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10 Euclidean Algorithm

EBook Name10 Euclidean Algorithm
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EBook Category Category: Mathematics eBooks

EBook Description: Number theory is the mathematics of integer arithmetic. In this chapter we will restrict ourselves to integers,

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Electronic Transactions on Numerical Analysis

its book of mathematics on numerical analysis.

11 Division Mod n

Let's look at the values of 4x + 6y when x and y are integers. If x is -6 and y is 4 we get zero. If x is !1 and y is 1 we get 2. In fact a little experimentation will convince you that you can get all the even integers but only even integers.

12 Chinese Remainder Theorem

Many classroom exercises involve dealing cards. In this chapter we will focus on a simple problem: Write an algorithm to randomly select one card out of an ordinary 52-card deck. My students frequently derive an efficient algorithm to solve this problem. The algorithm goes as follows we use a random number generator to select a number between 1 and 52 (or between 0 and 51; either way works fine).

13 Euler's Theorem and Fermat's Little Theorem

The formulas of this section are the most sophisticated number theory results in this book. The reason I am presenting them is that by use of graph theory we can understand them easily. Fermat was a great mathematician of the 17th century and Euler was a great mathematician of the 18th century.

14 Wilson's Theorem

Wilsonís Theorem is elegant. It is not very useful, but like a lot of other people, I like it. So that is why it is here. Consider an integer n > 1. If the integer n-1! + 1 is divided by any number from 2 to n-1, it yields a remainder of 1.

15 Counting

Let us examine a real-life problem. As a university professor I feel that teaching twenty students for one hour a week is too much of a demand on my time and that it cuts into my research (on sour mash). Therefore, in order to cut back on my teaching load in future classes, I have decided to flunk most of my students this semester. Specifically, I am going to give one A, one B, one C, one D, and I'll flunk everyone else.

16 Multinomials

Let us slightly generalize the problem of selecting r out of n objects without respect to order. Suppose, for example, that I have 20 students.

17 Stirling Numbers of the Second Type

Again: a partition of n objects is a division of these objects into separate classes. Each object must be in one and only one class and partitions with empty classes are not allowed.

18 Matrices

Matrix algebra is also known as linear algebra and it is one of the most important disciplines of mathematics. It pervades virtually all of mathematics, but it could be argued to be closest to geometry. In this chapter, we will look at the relation between matrices and graphs.

19 Boolean Algebra

It has been known for sometime in mathematics that set algebras and Boolean1 algebras are different perspectives on the same thing. The treatment of sets here is informal and is known as naive set theory. Most of the time naive set theory is sufficient for the purposes of even professional mathematicians.

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