Electronic Transactions on Numerical Analysis
its book of mathematics on numerical analysis.
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.
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.
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.
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.
20 Probability and the Law of Addition
We are interested in the probabilities of events. Whatever events are, they are not
numbers. Therefore, given events A and B, if I speak of the event A + B, it does not mean A
plus B. Rather it is shorthand for A or B. Elsewhere,
11 Division Mod n - Free eBook 11 Division Mod n - Download ebook 11 Division Mod n free