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# Monthly Archives: May 2011

## Free Online Calculus Text Books

Once I listed books on Algebra and Related Mathematics in this article, Since then I was recieving emails for few more related articles. Here I’ve tried to list almost all freely available Calculus texts. Here we go:

1. Elementary Calculus : An approach using infinitesimals by H. J. Keisler
2. Multivariable Calculus by Jim Herod and George Cain
3. Calculus by Gilbert Strang for MIT OPEN COURSE WARE
4. Calculus Bible by G S Gill [I found this link broken.]
5. Another Calculus Bible by Neveln
6. Lecture Notes for Applied Calculus [pdf] by Karl Heinz Dovermann
7. A Summary of Calculus [pdf] by Karl Heinz Dovermann
8. First Year Calculus Notes by Paul Garrett
9. The Calculus of Functions of Several Variables by Professor Dan Sloughter
10. Difference Equations to Differential Equations : An Introduction to Calculus by Professor Dan Sloughter
11. Visual Calculus by Lawrence S. Husch
12. A Problem Text in Advanced Calculus by John Erdman
13. Understanding Calculus by Faraz Hussain
14. Advanced Calculus [pdf] by Lynn Loomio and Schlomo Sternberg
15. The Calculus Wikibook [pdf] on Wiki Media
16. Stewart ‘s Calculus by James Stewert.
[Link removed due to copyright reasons.]
17. Vector Calculus
18. The Calculus for Engineers by John Perry
19. Calculus Unlimited by J E Marsden & A Weinstein
20. Advanced Calculus by E B Wilson
21. Differential and Integral Calculus by Daniel A Murray
22. Elements of the Differential and Integral Calculus[pdf] by W A Granville & P F Smith
23. Calculus by Raja Almukkahal, Victor Cifarelli, Chun Tuk Fan & L Jarvis

## Dedekind’s Theory of Real Numbers

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# Intro

Let $\mathbf{Q}$ be the set of rational numbers. It is well known that $\mathbf{Q}$ is an ordered field and also the set $\mathbf{Q}$ is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exists infinite number of elements of $\mathbf{Q}$. Thus the system of rational numbers seems to be dense and so apparently complete. But it is quite easy to show that there exist some numbers (?) (e.g., ${\sqrt{2}, \sqrt{3} \ldots}$ etc.) which are not rational. For example, let we have to prove that $\sqrt{2}$ is not a rational number or in other words, there exist no rational number whose square is 2. To do that if possible, purpose that $\sqrt{2}$ is a rational number. Then according to the definition of rational numbers $\sqrt{2}=\dfrac{p}{q}$, where p & q are relatively prime integers. Hence, ${\left(\sqrt{2}\right)}^2=p^2/q^2$ or $p^2=2q^2$. This implies that p is even. Let $p=2m$, then $(2m)^2=2q^2$ or $q^2=2m^2$. Thus $q$ is also even if 2 is rational. But since both are even, they are not relatively prime, which is a contradiction. Hence $\sqrt{2}$ is not a rational number and the proof is complete. Similarly we can prove that why other irrational numbers are not rational. From this proof, it is clear that the set $\mathbf{Q}$ is not complete and dense and that there are some gaps between the rational numbers in form of irrational numbers. This remark shows the necessity of forming a more comprehensive system of numbers other that the system of rational number. The elements of this extended set will be called a real number. The following three approaches have been made for defining a real number.

1. Dedekind’s Theory
2. Cantor’s Theory
3. Method of Decimal Representation

The method known as Dedekind’s Theory will be discussed in this not, which is due to R. Dedekind (1831-1916). To discuss this theory we need the following definitions:

Rational number A number which can be represented as $\dfrac{p}{q}$ where p is an integer and q is a non-zero integer i.e., $p \in \mathbf{Z}$ and $q \in \mathbf{Z} \setminus \{0\}$ and p and q are
relatively prime as their greatest common divisor is 1, i.e., $\left(p,q\right) =1$.

Ordered Field: Here, $\mathbf{Q}$ is, an algebraic structure on which the operations of addition, subtraction, multiplication & division by a non-zero number can be carried out.

