- Elementary Calculus : An approach using infinitesimals by H. J. Keisler
- Multivariable Calculus by Jim Herod and George Cain
- Calculus by Gilbert Strang for MIT OPEN COURSE WARE
- Calculus Bible by G S Gill [I found this link broken.]
- Another Calculus Bible by Neveln
- Lecture Notes for Applied Calculus [pdf] by Karl Heinz Dovermann
- A Summary of Calculus [pdf] by Karl Heinz Dovermann
- First Year Calculus Notes by Paul Garrett
- The Calculus of Functions of Several Variables by Professor Dan Sloughter
- Difference Equations to Differential Equations : An Introduction to Calculus by Professor Dan Sloughter
- Visual Calculus by Lawrence S. Husch
- A Problem Text in Advanced Calculus by John Erdman
- Understanding Calculus by Faraz Hussain
- Advanced Calculus [pdf] by Lynn Loomio and Schlomo Sternberg
- The Calculus Wikibook [pdf] on Wiki Media
- Stewart ‘s Calculus by James Stewert.
[Link removed due to copyright reasons.]
- Vector Calculus
- The Calculus for Engineers by John Perry
- Calculus Unlimited by J E Marsden & A Weinstein
- Advanced Calculus by E B Wilson
- Differential and Integral Calculus by Daniel A Murray
- Elements of the Differential and Integral Calculus[pdf] by W A Granville & P F Smith
- Calculus by Raja Almukkahal, Victor Cifarelli, Chun Tuk Fan & L Jarvis
Let be the set of rational numbers. It is well known that is an ordered field and also the set is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exists infinite number of elements of . 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., etc.) which are not rational. For example, let we have to prove that 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 is a rational number. Then according to the definition of rational numbers , where p & q are relatively prime integers. Hence, or . This implies that p is even. Let , then or . Thus is also even if 2 is rational. But since both are even, they are not relatively prime, which is a contradiction. Hence 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 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.
- Dedekind’s Theory
- Cantor’s Theory
- 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:
Ordered Field: Here, is, an algebraic structure on which the operations of addition, subtraction, multiplication & division by a non-zero number can be carried out.
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 , then the set of rational numbers is divided into two classes and . The rational numbers on the left, i.e. the rational numbers less than the number corresponding to the point of cut are all in and the rational numbers on the right, i.e. The rational number greater than the point are all in . If the point is not a rational number then every rational number either belongs to or . But if is a rational number, then it may be considered as an element of .
Let satisfying the following conditions:
Let . Then the ordered pair 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 is denote by .
Let then and are called Lower and Upper Class of respectively. These classes will be denoted by and respectively.
Few math problems, specially, problems on Numbers are very interesting. In this “Note”, I’ve added a classical problem, as follow:
Solve for , 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.
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.
- Abstract Algebra OnLine by Prof. Beachy
- Understanding Algebra by James Brennan
- Abstract Algebra : Theory and Applications by Tom Judson
- Elements of Abstract and Linear Algebra by E H Connell
- Linear Algebra by Jim Hefferon
- Elementary Linear Algebra by Keith Matthews
- Linear Algebra, Infinite dimensions and Mapleby James Herod
- Elementary Number Theory by William Stein
- 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.
- A Course in Universal Algebra by Stanley Burris & H. P. Sankappanvar
- An Introduction to the Theory of Numbers by Leo Moser
- A Computational introduction to Number Theory and Algebra by Victor Shoup
- Sets, Relations, Functions by Ivo Duentsch and Günther Gedigo
- Group Theory by Pedrag Civitanovic
- Linear & Multilinear Algebra by C C Wang & R M Bowen
- Abelian Categories by Peter Freyd
- Categories and Groupoids by P. J. Higgins
- 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.)