CALCULUS Understanding Its Concepts and Methods

Nicolas Chuquet (~1445--1488) --- Historical Sketch

Nicolas Chuquet was born in 1445 in Paris and died in Lyon at about age 43. The exact dates of his birth and death are not known. In fact, not very much at all seems to be known about him. His bachelor's degree was in medicine, but mathematics occupied much of his interest. He moved to Lyon at about age 35, where he described himself as an algoriste, a word we don't use today, but surely it meant a person who worked with algorithms.

Chuquet wrote a truly significant manuscript in 1484, called Triparty en la science des nombres, which means something like A three-part book on the science of numbers. Part one covers the Hindu-Arabic numeration system, zero, positive and negative numbers, fractions, order, averages, perfect numbers, progressions, etc. Chuquet even got around to the fact that which is today associated with what are called Farey fractions. Part two introduces roots of numbers and contains the first use of the radical sign, even getting to compound roots like , and indexed radicals like for cube roots. Part three is algebra, introducing unknowns and exponential notation, including negative exponents and the exponent , the equation , laws of exponents, quadratic equations and their roots. His discussion of powers of two amounted to what today are called logarithms base two. He treated what are now called imaginary numbers as solutions of equations. This 1484 manuscript did not appear in print until 1880.

There was a bit of ado concerning one of Chuquet's students, Estienne de La Roche, who taught arithmetic for 25 years and published some of Chuquet's Triparty as his own work. It was later discovered that La Roche borrowed from other authors as well, and acknowledged them namelessly as "masters, experts in the art."

Demonstrate with some examples, with positive integers, that if then . Can you prove this holds in general?

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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.