, such that any element of one class is on the left of any element of the other, then there exists one, and only one, point which produces such a partition. Abandoning geometric intuition, Dedekind then transfers this property to the numeric system, defining as a irrational number a section of rational numbers, that is a couple $(A_1 , A_2)$ of non-empty and disjointed subsets, whose union is the set of rational numbers, such that for every element $a$ of $A_1$ and $b$ of $A_2$, $a<b$ results. Sections that are not produced by any rational number "create" a new number, an irrational number. Thus to every section now corresponds, in analogy to the line, one and only one specific number, rational or irrational.
Starting from the sections, one can then verify that the numbers thus construed enjoy the usual properties, properly defining the ordering and the arithmetic operations.

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A passage from Heine's Die Elemente der Functionenlehre.

* Page VIII in the exhibition.

Edward Heine
Die Elemente der Functionenlehre

On numbers
   1. Numeric series
1. Definition. I call a numeric series a series of numbers $a_1$, $a_2$, ..., $a_n$, ... if for every given number $\eta$ different from zero, sufficiently small, there exists a value $n$ such that $a_n -a_{n+\nu}$ for every positive natural number $\nu$ is less than $\eta$.
Observation. The word number without any addition in Chapter A always stands for rational number. Zero will here be considered a rational number.
2. Definition. Every numeric series in which the numbers $a_n$, with a growing $n$ index, are smaller than a given amount, I will call an elementary series. [...]

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A passage from Cantor's Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen

* Page VIII.2 in the exhibition

Georg Ferdinand Cantor
Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen

Rational numbers form the basis for the definition of the next concept of numeric dimension; I will say that they form a dominion A (and in these I include zero). When I speak of numeric dimension in an extended sense, it is the case presented by an infinite sequence of rational numbers

$$(1)\qquad\qquad a_1,\quad a_2, \quad ...,\quad a_n, \quad...$$

having the property that the difference $a_{n+m} -a_n$ becomes infinitely small as $n$ grows, with any positive natural number $m$, or in other words that for every arbitrary $\varepsilon$ (positive natural) there is a natural $n_o$ such that $\vert a_{n+m} -a_n\vert<\varepsilon$, when $n\geq n_o$ and $m$ is an arbitrary natural number. I express the property of sequence (1) by saying that sequence (1) has a definite limit $b$ [...].

If there is a second sequence

$$(1')\qquad\qquad a'_1,\quad a'_2, \quad ...,\quad a'_n, \quad...$$

having a definite limit $b'$ , one finds that the two sequences (1) and (1') can be related to each other in one of the following three ways, which are mutually exclusive: either (i) $a_n -a'_n$ becomes infinitely smaller with the growth of $n$, or (ii) $a_n -a'_n$ from a certain $n$ onwards always remains greater than a positive (rational) quantity $\varepsilon$, or (iii) $a_n -a'_n$ from a certain $n$ onwards remains smaller than a positive (rational) quantity $- \varepsilon$.
If the first condition is verified I posit $b=b'$ , if the second is verified $b>b'$ , if the third is verified $b<b'$.

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A passage from Dedekind's Stetigkeit und irrationale Zahlen.

* Page VIII.3 in the exhibition

Richard Dedekind
Stetigkeit und irrationale Zahlen

These last words clearly light up the path by which we can reach a continuous field, enlarging the discontinuous field R of rational numbers. In paragraph I we have seen how every rational number a determines a partition of the R system into two classes $A_1$, $A_2$ such that any number $a_1$ of the first class $A_1$ is smaller than any number $a_2$ of the second class; the number $a$ itself is either the highest number of the first class or the lowest of the second. Now, we shall call a section and indicate with the symbol $(A_1 , A_2)$ any partition of the R system into two classes $A_1$, $A_2$ which only enjoys this characteristic property, that any number of the class $A_1$ is smaller than any number of the class $A_2$.
We can then say that every rational number a determines a section or rather two sections, which we will not however consider as essentially distinct. This section also enjoys the property that either there is a largest number among the numbers of the first class, or there is a smallest number among the numbers of the second class. Inversely, if a section enjoys this latter property, then it is produced by this largest or smallest rational number.
But it is easy to prove that there exist infinite sections not produced by any rational number. The simplest example is the following. [...]

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A passage from Meray's Nouveau précis d'analyse infinitésimale.

* Page VIII.4 of the exhibition

Charles Meray
Nouveau précis d'analyse infinitésimale

1. We will call variant a variable number (natural or fractional, positive or negative), $v_{m,n,...}$ whose value depends on the natural numbers m, n, ... that assume all possible combination of values, and which we will call its indices. [...]

2. If there exists a number $V$ such that one can choose m, n, ... large enough that the difference $V- v_{m,n,...}$ is, in absolute value, smaller than any desired quantity, for some values of the indices and for all larger values, then we say that the variant $v_{m,n,...}$ tends or converges towards the limit $V$.
When $V=0$, variant $v_{m,n,...}$ is called an infinitely small quantity; such is for example the difference between a variant and its limit.[...]

(here follows Meray's construction, in which variants play a similar role than Cantor's sequences)

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