Moebius transformation

Moebius transformation of the plane is a rational function of the form
f(z)=(az+b)/(cz+d)
The variable z and the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0. Moebius transformations are named in honor of August Ferdinand Moebius.

Here is a short movie depicting the beauty of Moebius Transformations in mathematics, it shows how moving to a higher dimension can make the transformations easier to understand.

A Very Short History of Mathematics

70,000 BC there are drawings that indicate some knowledge of elementary mathematics and of time measurement based on the stars. Paleontologists have discovered ochre rocks adorned with scratched geometric patterns.

20,000 BC - The Ishango bone, found in northeastern Congo, is the earliest known demonstration of sequences of prime numbers and of Ancient Egyptian multiplication.

3,000 BC - The Indus Valley Civilization of North India and Pakistan developed a system of measures that used the decimal system, and an advanced brick technology which utilized ratios.

2,500 BC - The Sumerians wrote multiplication tables on clay tablets and dealt with division problems. The traces of the Babylonian numerals also date back to this period.

1,650 BC - The Rhind papyrus, a major Egyptian mathematical text, is an instruction manual in arithmetic and geometry. It gives area formulas, multiplication methods, working with unit fractions, composite and prime numbers, arithmetic, geometric and harmonic means.

550 BC - Pythagoras of Samos is credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history. He expressed the theorem algebraically rather than geometrically.

400 BC - Jaina mathematicians from ancient India began studying mathematics for the sole purpose of mathematics. They developed transfinite numbers, logarithms, fundamental laws of indices, cubic equations, quartic equations, set theory, sequences and progressions, permutations and combinations, etc.

370 BC - Eudoxus developed the method of exhaustion, a precursor of modern integration. The Pythagoreans proved the existence of irrational numbers.

300 BC - Euclid wrote Elements, the most important mathematics book ever written. It is the first example of the format still used in mathematics today: definition, axiom, theorem, proof.

230 BC - Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.

400 - The Surya Siddhanta, classical Indian mathematician, introduced the trigonometric functions of sine, cosine, and inverse sine, and laid down rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky.

650 - Brahmagupta lucidly explained the use of zero as both a placeholder and decimal digit, and explained the Hindu-Arabic numeral system.

825 - Muhammad ibn Musa al-Kwarizmi wrote several books on the Hindu-Arabic numerals and on methods for solving equations. The word algorithm is derived from the Latinization of his name.

1000 - Al-Karaji, a Persian mathematician, gives the first known proof by mathematical induction. He proved the binomial theorem, Pascal's triangle, and the sum of integral cubes.

1170 - Bhaskara, another Indian mathematician first conceived differential calculus, the concept of the derivative, differential coefficient and differentiation. He also stated Rolle's theorem and investigated the derivative of the sine function.

1202 - Fibonacci produced the first significant mathematics in Europe since the time of Eratosthenes, a gap of more than a thousand years. His book introduced Hindu-Arabic numerals to Europe, and discussed many other mathematical problems.

1654 - Blaise Pascal and Pierre de Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling.

1665 - Isaac Newton brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz developed calculus and much of the calculus notation still in use today.

1736 - Leonhard Euler, the most influential mathematician of the 18th century, solved the Koenigsberg bridge problem. He founded the study of graph theory named the square root of -1 with the symbol i, made contributions to the study of topology, etc.

1799 - Karl Friedrich Gauss proves that every polynomial equation has a solution among the complex numbers. Gauss did revolutionary work on functions of complex variables, in geometry, and on the convergence of series.

1807 - Joseph Fourier announced his discoveries about the trigonometric decomposition of functions, but the demonstration was not altogether satisfactory. The final solution of the problem was given in 1829 by Jacques Charles François Sturm.

1822 - Augustin Louis Cauchy proved the Cauchy integral theorem for integration around the boundary of a rectangle. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner and was thus a pioneer of analysis.

1829 - Nikolai Ivanovich Lobachevsky publishes his work on hyperbolic non-Euclidean geometry, where uniqueness of parallels no longer holds.

1832 - Evariste Galois presents a general condition for the solvability of algebraic equations. Galois and Niels Henrik Abel proved that there is no general algebraic method for solving polynomial equations of degree greater than four.

