A Common Book of pi

8/2/2019 A Common Book of pi

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A Common Book of

3.1415926535897932384626433832795028841971693993751058209749445923078164062862

089986280348253421170679...

The mysterious and wonderful is reduced to a gargle that helps computing machines clear their throats.

-- Philip J. Davis

In recent years, the computation of the expansion of has assumed the role of a standard test of computer integrity.

-- David H. Bailey

It requires a mere 39 digits of in order to compute the circumference of a circle of radius (an upper bound

on the distance travelled by a particle moving at the speed of light for 20 billion years, and as such an upper bound

for the radius of the universe) with an error of less than meters (a lower bound for the radius of a hydrogen

atom).

-- Jonathan and Peter Borwein

The number has been the subject of a great deal of mathematical (and popular) folklore. It's

been worshipped, maligned, and misunderstood. Overestimated, underestimated, and legislated.Of interest to scholars, crackpots, and everyday people.

Pretty amazing accomplishments for a number!

The next few pages will attempt to teach you a few facts about .

You will find here, among other things, abrief historyof extended precison approximations of ,includingArchimedes' methodfor estimating , a page full of"oh, wow!" formulasused to

estimate over the centuries, and a brief look at a modern algorithm used to compute . I also

have alist of referencesfor further reading and a list ofother pages devoted to pion the Web.

Before we begin, it might not hurt to remind you that is defined as the (constant) ratio of thecircumference to the diameter in any circle. In other words, the circumference and diameter

of every circle are known to be related by .

It's not hard to see (using only elementary geometry) that is bigger than 3 but less than 4.
http://personal.bgsu.edu/~carother/pi/Pi2.htmlhttp://personal.bgsu.edu/~carother/pi/Pi2.htmlhttp://personal.bgsu.edu/~carother/pi/Pi2.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3a.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3a.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3a.htmlhttp://personal.bgsu.edu/~carother/pi/Pi5.htmlhttp://personal.bgsu.edu/~carother/pi/Pi5.htmlhttp://personal.bgsu.edu/~carother/pi/Pi5.htmlhttp://personal.bgsu.edu/~carother/pi/Pi-refs.htmlhttp://personal.bgsu.edu/~carother/pi/Pi-refs.htmlhttp://personal.bgsu.edu/~carother/pi/Pi-refs.htmlhttp://personal.bgsu.edu/~carother/pi/Pi-other.htmlhttp://personal.bgsu.edu/~carother/pi/Pi-other.htmlhttp://personal.bgsu.edu/~carother/pi/Pi-other.htmlhttp://personal.bgsu.edu/~carother/pi/Pi-other.htmlhttp://personal.bgsu.edu/~carother/pi/Pi-refs.htmlhttp://personal.bgsu.edu/~carother/pi/Pi5.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3a.htmlhttp://personal.bgsu.edu/~carother/pi/Pi2.html


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The next era in the history of the extended calculation of was ushered in by James Gregory (c.

1671), who provided us with the series

Using Gregory's series in conjunction with the identity

John Machin (c. 1706) calculated 100 decimal digits of . Methods similar to Machin's would

remain in vogue for over 200 years.

William Shanks (c. 1807) churned out the first 707 digits of . This feat took Shanks over 15

years -- in other words, he averaged only about one decimal digit per week! Sadly, only 527 of

Shanks' digits were correct. In fact, Shanks published his calculations 3 times, each time

correcting errors in the previously published digits, and each time new errors crept in. As ithappened, his first set of values proved to be the most accurate.

In 1844, Johann Dase (a.k.a., Zacharias Dahse), a calculating prodigy (or "idiot savant") hired by

the Hamburg Academy of Sciences on Gauss's recommendation, computed to 200 decimal

places in less than two months.

In the era of the desktop calculator (and the early calculators truly required an entire desktop!),

D. F. Ferguson (c. 1947) raised the total to 808 (accurate) decimal digits. In fact, it was Ferguson

who discovered the errors in Shanks' calculations.

