- Geometry:The first book in this field is Euclid's The Elements: http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf This book, --thirteen books, indeed; now considered one book divided into thirteen chapters— constitutes not only the best account on plane geometry, but also an implementation of the deductive method in science. Euclid's work also happens to be the first text in number theory. Geometry embodies the most influential branch of human knowledge that has modeled our vision of the universe. Take a look to Copernicus De revolutionibus orbium coelestium ( On the Revolutions of the Celestial Spheres ) at http://www.webexhibits.org/calendars/year-text-Copernicus.html --full of geometric constructions –this is the single work of an author that indeed revolutionized the modern concept of the place of man in the universe. Next we have a book regarded as the most important work in the history of science : Newton's Principia (The mathematical principles of natural philosophy) http://www.maths.tcd.ie/pub/HistMath/People/Newton/Principia/Bk1Sect1/ Newton in the first page of the author's preface asserts that he will use geometry to solve mechanical problems. Most recently, Albert Einstein first page of his “The Special Theory of Relativity” wrote: In your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember — perhaps with more respect than love — the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. See the complete work here: http://www.gutenberg.org/cache/epub/5001/pg5001.html
Einstein himself as a child was very impressed for two major events: one, when his father gave him a compass (that instrument whose needle follows the magnetic poles of the earth –there is another compass, the instrument to draw circles—mysteries of languages); and, the second event in Einstein's childhood was when he received a copy of Euclid's Elements.
The Foundations of Geometry By David Hilbert. Hilbert was an influential German Mathematician who presented the 23 problems in Paris in 1900. He was an advocate for the complete axiomatization of mathematics. Euclidean’s Geometry in Hilbert’s view:
- Trigonometry: This subject was developed primarily by Hipparchus around 140 BC. Anyone who dares to learn mathematics need to know a great deal of trigonometry. See Dave's Short Trigonometry Course: http://www.clarku.edu/~djoyce/trig/
- Algebra: This subject historically was first touched by Diophantus. We should introduce here a text on History of mathematics. Since Math is the only science where no attained knowledge becomes obsolete, a History of mathematics provide us with a general perspective of this very interconnected and complex edifice. So we start algebra by reading this monograph, Diophantus of Alexandria A Text and its History –32 pages,
http://www-irma.u-strasbg.fr/~schappa/NSch/Publications_files/Dioph.pdf and also Diophantus of Alexandria, a study in the history of Greek algebra: http://www.archive.org/details/diophantusofalex00heatiala Diophantus was the first mathematician to write equations with symbols (before and after him, including Arabs authors in the late middle ages, wrote equations as statements in words, sentences.)
- Also in Algebra, we should read Euler's course of Algebra, on the web the entire text: http://web.mat.bham.ac.uk/C.J.Sangwin/euler/index.html
- College Algebra: http://tutorial.math.lamar.edu/pdf/Alg/Alg_Complete.pdf
- Advanced Algebra, a course of Abstract Algebra is available at
http://shell.cas.usf.edu/~wclark/Elem_abs_alg.pdf and http://www.math.umn.edu/~garrett/m/intro_algebra/notes.pdf
- A Course in Universal Algebra (Abstract Algebra). Hawaii University:
- Analytic Geometry: The subject was developed by Rene Descartes –a French philosopher and mathematician who was also known in Latin as Renatus Cartesius so after him was named the Cartesian plane. Plane geometry was for the first time combined with algebra; A nice text on this subject,
- Calculus. Before this point in mathematics everything was static. The subject was mainly developed by Leibniz and Newton, although main contributions had been introduced by Fermat and Barrow. MIT open course, an introduction to Calculus: http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/
- Calculus I: http://tutorial.math.lamar.edu/pdf/CalcI/CalcI_Complete.pdf
- Calculus II: http://tutorial.math.lamar.edu/pdf/CalcII/CalcII_Complete.pdf
- Calculus III: http://tutorial.math.lamar.edu/pdf/CalcIII/CalcIII_Complete.pdf
- Community Calculus: Whitman College, ebooks:
- Single variable calculus, early transcendentals, in PDF format.
