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1. Usefulness of Pi
Pi is an extremely interesting number that is important to
all sorts of mathematical calculations. Anytime you find yourself working
with circles, arcs, pendulums (which swing through an arc), etc. you find Pi
popping up. We have run into Pi when looking at
gears,
spherical
helium balloons and
pendulum
clocks. But you also find it in many unexpected places for reasons
that seem to have nothing at all to do with circles.
On one level pi is simple: It is the ratio of a circle's
circumference divided by its diameter. This ratio, for any circle, is always
the same - 3.14 or so. You can prove this to yourself with a circle, a piece
of tape and a ruler. Look around your house and find something circular: a
jar lid, a CD, a plate - whatever you can find that is circular, the bigger
the better. Measure its diameter (the width across the center of circle) with
the ruler. Now wrap a piece of tape around the circle and cut or mark the
tape so that it is exactly as long as the outer edge (the circumference) of
the circle you are measuring. Measure the piece of tape. With a calculator
divide the length of the tape by the diameter you measured for the circle.
The answer you will get, if you have measured accurately, is always
3.14.
The following figure shows how the circumference of a circle
with a diameter of 1.27 inches is equal to a linear distance of 4
inches:
As you might imagine, 4.0 (the circumference) / 1.27 (the
diameter) = 3.14.
As you can see, on this level Pi is a basic fact of life for all
circles. It is a constant, 3.14, for any circle you find. The funny thing
about Pi is that it also has another level. Pi is an irrational number (it
cannot be expressed by any simple fraction of two integers) that has an
infinite number of non-repeating digits. There are ways to calculate Pi that
have nothing to do with circles. Using these techniques, Pi has been
calculated out to millions of digits.
The following links give you more information about calculating
Pi and different things that you can do with it:
II. History
of Pi
A little known verse of the Bible reads,
And he made a molten sea, ten cubits
from the one brim to the other: it was round all about, and his height was
five cubits: and a line of thirty cubits did compass it about. (I Kings 7,
23)
Pi was once 3 now 3.14 and soon to be
4. As we drift away from our
floating point (center, mode or
Ra, and/or a
constant time space continuum. The further away from center the more pi will
change in relation to time and space.
The same verse can be found in II Chronicles 4, 2. It occurs in
a list of specifications for the great temple of Solomon, built around 950 BC
and its interest here is that it gives = 3. Not a very accurate value of course
and not even very accurate in its day, for the Egyptian and Mesopotamian
values of 25/8 = 3.125 and  10 = 3.162 have been traced to much
earlier dates: though in defense of Solomon's craftsmen it should be noted
that the item being described seems to have been a very large brass casting,
where a high degree of geometrical precision is neither possible nor
necessary. There are some interpretations of this which lead to a
much better value.
The
fact that the ratio of the circumference to the diameter of a circle is
constant has been known for so long that it is quite untraceable. The
earliest values of including the 'Biblical' value of 3, were almost certainly
found by measurement. In the Egyptian
Rhind Papyrus, which is dated about 1650 BC,
there is good evidence for 4(8/9)2 = 3.16 as a value for .
The earliest records of Egyptian fractions date to nearly 3900 years ago in
the papyrus copied by Ahmes
(sometimes called Ahmos - ref1, ref2) purportedly from records at least 300
years earlier. It is
conjectured that the mysterious, so called, meaningless, Egyptian triple 13,
17, 173 actually means
3 + 1/13 + 1/17 + 1/173 = 3.141527 which
approximates to 4 places!!!
(considerably better than the usual 3.16 credited to the Egyptians)
The first theoretical calculation seems to have
been carried out by
Archimedes of Syracuse (287-212 BC). He
obtained the approximation
223/71
< < 22/7.
Before giving an indication of his proof, notice that very
considerable sophistication involved in the use of inequalities here.
Archimedes knew, what so many people to this
day do not, that does not equal 22/7, and made no claim to have discovered
the exact value. If we take his best estimate as the average of his two
bounds we obtain 3.1418, an error of about 0.0002.
Here
is
Archimedes' argument.
Consider
a circle of radius 1, in which we inscribe a regular polygon of 3  2n-1
sides, with semiperimeter bn, and ascribe a regular polygon
of 3 2n-1 sides, with semiperimeter an.
 The diagram for the case n = 2
is on the Left.
The effect of this procedure is to define an increasing sequence
b1, b2,
b3, ...
and a decreasing sequence
a1, a2,
a3, ...
such that both sequences have limit .
