48
II - Convergence: Discrete variables
way of representing an increasing sequence of decimal numbers 3 , namely Xo;
XO.Xl; XO.XIX2; •.• and agree that, by definition, a real number is such an
expansion or such an increasing sequence. To say that it is the limit of such
an increasing sequence would be a perfect vicious circle: one would first have
to define what a limit is, which is precisely the first aim of analysis (and of
this chapter), and further to prove that it exists, which assumes that all the
problems have been solved. Some mediaeval theologians "proved" the existence of God by observing that existence is one of the divine qualities that
figure in His definition. One would not get very far in mathematics by such
methods.
This definition of the real numbers brings up other tiresome problems. The
first is that one has to explain why, for example, the expansions 1.0000 ...
and 0.9999 ... define the same number 1. The second, much more serious, is
that one has to define the sum and the product of two real numbers.
Everyone knows how to add or multiply decimal numbers that have a finite
number of nonzero digits. For the sum, for example, one adds the decimals
of the same rank, starting with the last digits to the right, and carrying to
the next place when the sum exceeds 9. If one tries to apply this rule of
commercial arithmetic to nonterminating expansions, one comes against a
little obstacle: there is no last digit to the right. One might try to start from
the left, but each new addition may have repercussions on all the preceding. In
short, it is a practical impossibility to define addition in this way, and even
less multiplication; nor to proving the rules of calculation which everyone
expects - associativity, distributivity, etc. -, without relying on an "it is
obvious that ... " or an "everyone knows that ... "; but mathematics is not
based on rumours, no matter how ancient. This will not prevent us, in the
sequel, from sometimes using decimal expansions to explain - explain, and
not to prove - some theorem or proof, but one cannot deduce anything more
without ridiculous contortions - or to resorting to swindles as I myself very
deliberately did, to avoid complications, in my course in Algiers in 19644.
One mathematically correct method - there are others - for defining the
positive real numbers (the negative numbers then come from the usual algebraic procedures) consists of introducing particular sets in the set Q+ of
rational numbers 2: 0, the cuts: a nonempty subset X of Q+ is a cut if it
satisfies the following conditions 5 :
3 A decimal number is the quotient of an integer by a power of 10, so its decimal
expression has only a finite number of nonzero digits. 2/10 is a decimal number,
but 2/3 = 0.6666 ... not, even though it is rational.
4 Introduction d l'analyse mathematique (Union nationale des etudiants algeriens,
1964). See also Calcul infinitesimal in the Encyclopaedia Universalis.
5 The condition (c) is automatically satisfied for every cut that defines an irrational
number, but if one omits it one finds that the number 1, for example, corresponds
to two different cuts: the set of x E Q+ such that x < 1, and the set of x E Q+
such that x ::; 1. The definition adopted here finesses the need to distinguish the
II - Convergence: Discrete variables
way of representing an increasing sequence of decimal numbers 3 , namely Xo;
XO.Xl; XO.XIX2; •.• and agree that, by definition, a real number is such an
expansion or such an increasing sequence. To say that it is the limit of such
an increasing sequence would be a perfect vicious circle: one would first have
to define what a limit is, which is precisely the first aim of analysis (and of
this chapter), and further to prove that it exists, which assumes that all the
problems have been solved. Some mediaeval theologians "proved" the existence of God by observing that existence is one of the divine qualities that
figure in His definition. One would not get very far in mathematics by such
methods.
This definition of the real numbers brings up other tiresome problems. The
first is that one has to explain why, for example, the expansions 1.0000 ...
and 0.9999 ... define the same number 1. The second, much more serious, is
that one has to define the sum and the product of two real numbers.
Everyone knows how to add or multiply decimal numbers that have a finite
number of nonzero digits. For the sum, for example, one adds the decimals
of the same rank, starting with the last digits to the right, and carrying to
the next place when the sum exceeds 9. If one tries to apply this rule of
commercial arithmetic to nonterminating expansions, one comes against a
little obstacle: there is no last digit to the right. One might try to start from
the left, but each new addition may have repercussions on all the preceding. In
short, it is a practical impossibility to define addition in this way, and even
less multiplication; nor to proving the rules of calculation which everyone
expects - associativity, distributivity, etc. -, without relying on an "it is
obvious that ... " or an "everyone knows that ... "; but mathematics is not
based on rumours, no matter how ancient. This will not prevent us, in the
sequel, from sometimes using decimal expansions to explain - explain, and
not to prove - some theorem or proof, but one cannot deduce anything more
without ridiculous contortions - or to resorting to swindles as I myself very
deliberately did, to avoid complications, in my course in Algiers in 19644.
One mathematically correct method - there are others - for defining the
positive real numbers (the negative numbers then come from the usual algebraic procedures) consists of introducing particular sets in the set Q+ of
rational numbers 2: 0, the cuts: a nonempty subset X of Q+ is a cut if it
satisfies the following conditions 5 :
3 A decimal number is the quotient of an integer by a power of 10, so its decimal
expression has only a finite number of nonzero digits. 2/10 is a decimal number,
but 2/3 = 0.6666 ... not, even though it is rational.
4 Introduction d l'analyse mathematique (Union nationale des etudiants algeriens,
1964). See also Calcul infinitesimal in the Encyclopaedia Universalis.
5 The condition (c) is automatically satisfied for every cut that defines an irrational
number, but if one omits it one finds that the number 1, for example, corresponds
to two different cuts: the set of x E Q+ such that x < 1, and the set of x E Q+
such that x ::; 1. The definition adopted here finesses the need to distinguish the