Ancient and Modern Mathematics: 1 - Ancient Problems 2 - Partial Permutations

By DAT PHUNG TO

Discover modern solutions to ancient mathematical problems with this engaging guide, written by a mathematics enthusiast originally from South Vietnam. Author Dat Phung To provides a theory that defines the partial permutations as the compositions of the permutations nPn=n!. To help you apply it, he looks back at the ancient mathematicians who solved challenging problems.

Unlike people today, the scholars who lived in the ancient world didnt have calculators and computers to help answer complicated questions. Even so, they still achieved great works, and their methods continue to hold relevance.

In this textbook, youll find fourteen ancient problems along with their solutions. The problems are arranged from easiest to toughest, so you can focus on building your knowledge as you progress through the text.

Fourteen Ancient Problems also explores partial permutations theory, a mathematical discovery that has many applications. It provides a specific and unique method to write down the whole expansion of nPn = n! into single permutations with n being a finite number.

Take a thrilling journey throughout the ancient world, discover an important theory, and build upon your knowledge of mathematics with Fourteen Ancient Problems.

• Language: English
• Publisher: Trafford Publishing
• Release date: Aug 27, 2012
• ISBN: 9781466900950

DAT PHUNG TO

Dat Phung To studied mathematics at Saigon University and is a mathematics enthusiast. A native of South Vietnam, he earned the rank of major in the South Vietnamese Air Force during the Vietnam War before being imprisoned by the communists for almost ten years. He moved to the United States in 1991 via a refugee program and lives with his family in New Jersey.

Book preview

Ancient and

Modern

Mathematics

1)- Ancient problems

2)- Partial permutations

Dat Phung To

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© Copyright 2012 Dat Phung To.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written prior permission of the author.

ISBN:

978-1-4669-0094-3 (sc)

ISBN:

978-1-4669-0093-6 (hc)

ISBN:

978-1-4669-0095-0 (e)

Library of Congress Control Number: 2012909213

Trafford rev. 02/07/2013

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ForeWord Reviews

Clarion Review

EDUCATION

Ancient and Modern Mathematics: The Partial Permutations

Dat Phung To

Trafford

978-1-4669-0094-3

Five Stars (out of Five)

In Ancient and Modern Mathematics, Dat Phung To offers a refreshing postulation that mathematics can be appreciated on a more fundamental level than how it is often presented in these modern times of advanced-function calculators and whizbang computers. Dat Phung To is, in his own words, the man who loves mathematics. And while he surely is not the only one, he does prove true to his self-description throughout the pages of this gem of a book.

Ancient and Modern Mathematics is divided into two sections, Arithmetic and Geometric Problems and The Partial Permutations, two of no doubt many areas of mathematics of which the author has made a study for his own enjoyment.

The arithmetic problems include classic challenges such as The Sum of Rice in Sixtyfour Squares of Chessboard, in which a man in the Middle Ages invents the game of chess and shows the king how to play it. To reward the man, the king grants him a wish. Having always lived in poverty, the man asks for an amount of rice to be derived from placing one grain on the first square on the chessboard, two on the second, and so forth, each time doubling the number of grains of rice over the sixty-four squares. The king’s mathematician makes the calculation—without the advantage of logarithmic tables or an electronic calculator, of course. Not only is the final sum arrived at as astounding as the reader might expect, but Dat Phung To’s explanation of the most efficient calculation the king’s mathematician might have used is presented precisely and cleary.

The geometric problems presented are also classics. The first is whether a triangle having two angles bisected by line segments of equal length must be an isosceles triangle. The discussion then moves on to the euclidean theory that a straight line and a circle intersect at two points. The latter may seem intuitively obvious, and readers are told that Euclid considered it so. But the author is not satisfied until he can prove the case, and he does. His hand-drawn diagrams add authenticity to his work and again demonstrate the simple beauty of pure, diligent mathematical work.

The second section of the book begins with a look at partial permutations. The author clearly lays out the main rules and then goes on to develop his own corollary to a conventional permutations theorem, which he then applies to further expansions. More handwritten charts appear, meticulously written and very readable.

There are a few typos in the text and a couple of other flaws, such as an occasional missing word, none of which alters the book’s readability. Another round of editing would eliminate these errors for a second edition. And, the reader should be warned that the word nominator is used in place of numerator, a rare but not actually incorrect usage. The author might consider altering the subtitle to include a reference to the first section of the book as well as the second. Finally, an overall table of contents, rather than one just for each section, would be helpful.

