Pi (π) is the number you get when you divide a circle's circumference by its diameter—a number that is the same for a circle of any size. Pi can't be expressed exactly as a ratio of whole numbers. Indeed, starting with 3.14159…, the decimal digits of pi go on forever.
Statistically, the digits of pi appear to behave like a sequence of random numbers. Over the years, those digits have been the subject of considerable scrutiny and an astonishing amount of dedicated memory work.
Some of you may be familiar with the sentence: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics! If you write under each word the number of letters it contains, you end up with 3.14159265358979, the first 15 digits of pi.
This sentence appeared in a short news item printed in the Dec. 14, 1929, issue of Science News-Letter (which later became simply Science News). The weekly magazine, in turn, cited The Observatory, an English astronomical journal, as the sentence's source, where the author was given as J.H.J. The initials happen to be the same as those of James H. Jeans, a leading British astronomer.
This item was not the first mention in Science News-Letter of a mnemonic device for remembering the digits of pi. The Oct.. 9, 1926, issue featured the following poetic tribute to Archimedes. It gives pi to 30 decimals.
Now I, even I, would celebrate
In rhymes inept, the great
Immortal Syracusan, rivaled nevermore
Who, in his wondrous lore,
Passed on before,
Left men his guidance how to circles mensurate.
Readers were then invited to contribute any other such "memory rimes" they had composed or found useful. One contributor sent in a French sentence for remembering the value of pi to 10 digits: Que j'aime à faire apprendre le nombre utile aux sages.
As noted in the Nov. 27, 1926, issue of Science News-Letter, this example is actually the first line of a four-line poem encoding 30 decimal places of pi. The full poem originally appeared in a Belgian mathematics journal in 1879.
Over the years, pi enthusiasts have created mnemonic devices encoding pi in just about any language you can imagine—from ancient Greek to modern Armenian. These sentences, poems, miniature dramas, comic episodes, and so forth reflect not only the digits of pi but also the considerable ingenuity of their authors.
Many go no further than the 30th decimal digit, perhaps because of the first appearance of zero, as the 32nd decimal digit, necessitates some new rule—such as using 10-letter words—to continue the sequence.
Nevertheless, some inventive souls have ventured well beyond the 30th decimal digit of pi. In one astounding effort, software engineer, amateur mathematician, and pi fanatic Mike Keith encoded 740 digits in a lengthy poem modeled on Edgar Allan Poe's "The Raven." He later topped that effort with a complete short story in which the number of letters of each successive word gives the first 3,835 digits of pi.
Nevertheless, some inventive souls have ventured well beyond the 30th decimal digit of pi. In one astounding effort, software engineer, amateur mathematician, and pi fanatic Mike Keith encoded 740 digits in a lengthy poem modeled on Edgar Allan Poe's "The Raven." He later topped that effort with a complete short story in which the number of letters of each successive word gives the first 3,835 digits of pi.
New mnemonics continue to surface. In the April 1999 issue of Math Horizons, Mimi Cukier suggested the following sentence for remembering the first 22 digits of pi: Wow! I have a great technique to recall those fun, crazy numerals composing perhaps everyone's all-in-all favorite real number—Pi!
Mathematician and magician Arthur T. Benjamin responded to that article with his own suggestion for a better way to memorize pi, which appeared in an article in the February 2000 Math Horizons. Benjamin began his article with the sentence: How I wish I could elucidate to others: there are often superior mnemonics!
Benjamin then went on to suggest how a phonetic code, which replaces digits with consonant sounds, is superior to traditional mnemonic devices for remembering strings of digits.
Benjamin recommended the following phonetic code, which has been around for more than 140 years: 1 = t or th or d; 2 = n; 3 = m; 4 = r; 5 = l; 6 = sh, ch, or j; 7 = k or hard g (as in goat); 8 = f or v; 9 = p or b; 0 = s or z.
"A quick way to memorize the code was suggested to me by Tony Marloshkovips," Benjamin hinted.
By placing vowel sounds between consonants, numbers can be turned into words. For example, the first 24 digits of pi can be translated into My turtle Pancho will, my love, pick up my new mower Ginger.
"Invest just a little bit of time to master the code…, and soon you will be able to rattle off 60 digits of pi in no time!" Benjamin insisted.
I came across a remarkable memory feat involving pi when I was researching the 1995 discovery by David H. Bailey, Peter Borwein, and Simon Plouffe of a truly fantastic formula for computing any given hexadecimal (base 16) digit or binary digit of pi without being forced to calculate the preceding digits. Plouffe, Borwein, and Bailey then used the novel algorithm to establish that the 400 billionth binary digit of pi is 0.
