A brief history of Mathematics

1. A brief history of Mathematics • Before the Ancient Greeks: • Egyptians and Babylonians (c. 2000 BC): • Knowledge comes from “papyri” • Rhind Papyrus

2. Babylonian Math • Main source: Plimpton 322 • Sexagesimal (base-sixty) originated with ancient Sumerians (2000s BC), transmitted to Babylonians … still used —for measuring time, angles, and geographic coordinates

3. Greek Mathematics • Thales (624-548) • Pythagoras of Samos (ca. 580 - 500 BC) • Zeno: paradoxes of the infinite • 410- 355 BC- Eudoxus of Cnidus (theory of proportion) • Appolonius (262-190): conics/astronomy • Archimedes (c. 287-212 BC)

4. Archimedes, Syracuse

5. Euclid (c 300 BC), Alexandria

6. Ptolemy (AD 83–c.168), Roman Egypt • Almagest: comprehensive treatise on geocentric astronomy • Link from Greek to Islamic to European science

7. Al-Khwārizmī (780-850), Persia • Algebra, (c. 820): first book on the systematic solution of linear and quadratic equations. • he is considered as the father of algebra: • Algorithm: westernized version of his name

8. Leonardo of Pisa (c. 1170 – c. 1250) aka Fibonacci • Brought Hindu-Arabic numeral system to Europe through the publication of his Book of Calculation, the Liber Abaci. • Fibonacci numbers, constructed as an example in the Liber Abaci.

9. Cardano, 1501 —1576) • illegitimate child of Fazio Cardano, a friend of Leonardo da Vinci. • He published the solutions to the cubic and quartic equations in his 1545 book Ars Magna. • The solution to one particular case of the cubic, x3 + ax = b (in modern notation), was communicated to him by Niccolò Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long fight), and the quartic was solved by Cardano's student Lodovico Ferrari.

10. John Napier (1550 –1617) • Popularized use of the (Stevin’s) decimal point. • Logarithms: opposite of powers • made calculations by hand much easier and quicker, opened the way to many later scientific advances. • “MirificiLogarithmorumCanonisDescriptio,” contained 57 pages of explanatory matter and 90 of tables, • facilitated advances in astronomy and physics

11. Galileo Galilei (1564-1642) • “Father of Modern Science” • Proposed a falling body in a vacuum would fall with uniform acceleration • Was found "vehemently suspect of heresy", in supporting Copernican heliocentric theory … and that one may hold and defend an opinion as probable after it has been declared contrary to Holy Scripture.

12. René Descartes (1596 –1650) • Developed “Cartesian geometry” : uses algebra to describe geometry. • Invented the notation using superscripts to show the powers or exponents, for example the 2 used in x2 to indicate squaring.

13. Blaise Pascal (1623 –1662) • important contributions to the construction of mechanical calculators, the study of fluids, clarified concepts of pressure and. • wrote in defense of the scientific method. • Helped create two new areas of mathematical research: projective geometry (at 16) and probability theory

14. Pierre de Fermat (1601–1665) • If n>2, then a^n + b^n = c^n has no solutions in non-zero integers a, b, and c.

15. Sir Isaac Newton (1643 – 1727) • conservation of momentum • built the first "practical" reflecting telescope • developed a theory of color based on observation that a prism decomposes white light into a visible spectrum. • formulated an empirical law of cooling and studied the speed of sound. • And what else? • In mathematics: • development of the calculus. • demonstrated the generalised binomial theorem, developed the so-called "Newton's method" for approximating the zeroes of a function....

16. Euler (1707 –1783) • important discoveries in calculus…graph theory. • introduced much of modern mathematical terminology and notation, particularly for mathematical analysis, • renowned for his work in mechanics, optics, and astronomy. • Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time

17. David Hilbert (1862 –1943) • Invented or developed a broad range of fundamental ideas, in invariant theory, the axiomatization of geometry, and with the notion of Hilbert space

18. John von Neumann ) (1903 –1957) • major contributions set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics and statistics, as well as many other mathematical fields. • Regarded as one of the foremost mathematicians of the 20th century • Jean Dieudonné called von Neumann "the last of the great mathematicians.”

19. Norbert Wiener (1894-1964). • American theoretical and applied mathematician. • pioneer in the study of stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems. • founded “cybernetics,” a field that formalizes the notion of feedback and has implications for engineering, systems control, computer science, biology, philosophy, and the organization of society.

20. Claude Shannon (1916 –2001)] • famous for having founded “information theory” in 1948. • digital computer and digital circuit design theory in 1937 • demonstratedthat electrical application of Boolean algebra could construct and resolve any logical, numerical relationship. • It has been claimed that this was the most important master's thesis of all time

21. What does the future hold? • Applications.. • Biology and Cybernetics

22. Clay Millenium Prizes • Birch and Swinnerton-Dyer Conjectureif ζ(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if ζ(1) is not equal to 0, then there is only a finite number of such points. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles. • Navier-Stokes Equationhe challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. • P vs NP Problem • Poincaré Conjecture • The Riemann hypothesis asserts that all interesting solutions of the equation • ζ(s) = 0 • Yang-Mills and Mass Gap P vs NP Problem