§ 1. Convergent sequences and series
47
But for them, in reality only the first four, ten or twenty-five decimals of 11"
matter, since their calculations are intended only to lead to experimentally
verifiable formulae. The value of mathematics to its users is to provide systematic procedures for calculating the numbers they need to an arbitrarily
high accuracy. At the end of the XVIIth century, an English astronomer, John
Machin, used the formula
11"/4
4. arctan(1/5) - arctan(1/239) =
4(1/5 - 1/3.5 3 + 1/5.5 5 - 1/7.5 7 + ... )
- (1/239 - 1/3.239 3 + 1/5.239 5 - ... )
to calculate 11" to 100 decimal places: here there are two sums with infinitely
many terms, i.e. two series; clearly one cannot calculate their sums exactly:
one takes only a finite, though sufficiently large, number of terms in these
sums. This formula, mathematically exact if one knows how to give a precise
meaning to an infinite sum, provides a systematic procedure for calculating
as many decimal places as one wishes of the number 11". In other words, it
is the scheme of a numerical calculation pushed to infinity, a scheme which,
again, exists only in the minds of mathematicians and which the "users"
can always interrupt when it has provided the needed places of the complete
result. Those who complain of the "pedantry" of mathematicians have simply
not understood the problem. Without us, they would drag the letter 11" along
in their formulae (or, better still, its 25 first decimals until they need the 150
following - after all, there exist tables of log to 100 decimals, probably not
calculated simply for pleasure) without knowing what it represents.
At the present time the simplest way to define the real numbers might be
to say that they are "nonterminating decimal expansions", like the number
7r = 3.14159 ... This supposes that we know all the decimals of the number 7r;
a vast programme which some pursue, not in the hope of arriving at the end,
to be sure, but in the hope, probably also illusory, of showing that the statistical distribution of the decimals of 7r is not the result of a process analogous
to a random draw: if such were the case, one could deduce mathematically
demonstrable conjectures.
Apart from this particular case, which, after all, has never brought anybody to a halt, what sort of mathematical object does a nonterminating
decimal expansion
represent? So long as one has not defined the real numbers clearly and precisely and proved their existence as mathematical objects, such an expansion
is only a sequence of numbers Xo, Xl, .•• , where Xo is any rational integer, the
"integer part" of the pseudo-number X, and the rest are integers between 0
and 9, the "decimals" of X; as much to say a pattern on a strip of paper of
infinite length. One might also consider the preceding formula as a condensed
47
But for them, in reality only the first four, ten or twenty-five decimals of 11"
matter, since their calculations are intended only to lead to experimentally
verifiable formulae. The value of mathematics to its users is to provide systematic procedures for calculating the numbers they need to an arbitrarily
high accuracy. At the end of the XVIIth century, an English astronomer, John
Machin, used the formula
11"/4
4. arctan(1/5) - arctan(1/239) =
4(1/5 - 1/3.5 3 + 1/5.5 5 - 1/7.5 7 + ... )
- (1/239 - 1/3.239 3 + 1/5.239 5 - ... )
to calculate 11" to 100 decimal places: here there are two sums with infinitely
many terms, i.e. two series; clearly one cannot calculate their sums exactly:
one takes only a finite, though sufficiently large, number of terms in these
sums. This formula, mathematically exact if one knows how to give a precise
meaning to an infinite sum, provides a systematic procedure for calculating
as many decimal places as one wishes of the number 11". In other words, it
is the scheme of a numerical calculation pushed to infinity, a scheme which,
again, exists only in the minds of mathematicians and which the "users"
can always interrupt when it has provided the needed places of the complete
result. Those who complain of the "pedantry" of mathematicians have simply
not understood the problem. Without us, they would drag the letter 11" along
in their formulae (or, better still, its 25 first decimals until they need the 150
following - after all, there exist tables of log to 100 decimals, probably not
calculated simply for pleasure) without knowing what it represents.
At the present time the simplest way to define the real numbers might be
to say that they are "nonterminating decimal expansions", like the number
7r = 3.14159 ... This supposes that we know all the decimals of the number 7r;
a vast programme which some pursue, not in the hope of arriving at the end,
to be sure, but in the hope, probably also illusory, of showing that the statistical distribution of the decimals of 7r is not the result of a process analogous
to a random draw: if such were the case, one could deduce mathematically
demonstrable conjectures.
Apart from this particular case, which, after all, has never brought anybody to a halt, what sort of mathematical object does a nonterminating
decimal expansion
represent? So long as one has not defined the real numbers clearly and precisely and proved their existence as mathematical objects, such an expansion
is only a sequence of numbers Xo, Xl, .•• , where Xo is any rational integer, the
"integer part" of the pseudo-number X, and the rest are integers between 0
and 9, the "decimals" of X; as much to say a pattern on a strip of paper of
infinite length. One might also consider the preceding formula as a condensed