Using
trigonometrical notation, we see that the two semiperimeters are given
by
an = K
tan(/K),
bn = K sin(/K),
where K = 3 2n-1.
Equally, we have
an+1 = 2K
tan(/2K),
bn+1 = 2K sin(/2K),
and it is not a difficult exercise in trigonometry to show
that
(1) . . . (1/an + 1/bn)
= 2/an+1
(2) . . . an+1bn
= (bn+1)2.
Archimedes, starting from a1 = 3 tan(/3) = 33 and b1
= 3 sin(/3)
= 33/2,
calculated a2 using (1), then b2 using
(2), then a3 using (1), then b3 using
(2), and so on until he had calculated a6 and b6.
His conclusion was that
b6 < < a6.
It is important to realize that the use of trigonometry here is
unhistorical:
Archimedes did not have the advantage of an
algebraic and trigonometrically notationed, and had to derive (1) and (2) by
purely geometrical means. Moreover he did not even have the advantage of our
decimal notation for numbers, so that the calculation of a6
and b6 from (1) and (2) was by no means a trivial task. So
it was a pretty stupendous feat both of imagination and of calculation and
the wonder is not that he stopped with polygons of 96 sides, but that he went
so far.
For
of course there is no reason in principle why one should not go on. Various
people did, including:
Except for
Tsu Ch'ung Chi, about whom next to nothing is
known and who is very unlikely to have known about
Archimedes' work, there was no theoretical
progress involved in these improvements, only greater stamina in calculation.
Notice how the lead, in this as in all scientific matters, passed from Europe
to the East for the millennium 400 to 1400 AD.
Al-Khwarizmi lived in Baghdad, and incidentally
gave his name to 'algorithm', while the words al jabr in the title of
one of his books gave us the word 'algebra'.
Al-Kashi lived still further east, in
Samarkand, while
Tsu Ch'ung Chi, one need hardly add, lived in
China.
The
European Renaissance brought about in due course a whole new
mathematical world. Among the first effects of this reawakening was the
emergence of mathematical formulae for . One of the earliest was that of
Wallis (1616-1703)
2/ = (1.3.3.5.5.7. ...)/(2.2.4.4.6.6. ...)
and one of the best-known is
/4 = 1 - 1/3
+ 1/5 - 1/7 + ....
This formula is sometimes attributed to
Leibniz (1646-1716) but is seems to have been
first discovered by James
Gregory (1638- 1675).
These
are both dramatic and astonishing formulae, for the expressions on the right
are completely arithmetical in character, while arises in the first instance
from geometry. They show the surprising results that infinite processes can
achieve and point the way to the wonderful richness of modern
mathematics.
From
the point of view of the calculation of , however, neither is
of any use at all. In
Gregory's series, for example, to get 4
decimal places correct we require the error to be less than 0.00005 =
1/20000, and so we need about 10000 terms of the series. However,
Gregory also showed the more general
result
(3) . . . tan-1 x = x
- x3/3 + x5/5 - ... (-1  x 1)
from which the first series results if we put x = 1. So using
the fact that
tan-1(1/3)
= /6
we get
/6 = (1/3)(1
- 1/(3.3) + 1/(5.3.3) - 1/(7.3.3.3) + ...
Which converges much more quickly. The 10th term is 1/19 393,
which is less than 0.00005, and so we have at least 4 places correct after
just 9 terms.
An
even better idea is to take the formula
(4) . . . /4 = tan-1(1/2)
+ tan-1(1/3)
and then calculate the two series obtained by putting first 1/2
and the 1/3 into (3).
Clearly
we shall get very rapid convergence indeed if we can find a formula something
like
/4 = tan-1(1/a)
+ tan-1(1/b)
with a and b large. In 1706 Machin found such a
formula:
(5) . . . /4 = 4 tan-1(1/5)
- tan-1(1/239)
Actually this is not at all hard to prove, if you know how to
prove (4) then there is no real extra difficulty about (5), except that the
arithmetic is worse. Thinking it up in the first place is, of course, quite
another matter.
With
a formula like this available the only difficulty in
computing is the sheer boredom of continuing the calculation. Needless to say,
a few people were silly enough to devote vast amounts of time and effort to
this tedious and wholly useless pursuit. One of them. an Englishman named
Shanks, used Machin's formula to
calculate to 707 places, publishing the results of many years of labour in
1873. Shanks has achieved immortality for a very curious reason which we
shall explain in a moment.
