 |
| Gotfried Wilhelm von Leibniz |
Two posts ago I
mentioned that the best simple fractional approximation of the value of p is
355/113 = 3.14159292..., which was discovered in the West in the 16th century. Later,
better approximations were obtained, but no
longer in the form of a fraction, rather as the sum of a series.
Several infinite series of terms are known whose sum is p. So it
is enough to add a sufficiently large number of terms to obtain as many digits
of p as we
want, as long as we have time to do the sums. The first to propose one of these
series was the French mathematician François Vieta. As his series was quite complicated,
we give here the much better known series proposed in 1673 by the German mathematician
and philosopher Gotfried Wilhelm von Leibniz:
The more terms we add of this series, the
closer we will come to the value of p. The following table shows the
advances made over time in the calculation of the successive approximations of
this number, using different series, formulas or procedures.
Year
|
Author
|
Number of digits of p
|
1593
|
Vieta
|
17
|
1615
|
Ludolf von Ceulen
|
35
|
1717
|
Abraham Sharp
|
72
|
1844
|
Zacharias Dase
|
200
|
1873
|
William Shanks
|
707 (527)
|
Let’s consider the calculation by William
Shanks. After several years computing, he obtained 707 digits of p, breaking the previous record. For three
quarters of a century, nobody could improve it. In 1949,
an electronic computer was used for the first time to calculate the digits of p. It was then found that Shanks had
made a mistake in digit 528, which he said was 5, but the computer –which couldn’t
be wrong– discovered was actually 4. From that point, all the remaining digits computed
by Shanks, up to digit 707, were wrong. Fortunately Shanks never knew, as he
had died in 1882, nine years after completing his calculation.
Now we consider the following question:
Between 1873 and 1949, what was
the value of the 528th digit of p? Was it 4 or was it 5?
It looks like a stupid question, but consider:
•
If
realists are right and mathematics exists outside the human mind, the value of
the 528th digit of p was always equal to 4. Shanks
simply made a mistake when he computed it.
•
If
anti-realists are right, and the value of p makes no sense outside the human
mind, then the 528th digit of p had no
value before 1873, was 5 between 1873 and 1949, and changed to 4 in 1949. Shanks did not make a mistake, he simply gave p a slightly different value from the
value we assign it today.
The reader must decide which of the two
possibilities looks more reasonable.
The next question is: Why do we
need to know so many digits of the value of p? Do we need them to compute the
diameter of a circle, knowing its circumference? Let us look at a practical
example.
Suppose that the Earth were a perfect sphere,
with a circumference equal to that of the meridian going through Paris: 40,000
km (this was the first definition of the meter). If we divide 40,000,000 by the
value of p, we will get the diameter of the
Earth, which is equal to 12,732,395 meters. If we use the simplest approximation
by Archimedes (22/7), we get 12,727,272 meters, i.e. an error a little over 5
kilometers.
If we use the best simple fractional
approximation (355/113) the error would be of the order of one meter,
even though this fraction only provides six exact decimal digits of p. If we use the value of p with 10 exact decimals (3.1415926536) the
error would be about 40 microns. And if we go to 20 exact digits
(3.14159265358979323846) we would get the diameter of the Earth with an error
of the order of a few femtometers, about the size of elementary particles, much
smaller than the atoms. Does anyone think we need to know the diameter of the
Earth with so much approximation? So, why waste time
calculating more digits of p?
One of the reasons why we have continued computing
digits of p has been to apply statistical
randomness tests to these digits. In fact, the digits of p seem to be random, for any sequence of digits that
we happen to test appears among the digits of p a number of times inversely
proportional to the length of the sequence. Let us look at a few examples:
•
If
the digits of p meet the conditions of randomness, every
one-digit number should appear approximately 10% of the time in any sequence of
digits of p arbitrarily large. Thus, among the first billion
digits of p, each one-digit number should
appear about 100 million times. Well: 0 appears 99,993,942 times; 1,
99,997,334; 2, 100,002,410; 3, 99,986,911; 4, 100,011,958; 5, 99,998,885; 6,
100,010,387; 7, 99,996,061; 8, 100,001,839; and 9, 100,000,273. As you can see,
all these numbers are very close to the expected 100 million.
•
The
same happens with the 100 possible two-digit sequences (from 00 to 99), each of
which appears approximately 10 million times. The randomness conditions are
also met by three-digit sequences (from 000 to 999), each of which appears
approximately one million times. And so on.
 |
| Andrey Kolmogorov |
If the digits of p satisfy the conditions of
randomness, it would seem reasonable to think that the value of p should be random. Nothing is further from the
truth. There is another measure –Kolmogorov
complexity– that analyzes the randomness of a number by checking
to what extent it can be compressed. Well, the value of p can be compressed enormously, for any
algorithm used to calculate billions of digits of p is much shorter than the value of p and represents it exactly (or would represent
it exactly if we let it run an infinite time). So in p we have the apparent contradiction
of a
number whose digits look random, but in fact they are not, for their
value is perfectly determined.
Computer scientists are also interested in
calculating many digits of p to compare the efficiency of different types
of computers. In fact, it is possible to compare them by measuring the
number of exact digits of p they can calculate in a given time, using the same
algorithm. Different algorithms can also be checked, by running them in the same
computer. Finally, the calculation of p has been used as a test problem, to verify that
new computers work properly, without making mistakes.
The same post in Spanish
Thematic Thread on Mathematics: Previous Next
Manuel Alfonseca
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