- Multivariable calculus, early transcendentals, in PDF format.
- Advanced Calculus: http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf
- Number Theory: this is a deeper look into Arithmetic. Arithmos, in Greek, is Number. Arithmetic is a kind of elementary Theory of numbers; however, only after Fermat and others in the sixteen hundreds we can properly speak about Number theory. Here I only recommend this text, An Introduction to the Theory of Numbers of G.H. Hardy, first edited in 1938 in one of the most important on the subject. This book, however, is a very demanding and it is not, in my opinion, the first text anyone should study on this matter. Neither it is not available free on the web; so a best approximation to the topic might be:
Analytic Number Theory: http://www.dms.umontreal.ca/~andrew/PDF/PrinceComp.pdf
Linear Algebra: Algebra becomes a very important tool in discrete mathematics. A course of Elementary Linear Algebra:
- Probability and Statistics: Indeed, mathematics in the only theoretical instrument that allows us to see the not-seen, the unveil-future or the probability of events that are about to happen. It is an obligated instrument in almost every field of modern science from Medicine to Sociology, Economics to Physics, including Psychology and several other careers. An introduction to probability:
An Introduction to Statistic: http://www.artofproblemsolving.com/LaTeX/Examples/statistics_firstfive.pdf
Collaborative Statistics, pdf download, by Barbara Illowsky PhD.
Introductory Statistics, Vol 1 & Introductory Statistics, Vol 2: textbookequity.org
Introduction to Probability, Dartmouth College: 1. Ebook; 2. Solutions.
- Set Theory: the rules and properties of collections of objects, points or intervals in any numerical field are proven and demonstrated by set theory. It includes the study of the infinity or infinities, since not all infinities are the same. Some texts on the subject are: http://www.math.miami.edu/~ec/book/ch01.pdf Also: Set, Relations, Functions: http://www.cosc.brocku.ca/~duentsch/archive/methprimer1.pdf
- Logic: For some authors, mathematics can be reduced to Logic, for other, Logic is just the language. There is a well founded website dedicated to the subject: http://www.math.psu.edu/simpson/papers/philmath/
Also, free on the web:
A Problem Course in Mathematical Logic, http://euclid.trentu.ca/math/sb/pcml/pcml-16.pdf
- Mathematical Analysis: the mathematical knowledge requires proofs, verification of every single statement. The deep understanding of the nature of numbers is provided by a math Analysis textbook. Free on the web Introduction to real Analysis: http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF
And http://math.arizona.edu/~faris/real.pdf and
Revisiting the Introduction to Cauchy's Cours d'Analyse:
Amazing and Aesthetic Aspects of Analysis by Paul Loya (see chapters on continued fractions, excellent!!)
A First Course in Complex Analysis: http://math.sfsu.edu/beck/papers/complex.pdf
- Miscellanea of Mathematics texts:
Abstract Algebra Done Concretely: http://www.math.purdue.edu/~dvb/algebra/algebra.pdf
Also, A brief History of Mathematics.
Notes on Algebraic Structures: http://www.maths.qmul.ac.uk/~pjc/notes/algstr.pdf
Differential Equations: http://tutorial.math.lamar.edu/pdf/DE/DE_Complete.pdf
Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler at http://www.math.wisc.edu/~keisler/calc.html
Measure Theory and Lebesgue Integration by Joshua H. Lifton at:
Lectures on the Geometry of Manifolds by Liviu I. Nicolaescu
Street fighting mathematics, MIT. By Sanjoy Mahan.
- Web page: The History of the Calculus and the Development of Computer Algebra Systems by Dan Ginsburg, Brian Groose, Josh Taylor and Prof. Bogdan Vernescu. An Interactive Qualifying Project @ WPI
- Street-Fighting Mathematics, The Art of Educated Guessing and Opportunistic Problem Solving By Sanjoy Mahajan
- The OPEN textbook LIBRARY: the most complete source of free textbook in the world wide web!! Algebra, Liner Algebra, Calculus, Statistics, Abstract Algebra, Number Theory etc ...
- Arkansas Tech University, Department of Mathematics, electronic textbooks collection.