Here
is a summary of how the improvement went:
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1699:
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Sharp used
Gregory's result to get 71 correct digits
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1701:
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Machin used an improvement to
get 100 digits and the following used his methods:
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1719:
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de
Lagny found 112 correct digits
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1789:
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Vega got 126 places and in 1794 got 136
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1841:
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Rutherford calculated 152
digits and in 1853 got 440
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1873:
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Shanks calculated 707 places of which 527
were correct
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A more detailed a
Chronology is available.
Shanks knew that
was irrational since this had been
proved in 1761 by
Lambert. Shortly after
Shanks' calculation it was shown by
Lindemann that
is transcendental, that
is,
is not the solution of any polynomial equation with integer
coefficients. In fact this result of
Lindemann showed that 'squaring the circle' is
impossible. The transcendentality of implies that there is no ruler and
compass construction to construct a square equal in area to a given
circle.
Very
soon after
Shanks' calculation a curious statistical
freak was noticed by
De Morgan, who found that in the last of 707
digits there was a suspicious shortage of 7's. He mentions this in his Budget
of Paradoxes of 1872 and a curiosity it remained until 1945 when Ferguson
discovered that
Shanks had made an error in the 528th place,
after which all his digits were wrong. In 1949 a computer was used to
calculate to 2000 places. In this and all subsequent computer expansions the
number of 7's does not differ significantly from its expectation, and indeed
the sequence of digits has so far passed all statistical tests for
randomness.
You
can see
2000 places of .
We
should say a little of how the notation arose.
Oughtred in 1647 used the symbol d/ for the
ratio of the diameter of a circle to its circumference.
David Gregory (1697) used /r for the
ratio of the circumference of a circle to its radius. The first to use with its
present meaning was an Welsh mathematician William
Jones in 1706 when he states 3.14159 and c.
= .
Euler adopted the symbol in 1737 and it
quickly became a standard notation.
We
conclude with one further statistical curiosity about the calculation
of , namely
Buffon's needle experiment. If we have a
uniform grid of parallel lines, unit distance apart and if we drop a needle
of length k < 1 on the grid, the probability that the needle falls across
a line is 2k/. Various people have tried to calculate by throwing needles.
The most remarkable result was that of Lazzerini (1901), who made 34080
tosses and got
= 355/113 =
3.1415929
which, incidentally, is the value found by
Tsu Ch'ung Chi. This outcome is suspiciously
good, and the game is given away by the strange number 34080 of tosses.
Kendall and Moran comment that a good value can be obtained by stopping the
experiment at an optimal moment. If you set in advance how many throws there
are to be then this is a very inaccurate way of computing . Kendall and Moran
comment that you would do better to cut out a large circle of wood and use a
tape measure to find its circumference and diameter.
Still
on the theme of phoney experiments, Gridgeman, in a paper which pours scorn
on Lazzerini and others, created some amusement by using a needle of
carefully chosen length k = 0.7857, throwing it twice, and hitting a line
once. His estimate for was thus given by
2 0.7857 / = 1/2
from which he got the highly creditable value of = 3.1428. He
was not being serious!
It is
almost unbelievable that a definition of was used, at least as
an excuse, for a racial attack on the eminent mathematician Edmund
Landau in 1934.
Landau had defined in this textbook
published in Göttingen in that year by the, now fairly usual, method of
saying that /2 is the value of x between 1 and 2 for which cos x
vanishes. This unleashed an academic dispute, which was to end in
Landau's dismissal from his chair at
Göttingen.
Bieberbach, an eminent number theorist who
disgraced himself by his racist views, explains the reasons for
Landau's dismissal:-
Thus the valiant rejection by the
Göttingen student body which a great mathematician, Edmund
Landau, has experienced is due in the final
analysis to the fact that the un-German style of this man in his research and
teaching is unbearable to German feelings. A people who have perceived how
members of another race are working to impose ideas foreign to its own must
refuse teachers of an alien culture.
G H
Hardy replied immediately to
Bieberbach in a published note about the
consequences of this un-German definition of
There are many of us, many Englishmen
and many Germans, who said things during the War, which we scarcely meant and
are sorry to remember now. Anxiety for one's own position, dread of falling
behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic
excuses. Professor
Bieberbach's reputation excludes such
explanations of his utterances, and I find myself driven to the more
uncharitable conclusion that he really believes them true.