This is without doubt the work of an inspired man. Any math teacher, especially of algebra (heavily relied on in the first several problems), geometry, or discrete mathematics should have this thought-provoking book in his or her personal library and can recommend its reading to students as preparation for some interesting class discussion. Additionally, instructors of computer science might challenge their students to convert Dat Phung To’s work into ditigal algorithms, just for the fun of it.

Tricia Morrow

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Ancient and Modern Mathematics: The Partial Permutations

Dat Phung To

Trafford Publishing, 223 pages,

(paperback) $15.86, 978-1-4669-0094-3

(Reviewed: January 2013)

In a market glutted with mathematics textbooks, Ancient and Modern Mathematics is a refreshing approach to mathematical problems that have been around since ancient days. Author Dat Phung To says, Studying the ancient problems is my great pleasure; pursuing solving them is my favorite hobby.

He gives readers more than a cursory look at these iconic mathematical challenges. At the same time, he convinces readers that the solutions can be determined through the use of paper and pencil rather than programmable calculators. After all, if the ancient mathematicians could do it sans electronics, so can the modern practitioner.

In the book’s first half, Arithmetic and Geometric Problems, the author works 11 classic arithmetic problems before delving into geometry. The first two, A Basket of Eggs and The Sum of Rice in Sixty-four Squares of Chessboard, will be familiar to most mathematicians, even students. One problem requires only a firm grasp of algebra. The other applies binomial expansion using Chinese mathematician Chu Shih-Chieh’s triangle of coefficients, which evidently predates Pascal’s by some 360 years.

In the book’s second half, The Partial Permutations, the author introduces the relevant rules, definitions, and symbols in a logical, reader-friendly manner. He then navigates through partial permutations in a traditional way and through his own unique method of expansion.

The book is charmingly laced with bits of history, but it’s the math, the meat of the book, that makes it a worthy addition to any mathematician’s library. By offering multiple solutions to each problem, the author aims to offer readers some alternative thinking. He hits the mark again and again. His diagrams befit his sensible explanations, and his charts are nothing less than a labor of love.

Students of algebra, geometry, and discrete math will appreciate Ancient and Modern Mathematics, and teachers should consider adding it as a supplement to their standard textbook fare. One hopes there are more books to come from this man who loves mathematics.

Also available in hardcover.

Preface

Dear readers, I am Dat Phung To, the man who loves mathematics. Throughout my life, I used most of my spare time to research mathematics, mostly the ancient arithmetic problems. The more I worked on them, the more I admired the ancient mathematicians who invented them. Unlike humans today who live in favorable conditions with advanced technological products, ancient people had to live in the reverse situation, but they could still achieve their great works. Therefore, I consider them to be my admirable teachers. Studying the ancient problems is my great pleasure; pursuing solving them is my favorite hobby.

I am writing this text to do something that might be interesting for those who love mathematics. My textbook consists of two parts. In the first part, there are fourteen ancient problems. In the second part, I introduce the partial permutations theory. The problems in the first part are set up in order from the easiest to the toughest. My own method improves and solves them.

I unexpectedly discovered the partial permutations in the second part while I searched the base e in the website, in which I fortunately read the Bernoulli problem about the derangement of the hats that brought me the ideas I used to develop the partial permutations theory. This theory proves that 0! = 1 is no longer a convention but a corollary of Theorem 1. It astonishes us. The theory also provides us a specific and unique method to write down the whole expansion of nPn = n! into single permutations with n being a finite number.

I hope the subject matters in my textbook do not coincide with those of other authors worldwide. Also, I hope my work will provide some pleasure to those who love mathematics, or at least it will help them to not despair.

Dat Phung To

April 30, 2011

Part 1

Ancient Problems

Contents

Chapter 1

Problem 1: A Basket of Eggs

Problem 2: The Sum of Rice in Sixty-four Squares of Chessboard

Chapter 2

Problem 3: The Hundred Fowls (1)

Problem 4: The Hundred Fowls (2)

Problem 5: The Thousand Fowls

Chapter 3

Problem 6: The Problem of Sunzi

Problem 7: The Improved Problem of SUNZI to Inspect Battalion

Problem 8: The Improved Problem of SUNZI to Inspect Regiment

Problem 9: The Improved Problem of Sunzi to Inspect Division

Problem 10: The Improved Problem of Sunzi to Inspect Corps

Problem 11: The Improved Problem of Sunzi to Inspect a Group