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Pascal's triangle, first six rowsPascal'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

"Pascal's" triangle from Chu Shih Chieh, 1303

and in our arabic numerals:

Chuh Shih Chieh's triangle, with 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.


Portrait of Blaise Pascal

 

Blaise Pascal.

(Source: http://www-groups.dcs.st-andrews.ac.uk/~history/PictDisplay/Pascal.html)


 

The Pascaline calculating machinePascal made several other important contributions to the history of mathematics, including the first digital calculator, which he designed to help his father, who was a tax collector. Adding French currency was difficult, because the currency consisted of livres, sols, and deniers, with 12 deniers in a sol and 20 sols in a livre.
Pascal's machine, called the Pascaline, never was a great success, however. Fifty prototypes were manufactured, but the machine did not sell well--perhaps because the only arithmetical function it could perform was addition!

(Illustration source: http://starform.infj.ulst.ac.uk/Billsweb/PGCert/intranets/Graham/Assignment/GIFS/Pascul.htm)


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).

I have made this letter longer than usual, because I lack the time to make it short.

(Source for quotations: http://www-groups.dcs.st-andrews.ac.uk/~history/Quotations/Pascal.html)

 


Fun links:

Patterns in Pascal's triangle, with an applet to allow you virtually to color entries divisible by x: http://www.mat.bham.ac.uk/funfair/java/pascal/

Probability Plaza, with applets for calculating permutations and combinations and a Pascal's triangle row generator: http://library.thinkquest.org/C006087/english/interactive.shtml

More coloring: http://ted.educ.sfu.ca/people/students/jane/act1.html

And: http://www.cs.washington.edu/homes/jbaer/classes/blaise/bigblaise.html

And: http://www.cecm.sfu.ca/cgi-bin/organics/pascalform

Pascal's triangle generator; you specify how many rows: http://forum.swarthmore.edu/~ken/pascal.cgi

A poem on Pascal's triangle: http://www.contrib.andrew.cmu.edu/~dr4b/writing/newstuff/sestina

Generate your own Pascal's triangle and check it on line: http://www.educ.sfu.ca/people/students/jane/

Use Pascal's triangle to compute the number of presents received in the 12 days of Christmas: http://dimacs.rutgers.edu/~judyann/LP/lessons/12.days.pascal.html


For more information:

http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Pascal.html

http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/pascal_bla.html

http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html

http://www.britannica.com/eb/article?eu=114515

http://www.math.sfu.ca/histmath/Europe/17thCenturyAD/Blaise.htm

http://www.maa.org/mathland/mathland_2_10.html

http://www.math.usouthal.edu/~brick/teaching/math110/pascal-tri.html

Pascal's triangle and probability: http://www.peddie.org/Resources/MathJournal/binomial.htm

And: http://ted.educ.sfu.ca/people/students/jane/prob.html

On the Pascaline (or Pascal's calculating maching): http://starform.infj.ulst.ac.uk/Billsweb/PGCert/intranets/Graham/Assignment/History3.htm

And: http://www.maxmon.com/1640ad.htm


By Laura Smoller, UALR Department of History.

March 2001.