pi (π)

The ratio of a circle’s circumference to its diameter, about 3.14159.

drawing of a circle

 = π ×

 = π × 

Enter either a circumference or a diameter. The calculator is sensitive to the precision of the measurement. If you want more decimal places in the answer, add trailing zeroes to your entry to indicate a more precise measurement.

According to Cajori, the symbol π was first used for this number in 1706¹, but its widespread acceptance is due to Leonard Euler’s adopting it in 1736.

Calculating digits

3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067 982 148 086 513 282 306 647 093 844 609 550 582 231 725 359 408 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930 381 964 428 810 975 665 933 446 128 475 648 233 786 783 165 271 201 909 1..... It goes on forever.

A mnenomic for the first 15 digits of pi (count the letters in each word) is “How I like a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”²

In July 1997, Yasumasa Kanada and colleagues at the University of Tokyo Computer Centre announced that they had calculated pi to 51,539,600,000 decimal places. Actually, the number was calculated twice, by two different machines using different software, and the results compared as a check on accuracy. It took one machine 29 hours and the other 37.

Pi to many, many digits can be downloaded from a number of Web sites, formerly at Kanada’s own to 4,200,000,000 decimal digits. Fewer digits but more accessible is Simon Fraser University’s:

 www.cecm.sfu.ca/organics/papers/borwein/paper/html/local/billdigits.html.

Pi cannot be expressed as a fraction

In everyday life calculations involving pi have used a fractional approximation:

Early Values for Pi
Value Used where?
258 Babylon
the square of the fraction sixteen over nine Ancient Egypt
between 22371 and 227 Archimedes
377120 Ptolemy, about 160
square root of ten India
864275 Fibonacci, 1220

Even our decimals, to however many places, are really decimal fractions. These approximations are convenient, but in 1766 Lambert proved that pi cannot be expressed as the ratio between two whole numbers. This property makes it an irrational number, like the square root of two.

Pi is not a root of any algebraic equation

Some irrational numbers, such as the square root of two, can be the root of an algebraic equation (one like “ax² + bx + c = 0”; the square root of two is a root of x² - 2 = 0). Pi cannot, and that property makes it a transcendental number.

1. In:
William Jones.
Synopsis palmariorum matheseos.
London, 1706.
Page 263.

2. C. B. Boyer.
A History of Mathematics.
New York: Wiley, 1968.

Wonderful equations

François Viète (circa 1600)

2 over pi equals the product of an infinite series of fractions, all of whose denominators are 2. The numerator of the first fraction is the square root of 2; of the second fraction, the square root of the quantity 2 plus the square root of 2;the numerator of the third is the square root of the quantity 2 plus the numerator of the previous fraction, and so on

Press start to calculate pi as the result of an ever-increasing number of terms in Viete's equation.

after calculating .
The is over 2.

Note: JavaScript can handle a maximum of 17 digits in a number. By the 27th term the numerator has exceeded 0.9999999999999999 and is rounded off to 1, after which there is no further change in the output. That’s an effect of the program, not the equation. But compare how quickly pi’s digits appear using this equation with the performance of Wallis’ and Gregory’s.

William Brouncker, 1655

4 over pi equal 1 plus 1 squared over 2 plus 3 squared over 2 plus 5 squaredover 2 plus 7 squared over and so on

John Wallis, 1655

pi over 2 equals 2 over 1, times 2 over 3, times 4 over 3, times 4 over 5, times 6 over 5, times 5 over 7, and so on

Press start to calculate pi as the result of an ever-increasing number of terms in Wallis's equation.

after calculating .
The

James Gregory, 1671 (or maybe Leibniz)

pi over 4 equals 1 one minus one third plus one fifth minus one seventh plus one nineth minus one eleventh and so on

Press start to calculate pi as the result of an ever-increasing number of terms in Gregory's equation.

Known value: 3.141592653589793238462643383279

.

further reading

A nice collection of links is David Blattner's www.joyofpi.com.

Petr Beckmann.
The History of Pi.
Boulder, CO: The Golem Press, 1971.

Reprinted by St. Martins Press, 1976.  Opinionated and delightful.

David Blattner
The Joy of Pi.
New York: Walker and Co., 1997.

Alfred S. Posamentier and Ingmar Lehman.
π: A Biography of the World's Most Mysterious Number.
Prometheus Books, 2004.

Y. E. O. Adrian.
The Pleasures of Pi,e.
World Scientific, 2007.

Tamar Friedmann and C.R. Hagen.
Quantum mechanical derivation of the Wallis formula for pi.
Journal of Mathematical Physics, November 10, 2015.
doi:10.1063/1.4930800

www.numberworld.org/misc_runs/pi-5t/details.html

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