Today, of course, in the era of the supercomputer, hundreds of millions of digits are known. Theevolution of the machine-assisted approximations to is summarized ona table on the next page.
http://personal.bgsu.edu/~carother/pi/Pi2a.htmlhttp://personal.bgsu.edu/~carother/pi/Pi2a.htmlhttp://personal.bgsu.edu/~carother/pi/Pi2a.htmlhttp://personal.bgsu.edu/~carother/pi/Pi2a.html


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The Postcomputer History of

To further highlight the improvements in our abilities to compute in recent years, consider this:The 1961 computation of 100,000 decimal digits of required roughly 105,000 full-precision

operations, while a modern algorithm, devised by Jonathan and Peter Borwein in 1984, takes

only 112 full-precision operations to achieve the same accuracy. A mere 8 iterations of theiralgorithm (roughly 56 operations) will produce 694 digits of (thus reducing Wm. Shanks' 15

year calculation to a matter of seconds).


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Archimedes' Method of Exhaustion

InMeasurement of the Circle, the greatArchimedes(c. 287--212 BC) found an approximation

for the circumference of a circle of a given radius.

Since we know that the circumference and diameter of any circle are related by the formula

, this means that if we start with a circle of diameter 1, then Archimedes' approximation

for actually provides an approximation for .

Archimedes' idea was to approximate the circle using both inscribed and circumscribed (regular)

polygons. Below are pictured inscribed and circumscribed octagons.

More generally, we would consider inscribed and circumscribed -gons. The inscribed -gon

has sides, each of the same length , and the circumscribed -gon has sides, each of the

same length . (In truth, we should consider -gons, whereMis a positive integer. But, forsimplicity, forego this extra generality.)
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html


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The perimeter of the inscribed -gon, which we denote by , and the perimeter of the

circumscribed -gon, which we denote by , are approximations for and so, in this case, arealso approximations for :

By means of geometric (and what we would now call trigonometric) arguments, Archimedes was

able toderive iterative formulasfor and , which are reminiscent of theBabylonianalgorithmfor computing square roots.
http://personal.bgsu.edu/~carother/pi/Pi3b.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3b.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3b.htmlhttp://personal.bgsu.edu/~carother/babylon/Babylon1.htmlhttp://personal.bgsu.edu/~carother/babylon/Babylon1.htmlhttp://personal.bgsu.edu/~carother/babylon/Babylon1.htmlhttp://personal.bgsu.edu/~carother/babylon/Babylon1.htmlhttp://personal.bgsu.edu/~carother/babylon/Babylon1.htmlhttp://personal.bgsu.edu/~carother/babylon/Babylon1.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3b.html


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Archimedes' Method, Part II

Recall that we want to estimate the circumference of a circle of diameter 1 (which we know to be

). For each = 2, 3, 4, ..., we inscribe and circumscribe regular polygons having sides.

Recall, too, that the perimeters and satisfy .

In order to generate an iterative formula for the perimeters, we use a bit of geometry:

If we denote the central angle in our -gon by 2 , then is the side opposite the angle in a

right triangle with hypoteneuse 1. Hence, = sin .

Next we rotate our picture and concentrate on .
http://personal.bgsu.edu/~carother/pi/geometry.htmlhttp://personal.bgsu.edu/~carother/pi/geometry.htmlhttp://personal.bgsu.edu/~carother/pi/geometry.htmlhttp://personal.bgsu.edu/~carother/pi/geometry.html


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This time our right triangle has as the side opposite the angle , and as the adjacent side.

Hence, = tan .

Archimedes' iterative formulasfor and will now follow from two trig identities.
http://personal.bgsu.edu/~carother/pi/Pi3c.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3c.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3c.html


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Archimedes' Method, Part III

Recall our construction:

We have seen that = sin and = tan . Since we have used -gons, it follows that =

sin( ) and = tan( ).

