The study of Pi is old, going back at least
4000 years to the
Babylonians and Egyptians.
Around 250 BCE, Archimedes used
inscribed and circumscribed polygons to approximate
Pi.
Using a regular polygon with 96 sides, he
calculated
Pi to two places beyond the decimal point.
While the constant Pi, the ratio of the circumference
of a
circle to its diameter, has been studied for
millenia,
the name Pi, which is actually a Greek letter,
was only assigned
in 1706 by the mathematician William Jones.
The Greeks were also interested in geometrical
constructions
with compass and straight edge (what other
tools did they have?)
One question they asked was, "Using only compass
and ruler, can
one construct the square that has the same
area as a given circle?"
This problem, called ``squaring the circle",
became famous enough that it appeared in a
play by
Aristophanes called ``The Birds".
Oh, and if you were paying attention, you
will have noticed
that the same image was used for both Aristophones
and Archimedes.
The truth is I'm not exactly sure who this
is, though I suspect
it may be Pythagorus.
But it's a great image, so I'll
call him the ``generic ancient Greek guy."
We can recycle him
for other Greeks we may meet.
Moving on, can we calculate Pi exactly?
A verse from the Bible suggests that Pi is
roughly 3.
That's not a bad approximation for 3000 years
ago.
In 1897, an amateur mathematician
attempted to persuade the Indiana legislature
to pass the Indiana Pi Bill,
legislating that Pi is exactly 3.2.
The bill was passed by the Indiana
House of Representatives but was rejected
by the state Senate.
By the way, Pi Day, March 14, is officially
recognized
by the US congress today.
To generate digits of Pi, a superior method
to the polygon approach is
to use formulas based on the arctan function,
a function which is easy to approximate.
In 1706, John Machin constructed
such a formula from which he calculuated 100
digits of Pi.
Similar formulas have been constructed since
then, and for
over 200 years this was the principle approach
used to calculate Pi.
In 1844, the calculating prodigy Zacharias
Dase used one of these
formulas to set a Pi record by working out
200 decimals
in his head.
Every century in the modern era has witnessed
high level
mathematicians pursuing Pi's digits.
Even Isaac Newton
was lured into computing digits of Pi,
sheepishly confessing, “I am ashamed to
tell you to how many figures I
carried these computations, having no other
business at the time.”
By the way, "at the time" he was inventing
calculus, optics
and the theory of gravitation.
At this point you must be wondering if there
is an easy way to
memorize digits of Pi.
One could use a clever mnemonic expression,
for example,
May I have a large container of coffee?
Cream and sugar?
Some college students prefer
How I like a drink, alcoholic of course, after
the
heavy lectures involving quantum mechanics.
This is just fun, though, so let's be clear
that
I am neither endorsing drunkeness nor physics.
Note, however, that
this phrase CAN help you win money, since
it appeared on Jeopardy:
Lots of digits of Pi can be found in the
"Palais de la Decouverte" or Palace of Discovery,
a science museum
in Paris which contains the circular Pi room.
Memorizing and embracing Pi is raised to a
new level by
the European group "Die Freunde der Zahl Pi"
or
"The Friends of the Number Pi".
To join this group, you need to recite
--- with respect, fluency and smoothness --- the
first 100 digits of
Pi by memory.
They have made a Pi carpet, a holy book (the
Pibel),
and publicly proclaim their love for Pi.
Be aware, however, that,
as they have written in their bi-laws, "The
current members have
developed means - unpretty means - of dealing
with those who
trample upon our beloved number."
And while we're on the topic of reciting Pi,
let's talk about
records.
There are fictitious records such as when
Apu says to
Homer --- no, not that Homer --- yes that
one, "In fact I can recite
Pi to 40,000 places.
The last digit is one!"
A real life story involves Daniel Tammet,
an autistic savant who
in 2004 set a European record by reciting
Pi from memory
to 22,514 digits in five hours and nine minutes.
Mathematicians are mesmerized by the many
beautiful formulas for Pi.
These include
infinite series (where infinitely many terms
are added together),
infinite products (where infinitely many terms
are multiplied together),
and infinite continued fractions (where infinitely
many ratios are
taken).
The patterns one sees speak to the order within
Pi.
Some of these formulas have been used to calculate
HUGE numbers of digits of Pi.
Brothers
Gregory and David Chudnovsky used this formula
to shatter Pi records with 10 trillion digits.
In the early
1990s, the Chudnovskys built their own supercomputer
in
Gregory's living room, mainly to do Pi calculations.
Their computer, named "m zero"
1) was built with mail order parts
2) cost about $70,000, much of the money from
their wives
3) burned two thousand watts of power,
4) ran day and night (if they shut it off,
it may die)
and 5) required at least twenty-five fans
to keep the machine cool (otherwise something
might melt)
All this calculating for Pi is necessary since
the digits of Pi go on foreover without
repeating.
Lambert proved this in 1766 by showing that
Pi is irrational,
that is, it cannot be written in the form
a_1 Pi + a_0 = 0
for some non-zero integers a_0 and a_1.
Pi is actually much
more complicated; it is a transcendental number,
that is,
there is no quadratic, cubic, quartic, or
higher degree
formula which Pi satisfies.
This fact, proved by Lindemann
in 1882, can come in handy
if your spaceship's computer is taken over
by a malevolent alien,
as happened in a Star Trek episode:
Pi also appears in many unexpected mathematical
places.
The ratio of the area of the circle to that
of the square equals
Pi/4.
This number can be approximated by randomly
picking points
in the square and calculating what percentage
of these points also
land in the circle.
A similar experiment is called the Buffon
Needle Problem:
Given ruled paper with lines spaced 1 unit
apart and a needle of
length 1 which is dropped randomly onto the
paper, what is the
probability that a line is intersected?
2 over Pi.
Given two random positive integers, what is
the probability that
they share no common factors: 6 over Pi squared.
And have you heard of e, a number coming from
calculus which is
connected to lots of applications?
Pi and e are joined in
copious ways:
Stirlings formula gives a good approximation
for the number n!
if n is a large integer
The Indian genius Ramanujan found a beautiful
formula connecting Pi and
e which involves an infinite series and an
infinite continued fraction.
Lastly, how about "rounding up" to snag Pi.
Pick some starting
integer, say 10, and round it up to the smallest
multiple
of 9, which is 18.
Then round that up to the smallest multiple
of
8, which is 24.
Then the next multiple of 7, of 6, etc.
Keep going until you get to the next multiple
of
2 and stop.
leaving us with 34.
If you start with 50, you finish with
802, 1000 leaves us with 318,570.
Given an arbitrary starting number n, we'll
let f(n)
denote the ending number in the process.
It ends up that for large n, f(n) is approximately
n squared over Pi.
Pi has also walked the red carpet.
Whether its having a movie named after you
(PI), or a book title
with your name which is later made into a
hit movie (LIFE OF PI),
Pi is seemingly everywhere.
Must be pretty sweet.
