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Arithmetical Wonderland is intended as an unorthodox mathematics textbook for students in elementary education, in a contents course offered by a mathematics department. The scope is deliberately restricted to cover only arithmetic, even though geometric elements are introduced whenever warranted. For example, we showcase what the Euclidean Algorithm for finding the greatest common divisors of two numbers has to do with Euclid.
Many students find mathematics somewhat daunting. It is our belief that much of that is caused not by the subject itself, but by the language of mathematics. In this book, much of the discussion is in dialogues between Alice, of Wonderland fame, and the twins Tweedledum and Tweedledee who hailed from Through the Looking Glass. The boys are learning High Arithmetic or Elementary Number Theory from Alice, and the reader is carried along in this academic exploration. Thus many formal proofs are converted to soothing everyday language.
Nevertheless, the book has considerable depth. It examines many arcane corners of the subject, and raises rather unorthodox questions. For instance, Alice tells the twins that six divided by three is two only because of an implicit assumption that division is supposed to be fair, whereas fairness does not come into addition, subtraction or multiplication. Some topics often not covered are introduced rather early, such as the concepts of divisibility and congruence.
Great care is exercised in limiting the number of results labeled Theorems. This focuses the attention of the students on what is really important. Notation is introduced sparingly and with great care. It should be mentioned that one of the highlights of the book is the new notations for greatest common divisor and least (positive) common multiple. We leave it to the reader to look this up.
Preface to a Preliminary Edition
Introduction
Review of Arithmetic
Counting Numbers
Integers
Inequalities
Extras
Divisibility
Basic Properties of Divisibility
The Arithmetic of Divisibility
Divisibility Problems
Extras
Congruence
The Division Algorithm
Basic Properties and Arithmetic of Congruence
Congruence and Divisibility
Extras
Common Divisors and Multiples
Greatest Common Divisors and the Euclidean Algorithm
Relatively Prime Numbers
Least Common Multiples
Extras
Linear Diophantine Equations
B´ezoutian Algorithm
Homogeneous and Non-homogeneous Equations
Linear Diophantine Problems
Extras
Prime Factorizations
Prime and Composite Numbers
Fundamental Theorem of Arithmetic
Applications of the Fundamental Theorem of Arithmetic
Extras
Rational and Irrational Numbers
Fractions
Decimals
Real Numbers
Extras
Numeration Systems
Arithmetic in Other Bases
Conversion between Bases
Applications of Other Bases
Extras
Appendix
A Legacy of Martin Gardner
Solution to Odd-numbered Exercises
Index
About the Author