In order to relate and to and , we use two trig identities:

and

In terms of the perimeters of our polygons, this means:


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and

or

Notice that is theharmonic meanof and , while is thegeometric meanof and

. (Sound familiar?)

Finally, let's look ata simple example.
http://personal.bgsu.edu/~carother/babylon/Means.htmlhttp://personal.bgsu.edu/~carother/babylon/Means.htmlhttp://personal.bgsu.edu/~carother/babylon/Means.htmlhttp://personal.bgsu.edu/~carother/babylon/Means.htmlhttp://personal.bgsu.edu/~carother/babylon/Means.htmlhttp://personal.bgsu.edu/~carother/babylon/Means.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3d.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3d.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3d.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3d.htmlhttp://personal.bgsu.edu/~carother/babylon/Means.htmlhttp://personal.bgsu.edu/~carother/babylon/Means.html


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An Example

Let's try Archimedes' algorithm

starting with inscribed and circumscribed squares

Here are the results of 12 iterations of the algorithm.

2 2.82842712474619013 3.0614674589207182 3.31370849898476044 3.1214451522580523 3.18259787807452815 3.1365484905459393 3.15172490742925616 3.1403311569547529 3.1441183852459043

7 3.1412772509327729 3.14222362994245688 3.1415138011443011 3.14175036916896659 3.1415729403670914 3.141632080703181810 3.1415877252771597 3.141602510256808911 3.1415914215112 3.141595117749589112 3.1415923455701177 3.141593269629307313 3.1415925765848727 3.141592807599644614 3.141592634338563 3.1415926920922544

These computations were done using a spreadsheet with only limited accuracy (ostensibly, 15

decimal places). Nevertheless, notice that either of the last two entries agree with the actual value

of to at least 6 places. Another half dozen iterations would yield to 9 places. Not bad!

A better starting estimate would obviously have helped our situation here. Archimedes started

with regular hexagons (an inscribed perimeter of 3 and a circumscribed perimeter of

).

Archimedes' algorithm falls is one of a larger class ofrelated algorithms.
http://personal.bgsu.edu/~carother/pi/Pi3e.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3e.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3e.htmlhttp://personal.bgsu.edu/~carother/pi/Pi3e.html


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Related Algorithms

In the many years since Archimedes' discovery, algorithms of a similar type have come up

repeatedly. Perhaps no one understood them so well asCarl Friedrich Gauss(1777--1855);

Gauss considered the following iteration scheme:

Start with and . For , define

[This same algorithm is sometimes calledBorchardt's algorithm. It is equivalent to Archimedes'

algorithm; just substitute and .]

Gauss was considering a difficult problem when he encountered this sequence. He asked his

teacher, Pfaff, about the sequence; Pfaff showed that for any positive and , the common limit

of the two sequences is
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html


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Favorite Formulas for

= 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899

86280348253421170679...

"Name" Formulas

(Wallis, 1655)

(Brouncker, 1658)

(Newton, 1665)

(Gregory, 1671)

(Leibniz, 1674)

(Euler, 1748)

Integral Formulas
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Wallis.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Wallis.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Wallis.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Brouncker.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Brouncker.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Brouncker.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Brouncker.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Wallis.html


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Series

Odds and Ends

And everyone's favorite:


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To check that is a right angle, we will show that the Pythagorean theorem is satisfied for

this triangle (with the diameter of our circle as the hypoteneuse of the right triangle). In terms of

our coordinates:

Geometry Review, Part II

2. Given triangleDACinscribed in semicircle , as shown below, the central angleis twice the angle .

To prove this, we use the fact that is a right angle and the fact that the angles in any

triangle sum to .


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Summing the angles in triangleBDC, we get , or .

Since triangleADB is isosceles (two of its legs are radii of the circle), the missing angle must be

equal to .

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A Common Book of pi