Oct 26, 2007

Did you know?

Pascal's triangle is named after the French mathematician and philosopher Blaise Pascal (1623-62), who wrote a Treatise on the Arithmetical Triangle describing it. But Pascal was not the first to draw out this triangle or to notice its amazing properties!

Long before Pascal, 10th century Indian mathematicians described this array of numbers as useful for representing the number of combinations of short and long sounds in poetic meters. The triangle also appears in the writings of
Omar Khayyam, the great eleventh-century astronomer, poet, philosopher, and mathematician, who lived in what is modern-day Iran.

The Chinese mathematician Chu Shih Chieh depicted the triangle and indicated its use in providing coefficients for the binomial expansion of in his 1303 treatise The Precious Mirror of the Four Elements. Below is a reproduction of the triangle from Chu Shih Chieh, in Chinese numerals

and in our arabic numerals : (Both illustrations from Georges Ifrah, The Universal History of Numbers from Prehistory to the Invention of the Computer. New York: John Wiley & Sons, 1981, 1998.)

Pascal's work on the triangle stemmed from the popularity of gambling. A French nobleman had approached him with a question about gambling with dice. Pascal shared the question with another famous mathematician, Fermat, and Pascal's Arithmetical Triangle was the result.
Using Pascal's triangle, one can in fact find the number of ways of choosing k items from a set of n items simply by looking at the kth entry on the nth row of the triangle. So, to see how many different trios you could form using the 45 members of your jazz band, you would look at the 3nd entry on the 45th row. (The "1" at the top of the triangle is considered the "0"th row, and the first entry on each row is labeled the "0"th entry on the row.)

Since Pascal's time, mathematicians have found numerous patterns in Pascal's triangle. Some of the most interesting patterns are obtained by coloring in multiples of various numbers in Pascal's triangle; the results form endlessly repeating patterns called fractals.


In his own words:

We arrive at truth, not by reason only, but also by the heart. Pensees (1670)

It is not certain that everything is uncertain. Pensees (1670)

The excitement that a gambler feels when making a bet is equal to the amount he might win times the probability of winning it. N Rose Mathematical Maxims and Minims (Raleigh N C 1988).

Let us weigh the gain and the loss in wagering that God is. Let us consider the two possibilities. If you gain, you gain all; if you lose, you lose nothing. Hesitate not, then, to wager that He is.

The last thing one knows when writing a book is what to put first. Pensees (1670)

The more I see of men, the better I like my dog. H Eves Return to Mathematical Circles (Boston 1988).

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