CALCULUS Understanding Its Concepts and Methods
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Blaise Pascal (1623--1662) --- Historical Sketch
Blaise Pascal was born in the Summer of 1623 in Clermont (now Clermont-Ferrand), Province of Auvergne, in the center of France. And in Clermont-Ferrand, there is today the Université Blaise Pascal, a comprehensive university with faculties of Arts, Languages, The Humanities, Exact and Natural Sciences, and Technology. It is apparently the only university named in honor of a mathematician.
Young Blaise was the youngest of three children in the family, and the only boy. His mother died when he was but three years old. His father Étienne, a local judge who was himself interested in science and mathematics, retired and moved with his family to Paris, where he undertook to tutor his son, who had demonstrated early that he had exceptional mental abilities. The move was almost surely inspired by the fact that Étienne had several mathematician friends in Paris, including René Descartes, Gérard Desargues, Marin Mersenne, and Gilles Personne de Roberval. But Étienne did not want Blaise to study mathematics until he had studied classical languages, until one day Blaise at age eleven demonstrated that mathematics was his natural forte. So the young boy was then given Euclid's Elements, which he read and soon had conquered.
The geometric works of Desargues in projective geometry fascinated Pascal, and he wrote an original paper on conic sections, including the result now known as Pascal's Theorem concerning hexagons. Pascal also worked on problems in physics, invented a mechanical calculator, rediscovered what is today called Pascal's Triangle (which was known some 500 or so years earlier by a Chinese mathematician named Yanghui---in China today, this familiar array is called Yanghui's Triangle---and also by the Persian Omar Khayyam). And in relation to Pascal's Triangle, he also contributed to the mathematics of probability.
Studying the spiral of Archimedes, Pascal also contributed to ideas important to the development of integral calculus.
But Pascal had serious health problems, and he was in a devastating accident in a horse-drawn carriage which almost cost him his life. Convinced that this was a message from God, and always somewhat of a mystic, he devoted the remainder of his life to prayer. It is during this period that he developed what is now called Pascal's Wager: If God does not exist, one will lose nothing by believing so, whereas if God does exist, one will lose everything by not believing so.
When only 39 years old, an abdominal growth spread to his brain, and Pascal died.
Pascal's Triangle is usually written in the form that beginsand students are directed to look at the rows for patterns (how to construct each row from the previous one, binomial coefficients, etc.). But now look at Pascal's Triangle when written in this formand describe patterns that you now see and which weren't quite so visible to you when written in the first form. You will probably need to expand the second form to let some of the patterns reveal themselves.
If you compute the sum of the elements in each row of Pascal's Triangle, what do you get?
If you compute the alternating sum (i.e., alternating adding and subtracting: . . . ) of the elements in each row of Pascal's Triangle, what do you get?
You very likely never think of Pascal's Triangle as being associated with approximations, a subject that is found throughout calculus. But look at this: To find successive fractional approximations to where is a positive number, compute these fractions that are associated with each row of Pascal's Triangle where the numbers in the row occur alternately in the numerator and denominator of the corresponding fraction to the right of the Pascal row:
a.See what the successive approximations are for .
b.What are the next two rows in the above array?
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Factorial, Binomial coefficient, and Pascal's triangle
Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.