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The Whole Truth About Whole Numbers

Sylvia Forman and Agnes M. Rash
Publication Date: 
Number of Pages: 
[Reviewed by
Charles Ashbacher
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Several years ago when I was teaching college mathematics, I taught a course designed to give future teachers of elementary and middle school the basics of mathematics. It became clear very quickly how important that course was, for there were basic concepts covered in that class that were essential knowledge the students did not have. It was a very fun class for me to teach, for there were many times when the light bulb would shine and the students would say something like, “So that’s why that works.”

My favorite day was when I used the clock on the wall and a calendar showing the days of the week to introduce modulus notations and the operations. All I had to do was start the count at zero instead of one and the students had it down. I then moved on to say, what if there were five days in the week and 13 hours in a day?

I was reminded of that class when reading this book; it was once again clear how important these fundamentals are to teachers. The authors have developed a very readable introduction to number theory that will give the prospective teacher a solid understanding of why the integers and the operations on them have the behaviors we encounter.

This material is so fundamental and so well presented that if I was still teaching college mathematics I would recommend that a course based on this book be required for all potential teachers. For those on the track to teach high school math, this would be the first course in their math sequence.

One of the best features of this book is the section that introduces proofs, a topic that terrifies almost every education student and many math students. Showing how proofs are done using the integers allows the student to move into unfamiliar territory riding on a familiar mount. The chapter demonstrating the simple proofs would be a worthy start for any course that is based on developing and understanding mathematical proofs.

Review exercises and activities appear at the end of the chapters and solutions to the odds are included at the end. Basic number theory is something that nearly all people can learn and this book is a great resource for doing that.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, and teaching college classes. In his spare time, he reads about these things and helps his daughter in her lawn care business.


Conjectures, Proofs and Counterexamples

Pythagorean Triples

Prime Numbers

The Division Algorithm and the Euclidean Algorithm


Numerical Functions and Special Congruences