### Topics in Arithmetic and Elementary Mathematics

The materials below were used in a course for prospective elementary school teachers. I did not do any lecturing in the course. The students worked on the problem sets in groups of five, and I circulated among the groups and gave help as appropriate.

Many students in the course had a background consisting of only high school algebra, and many of them remembered very little even of that. In the past, the materials for the course had been not much more than a bunch of definitions. I was determined that my students should learn something of real substance.

My main objective, though, was not to teach my students any particular subject matter. What I hoped to accomplish was to teach them how to learn mathematics, and -- even more important -- how to be interested in mathematics. I tried to do as little as possible in the course that was routine, or boring.

About once every two weeks I gave an assignment consisting of non-routine word problems based on elementary algebra and logic. (For instance the problem of the missionaries and the cannibals.)

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One of the ironies of courses like these is that many of the topics -- such as the Euclidean algorithm for finding the greatest common divisor of two integers, or Horner's Method for evaluating polynomials -- are are never learned by many science majors because everything in the usual mathematics curriculum is oriented so strongly around calculus.

A lot of the files listed below are in PDF (Adobe Acrobat) format. Alternate versions are in DVI format (produced by TeX; see see here for a DVI viewer provided by John P. Costella) and postscript format (viewable with ghostscript.) Some systems may have some problem with certain of the documents in dvi format, because they use a few German letters from a font that may not be available on some systems. (Three alternate sites for DVI viewers, via FTP, are CTAN, Duke, and Dante, in Germany.)

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## Representing Numbers in Other Bases

The objective is for students to learn the most efficient algorithm for converting numbers from another base to base 10 (essentially Horner's Method) and from base 10 to an alien base, and to understand why these algorithms work.
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## Divisibility Tests

This explains why the simple tests for divisibility by 2, 3, 4, 5, 9, and 11 work.
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## More Divisibility Tests

Testing divisibility by 7. Divisibility by composite numbers. Determining whether a three-digit number is prime.
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## Periodic Decimal Expansions

The repeating pattern for some fractions, such as 1/13, begin immediately following the decimal point. But for some others, such as 1/15, there is some non-repeating stuff that comes first. The first objective in this problem set is to get students to see that what makes the difference is whether or not the denominator has a factor of 2 or 5 (i.e. whether or not it is relatively prime to 10).
Next, we see that if n is relatively prime to 10 and a/n has a decimal period of r, then n must divide 99...9, where there are r 9's. In other words, it must divide 10^r-1.
From here, it is easy to find all fractions 1/n which have a decimal expansion of length 1, 2, 3, 4, 5, 6, 8, or 9. (There are surprisingly few possible denominators n.)
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## The Period for the Decimal Expansion of a Fraction

Mostly I wanted to get students to see why the period of the decimal expansion for 1/p, where p is prime, must divide p-1.
The period of the decimal expansion is the same as the order of 10 module p, i.e. the smallest r such that 10^r is congruent to 1 modulo p. This is easily seen if you look at the long division that determines the decimal expansion and then ignore the decimal point.
From this, it is easy to see that the period divides p-1, provided that one knows Fermat's Little Theorem and a little group theory. My objective, though, was to be able to prove this without using either of these two results. The idea was to use the fact, easily shown, that if you do a circular permutation on the decimal expansion of a fraction a/p, with p prime and a strictly between 0 and p, then you get another such fraction b/p. For instance, 1/13=.076923... (repeats from this point). If one applies a circular permutation to get, say, .692307..., one finds that this is 9/13 (the reason being that 9 is 10^2 modulo 13).
It can be shown that for given p (prime) all the circular patterns for fractions a/p have the same length r. Thus if 1/p has period r, then each circular pattern for a decimal expansion with denominator p accounts for r of the total p-1 possible fractions a/p, with a between 1 and p-1. Thus if there are t different circular patterns, then tr=p-1. This shows that r divides p-1.
For instance, the decimal expansion for 1/37 has a circular pattern of length 3, since 1/37=.027027... (Note that 37 is a prime.) There are three different fractions 1/37 with this same circular pattern, namely 1/37, 10/37=.270270..., and 26/37=.702702... (Note that 26 is 100 modulo 37.) Thus for p=37, r is 3, and tr=37-1=36, so t, the number of different circular patterns for fractions of the form a/37 (with a between 1 and 36), must be 12. Besides 027, some of these 12 circular patterns are 054 (2/37=.054054...), 081 (3/37=.081081), and 135 (5/37=.135135...).
(That's about the best explanation I can give in HTML.)
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## Irrational Numbers in General and Square Roots in Particular

First, I want students to see that there exist numbers whose decimal expansions are not periodic. These numbers must therefore be irrational.
Next it is shown that the square root of an integer is irrational unless that integer is a perfect square. The key idea is that if one squares a fraction which is in lowest terms, then the result will still be in lowest terms. From this it is easy to see that the square of a rational number which is not an integer will never be an integer. But that's just another way of saying that the square root of an integer cannot be a rational number -- unless it's an integer.
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## The Lazy Man's Arithmetic

This problem set gives students a review of some basic principles of algebra by showing how they can be used as shortcuts in arithmetic.
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## Bride of the Lazy Man

``Russian Peasant'' multiplication.
Rather than explain this in terms of the binary representation of numbers, it is used to motivate binary notation.
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## The Lazy Man Helps His Neighbor

Horner's Method for evaluating polynomials. This is the same as synthetic division.
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## Mr. Tinker Learns About the Mouse People

Using Horner's method to change numbers from foreign bases to base 10.
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## The Lazy Man Finds Some Winning Combinations

Solving simple linear Diophantine equations.
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## The Lazy Man Gets to the Root of Things

Newton's method for computing square roots.
Proof that square roots of most integers are irrational.
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