Plouffe once held the world title for memorizing decimal digits of pi. He managed to commit a total of 4,096 digits to memory, an achievement that was duly recognized in the 1977 French edition of the Guinness Book of World Records.
Actually, Plouffe had memorized 4,400 digits but settled on 4,096 (212) as a nice round number to report to others interested in his feat. Back then, "I was young and I had not much else to do, so I did it," Plouffe recalled. He liked numbers and was fascinated by pi.
To Plouffe, memorizing the digits of pi was close to a mystical experience. He worked with blocks of 100 digits. He started by writing out a block five or six times. He then recited these digits in his head. To preserve the numbers in his long-term memory, he periodically isolated himself in a room—no lights, no noise, no coffee, no cigarettes. "Like a monk," Plouffe said.
As Plouffe recited the digits to himself, they would gradually seep into his mind. After a day or two, he would be ready to go on to the next block. When Plouffe got to 4,400 he decided to stop. "You can continue…forever," he explained. "You stop mainly because it is boring to do that all the time."
Two years later, the person who had held the previous record of 3,025 digits came back with 5,050 memorized digits. "I knew I could beat him, but…I had had enough," Plouffe said. In 2005, the record stood at 67,890 digits!Having a good memory for numbers and the ability to recognize them by sight proved useful to Plouffe in his mathematical work, which often involved looking for relationships between different mathematical series or among various number sequences.
Plouffe was the coauthor, with Neil J.A. Sloane, of The Encyclopedia of Integer Sequences, which contains nearly 6,000 examples of number sequences, collected from a variety of sources. Mathematicians and other researchers have used the book, now greatly expanded in an online database, as a reference for counting or tabulating things that involve number sequences, from the number of atoms in various molecules to different types of knots.
Plouffe has also developed software for doing automatically the kind of numerical pattern recognition that he himself did so naturally.
Suppose you happen upon the number 1.618033987. It looks vaguely familiar, but you can't quite place it. You can use Plouffe's Inverter (PI) to find whether this particular number is special in some way, perhaps as the output of a specific formula or the value of a familiar mathematical function or constant. In the case of 1.618033987, the PI database search produces a page of formulas and functions that generate the number. The most intriguing possibility is the expression (1 + √5)/2, which represents the golden ratio.
The PI database contained more than 200 million entries, making it possible to identify all kinds of "special" numbers. But there's a catch. Given a formula or expression such as 2 + 2, there's only one answer, 4. But, given the result 4, there are actually lots of different ways to get there besides 2 + 2.
Thus, it can become tricky to sift the "true" formula from a coincidental expression extracted from the database. The hazard is greatest when only a small number of digits is used and the number is truncated or rounded off.
Of all known mathematical constants, however, pi continues to attract the most attention. Indeed, the pi craze can sometimes take on unusual or unexpected forms. A while ago, the fragrance industry discovered "math appeal" when Parfum Givenchy introduced a men's cologne dubbed Pi.
Pi also made it to the big screen as the title of a thriller in which an eccentric mathematician unlocks the secret of the stock market in the digits of pi. The Exploratorium in San Francisco pioneered the celebration of Pi Day on March 14 each year, starting at 1:59 p.m., and continues the tradition.
There is something delightfully irrational about the enduring interest in—or perhaps obsession with—pi.
"Pi is one of the few concepts in mathematics whose mention evokes a response of recognition and interest in those not concerned professionally with the subject," Len Berggren, Jonathan Borwein, and Peter Borwein wrote in Pi: A Source Book. "It has been a part of human culture and the educated imagination for more than twenty-five hundred years."
"The computation of pi is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modern mathematical research," the authors continued. "And to pursue this topic as it developed throughout the millennia is to follow a thread through the history of mathematics that winds through geometry, analysis and special functions. numerical analysis, algebra, and number theory.
"It offers a subject which provides mathematicians with examples of many current mathematical techniques as well as a palpable sense of their historical development."
"This is a field of endeavor that has attracted some of the greatest minds of mankind," mathematician Dario Castellanos wrote in a 1998 article in Mathematics Magazine about the never-ending fascination with the number pi. (See Malcolm W. Browne's New York Times report about Castellanos's paper.)
"The studious pursuer of the many curious and fascinating properties which surround this number," Castellanos said, "will forever meet new results and new algorithms related to 'the mysterious and wonderful pi.'"
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