Not only in Germany did present problems. In the USA the value of gave rise to
heated political debate. In the State of Indiana in 1897 the House of
Representatives unanimously passed a Bill introducing a new mathematical
truth.
Be it enacted by the General Assembly
of the State of Indiana: It has been found that a circular area is to the
square on a line equal to the quadrant of the circumference, as the area of
an equilateral rectangle is to the square of one side.
(Section I, House Bill No. 246, 1897)
The Senate of Indiana showed a little more sense and postponed
indefinitely the adoption of the Act!
Open
questions about the number
1.
Does each of
the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in ?
2.
Brouwer's question: In the decimal expansion
of ,
is there a place where a thousand consecutive digits are all zero?
3.
Is simply
normal to base 10? That is does every digit appear equally often in its
decimal expansion in an asymptotic sense?
4.
Is normal to
base 10? That is does every block of digits of a given length appear equally
often in its decimal expansion in an asymptotic sense?
5.
Is normal
? That is does every block of digits of a given length appear equally often
in the expansion in every base in an asymptotic sense? The Borel introduced
the concept in 1909.
6.
Another normal
question! We know that is not rational so there is no point from which the digits
will repeat. However, if is normal then the first million digits
314159265358979... will occur from some point. Even if is not normal this
might hold! Does it? If so from what point? Note: Up to 200 million the
longest to appear is 31415926 and this appears twice.
As a postscript, here is a mnemonic for the decimal expansion
of .
Each successive digit is the number of letters in the corresponding word.
How I want a drink, alcoholic of
course, after the heavy lectures involving quantum mechanics. All of thy
geometry, Herr
Planck, is fairly hard...:
3.14159265358979323846264...
You can see more about the history of in the History topic:
squaring the circle "or view page (Squaring the circle )", that is
constructing with
ruler and compasses a square with area equal
to that of a given circle.
And you can see a
Chronology of how calculations of have
developed over the years.
Article by:
J J O'Connor and E F Robertson
The URL of this page:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html
Squaring a Circle in relation to Pi
Cartesian
equation: x2 + y2 = a2
or parametrically: x = a cos(t), y = a sin(t)
Polar equation: r = a
or add this page to ,
then Click
HERE to see one of the Associated
curves.
If your browser can handle JAVA code, Click
HERE to experiment interactively with this
curve and its associated curves.
The
study of the circle goes back beyond recorded history. The invention of the
wheel is a fundamental discovery of properties of a circle.
The
Greeks considered the Egyptians as the inventors of geometry. The scribe
Ahmes, the author of the Rhind papyrus, gives
a rule for determining the area of a circle which corresponds to =
256/81 or approximately 3.16.
The
first theorems relating to circles are attributed to
Thales around 650 BC. Book III of Euclid's
Elements, deals with properties of circles and problems of inscribing and
escribing polygons.
One
of the problems of Greek mathematics was the problem of finding a square with
the same area as a given circle. Several of the 'famous curves' in this stack
were first studied in an attempt to solve this problem.
Anaxagoras in 450 BC is the first recorded
mathematician to study this problem, by attempting
squaring the circle " or view page (squaring the circle )", that is
constructing with
ruler and compasses a square with area equal
to that of a given circle.
The
problem of finding the area of a circle led to integration. For the circle
with formula given above the area is a2 and the length of the curve is
2a.
The
pedal of a circle is a
cardioid if the pedal point is taken on the
circumference and is a
limacon if the pedal point is not on the
circumference.
The
caustic of a circle with radiant point on the circumference is a
cardioid, while if the rays are parallel then
the caustic is a
nephroid.
Apollonius, in about 240 BC, showed
effectively that the bipolar equation r = kr' represents a system of coaxial
circles as k varies. In terms of bipolar equations mr2 + nr'2 = c2 represents
a circle whose centre divides the line segment between the two fixed points
of the system in the ratio n to m.
JOC/EFR/BS January 1997
The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/history/Curves/Circle.html
References (30 books/articles)
Other Web sites:
You can find more information about calculating Waterloo,
Canada.
Some "useless" things about
Cupertino, USA together with a lot of
other links.
Another site is at
Hampshire, UK.
A site about University of Surrey, UK
The record for calculating Simon Fraser University, Canada
Another site about New Orleans
University_of_St_Andrews,_Scotland
Pi Links
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