Least or Smallest Element: Let $A \subseteq Q$ and $a \in Q$. Then $a$ is said to be a least element of $A$ if (i) $a \in A$ and (ii) $a \le x$ for every $x \in A$.

Greatest or Largest Element: Let $A \subseteq Q$ and $b \in Q$. Then $b$ is said to be a least element of $A$ if (i) $b \in A$ and (ii) $x \le b$ for every $x \in A$.

## Dedekind’s Section (Cut) of the Set of All the Rational Numbers

Since the set of rational numbers is an ordered field, we may consider the rational numbers to be arranged in order on straight line from left to right. Now if we cut this line by some point $P$, then the set of rational numbers is divided into two classes $L$ and $U$. The rational numbers on the left, i.e. the rational numbers less than the number corresponding to the point of cut $P$ are all in $L$ and the rational numbers on the right, i.e. The rational number greater than the point are all in $U$. If the point $P$ is not a rational number then every rational number either belongs to $L$ or $U$. But if $P$ is a rational number, then it may be considered as an element of $U$.

Def.

### Real Numbers:

Let $L \subset \mathbf{Q}$ satisfying the following conditions:

1. $L$ is non-empty proper subset of $\mathbf{Q}$.
2. $a, b \in \mathbf{Q}$ , $a < b$ and $b \in L$ then this implies that $a \in L$.
3. $L$ doesn’t have a greatest element.

Let $U=\mathbf{Q}-L$. Then the ordered pair $< L,U >$ is called a section or a cut of the set of rational numbers. This section of the set of rational numbers is called a real number.

Notation: The set of real numbers $\alpha, \beta, \gamma, \ldots$ is denote by $\mathbf{R}$.

Let $\alpha = \langle L,U \rangle$ then $L$ and $U$ are called Lower and Upper Class of $\alpha$ respectively. These classes will be denoted by $L(\alpha)$ and $U(\alpha)$ respectively.

## An Elementary Problem on Egyptian Fractions

Few math problems, specially, problems on Numbers are very interesting. In this “Note”, I’ve added a classical problem, as follow:

Solve $\dfrac{1}{w} + \dfrac{1}{x}+ \dfrac{1}{y} + \dfrac{1}{z}=1$ for $w \le x \le y \le z$, all being positive integers.

– This problem is quit easy to solve but interesting to understand steps, how it is solved. One with regular math knowledge would know that there are fourteen (14) solutions for the problem. Some where this problem is also called Egyptian Fractions Problem.

## Free Online Algebra Books

Internet is full of knowledge. There are many professors who have shared their works online. I have listed a few books on Algebra and Related Mathematics in this article. I am not very serious, but I think these are very useful for Undergraduate and Graduate Students, including me.

1. Abstract Algebra OnLine by Prof. Beachy
2. Understanding Algebra by James Brennan
3. Abstract Algebra : Theory and Applications by Tom Judson
4. Elements of Abstract and Linear Algebra by E H Connell
5. Linear Algebra by Jim Hefferon
6. Elementary Linear Algebra by Keith Matthews
7. Linear Algebra, Infinite dimensions and Mapleby James Herod
8. Elementary Number Theory by William Stein
9. Abstract Algebra: The Basic Graduate Year, A Course in Algebraic Number Theory, and, A Course in Commutative Algebra are three ebooks by Robert Ash and are available here on his website.
10. A Course in Universal Algebra by Stanley Burris & H. P. Sankappanvar
11. An Introduction to the Theory of Numbers by Leo Moser
12. A Computational introduction to Number Theory and Algebra by Victor Shoup
13. Sets, Relations, Functions by Ivo Duentsch and Günther Gedigo
14. Group Theory by Pedrag Civitanovic
15. Linear & Multilinear Algebra by C C Wang & R M Bowen
16. Abelian Categories by Peter Freyd
17. Categories and Groupoids by P. J. Higgins
18. Lie Algebras by Prof. Sternberg

All books, except few are in Acrobat Portable Document Format. You may need acrobat pdf reader to read these. However, if you don’t have that- or hate pdf readers – try online pdf/ps reader at view.samurajdata.se.

This list is expandable. If you know any other book on Algebra which is available online for free, then please give a few seconds and put that into the Comment-Box below, with the link. (It supports HTML.)