Slicing Pi Into Millions
February 13, 1986
A review of "Slicing Pi Into Millions" by Gene Gardner for my Honors Chemistry class.
Copyright © 1997 Property of Deborah K. Fletcher. All rights reserved.
This article was found in the January, 1985 issue of Discover, beginning
on page fifty. It deals with strictly mathematical science, rather than the "usual" sciences
(physics, chemistry, etc.). Once again, the volume is a part of my personal science library.
"If we take the world of geometrical relations, the thousandth decimal of pi sleeps
there, though no one may ever try to compute it." - William James, The Meaning of
Truth.
This article was primarily based on the above quote and the calculations which
have followed it.
The common formula for calculating pi is pi=4/1-4/3+4/5-4/7+4/9-.... This equation
is long and bulky. It is most accurate when it is carried out to rather small decimals.
In 1909, a British mathematician named William Shanks calculated pi to 707
decimals. His first 527 decimals were accurate. However, the 528th decimal of pi is 4, and
Shanks designated it 5. All of his following digits were wrong.
In 1949, John W. wrench Jr. and Levi B. Smith, both american mathematicians,
accurately extended the value of pi to 1,120 decimals. This was the last effort to compute pi on a
pre-electronic desk calculator.
Pi has been computed by large computers many times. The ENIAC was the first
computer to attempt pi. It achieved 2,037 decimal places in seventy hours. Five years later, the
NORC computed it to 3,089 places. In 1957 an IBM7090 computed pi to 100,265 places, taking
eight hours and 43 minutes to complete the task. In 1973 an IBM7600 calculated pi to a million
places in 23 hours and 18 minutes. In 1983 the HITAC M-280H computed pi to 16,777,216
places in less than 30 hours.
William James predicted that the thousandth decimal place of pi existed, but that it
was unlikely that it would be computed. That was in 1909. Today it is known that this decimal
place is occupied by the numeral "9." The question: "Was the thousandth decimal of pi 9 before
1940?" has been raised by mathematical philosophers. To some, this is a timeless truth,
regardless of man. To others, mathematical objects have no reality independent of the human
mind and, therefore, the thousandth decimal was not 9 until it was calculated to be
9.
A view between these two is that the uncalculated decimals of pi exist in an
abstraction in which they possess a pseudo-reality. They do not gain true reality until they are
calculated. Even then, their reality is of relative degree.
It may well be asked, "Why waste time figuring out thousands of decimal places for
a number, anyway?" In the case of pi, there are four reasons:
- Pi is there - wherever "there" is!
- Such calculations have useful spinoffs. much is learned about calculating and checking
large numbers on computers.
- The calculation of pi to tens of thousands of decimals provides useful exercises for testing
new computers and for training new programmers.
- The more digits of pi that are known, the more mathematicians hope to answer a major
unsolved problem of number theory: Is pi's sequence of digits totally patternless, or does it
exhibit a persistent, if subtle, deviation from randomness?"
Pi contains a number of remarkable numeric patterns. Beginning with pi's
710,100th decimal is the stutter 3333333. A matching run begins with the 3,204,765th decimal.
Among the first ten million decimals of pi, there are corresponding runs of every digit except 2
and 4. There are four runs of 9's, two runs each of 3's, 5's, 7's, and 8's, and one run each of 0's,
1's, and 6's. There are 87 runs of just six repetitions of the same digit.
The sequences 23456789 and 876543210 begin with decimals 995,998 and
2,747,956, respectively. The first six digits of pi occur six times. The first six digits of
e occur eight times, plus one occurance of the first seven digits. The first eight digits
of the square root of 2 appear beginning with the 52,638th decimal.
A pifor is a prime number found in pi while reading forward, or conventionally. A
piback is a prime number in pi, found by reading the digits backward. There are four pifors: 3,
31, 314159, and 31415926535897932384626433832795028841. There are seven known
pibacks through the 432nd decimal of pi. These are: 3,31, 51416, 951413, 2951413,
53562951413, and 979853562951413.
The fraction 355/113 was discovered in the fifth century A.D., by the Chinese
astronomer Tsu Ch'ung, to be pi to six decimals.
The square root of the square root of 2143/22 is pi to eight decimals. This was
discovered in 1914 by the Indian mathematician Srinivasa Ramanujan.
As a pinal note for the superstitious: "The first 144 decimals of pi add up to 666,
the New Testament's notorious number of the Beast, or anti-Christ (Revelation 13:18). Note that
144=(6+6)x(6+6). The three decimals of pi that begin with the 666th are 343=7x7x7."
This article has been very interesting, if a bit mind-boggling. The puzzles of
mathematics are always fascinating. I greatly enjoy them.
I have learned that pi is much more than a squiggly figure which students are
forced to use in math classes. I have learned that pi may be the key to the secrets of the
universe.
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