1843 - William Rowan Hamilton in Ireland discovers the calculus of quaternions and deduces that they are non-commutative.

1847 - George Boole devised Boolean algebra, in which the only numbers were 0 and 1 and in which, famously, 1 + 1 = 1. Boolean algebra is the starting point of mathematical logic and has important applications in computer science.

1854 - Bernhard Riemann introduces Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalize the ideas of curves and surfaces.

1899 - David Hilbert presents a set of self-consistent geometric axioms in Foundations of Geometry. In 1900, he set out a list of 23 unsolved problems in mathematics. These problems formed a central focus for much of 20th century mathematics.

1928 - John von Neumann, a Hungarian American mathematician who made major contributions to a vast range of fields, begins devising the principles of game theory and proves the minimax theorem.

1931 - Kurt Goedel shows that mathematical systems are not fully self-contained. One of the most significant logicians of all time, Goedel made an immense impact upon scientific and philosophical thinking in the 20th century.

1950 - Stanislaw Ulam and John von Neumann present cellular automata dynamical systems. Stanislaw Marcin Ulam was a Polish mathematician who participated in the Manhattan Project and proposed the Teller–Ulam design of thermonuclear weapons.

1961 - Daniel Shanks and John Wrench compute pi to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer. Shanks is best known for his book Solved and Unsolved Problems in Number Theory.

1983 - Gerd Faltings shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem.

1987 - Yasumasa Kanada, Jonathan Borwein, Peter Borwein, and David Bailey, use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to compute pi to 134 million decimal places.

1995 - Sir Andrew John Wiles, working in secrecy, proves Fermat's Last Theorem. This surprisingly lengthy proof has stood up to the scrutiny of the world's experts.


Sources & further reading:

MacTutor History of Mathematics archive
History of Mathematics Home Page
The History of Mathematics
Biographies of Women Mathematicians
Fred Rickey's History of Mathematics Page
Wikipedia.org: History of Mathematics

Grinstead and Snell's Introduction to Probability

This excellent introductory probability book is now available for free download in pdf format. It has been licensed under the terms of the GNU Free Documentation License: it is freely redistributable, and you can do whatever you like with the text, as long as you don't prevent anyone else from doing the same.

This book was written for an introductory probability course taken by sophomores, juniors, and seniors in mathematics, physical sciences, computer science, and engineering. It gives a thorough treatment of probability ideas and techniques necessary for a firm understanding of the subject.

Hypatia of Alexandria: the ancient Greek mathematician

Hypatia was the first woman who was studying and teaching advanced mathematics. During her lifetime, Alexandria had the world’s greatest library. She was a teacher and the school’s director at the Neoplatonic School of Philosophy. Hypatia was famous for her books on philosophy, mathematics, and other subjects. It is thought that Hypatia invented the astrolabe.

The comprehensive article on Hypatia at wikipedia

Andrew Baker's Mathematics Lecture Notes

I've just stumbled upon a set of interesting lecture notes by Andrew Baker, a member of the Department of Mathematics of the University of Glasgow. Algebraic Topology is the author's main research area, he also maintains the British Topology Home Page.

The notes are in PDF format and available for free download.

Titles:
An Introduction to p-adic Numbers and p-adic Analysis (final year Honours course)Representations of Finite Groups (BSc Honours degree)
Algebra and Number Theory (level 3 non-Honours degree)
An Introduction to Galois Theory (level 4 Honours degree)
Groups and symmetry (level 2)
Basic Linear Algebra (level 2)
Notes on chain complexes (postgraduate)

Here is the link to the Lecture notes.

Paul Dawkins' Online Math Notes

Paul Dawkins' web site at Lamar University is an excellent source of online math tutorials and lecture notes at the freshmen level. Most of the pages are available both for online reading and for download in PDF format. The texts cover algebra (pre-calculus topics), calculus (three parts), linear algebra, and differential equations. Dawkins has added some extra introductory material, if you need to review the basic algebra, trigonometry, exponential functions, complex numbers, etc. An essay on How To Study Math is my favorite.

Visit Paul's Online Math Notes