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A brief history of Mathematics

- Before the Ancient Greeks
- Egyptians and Babylonians (c. 2000 BC)
- Knowledge comes from papyri
- Rhind Papyrus

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

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)

Archimedes, Syracuse

Euclid (c 300 BC), Alexandria

Ptolemy (AD 83c.168), Roman Egypt

- Almagest comprehensive treatise on geocentric

astronomy - Link from Greek to Islamic to European science

Al-Khwarizmi (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

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.

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.

John Napier (1550 1617)

- Popularized use of the (Stevins) 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

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.

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.

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

Pierre de Fermat (16011665)

- If ngt2, then
- an bn cn has no solutions in non-zero

integers a, b, and c.

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....

Euler (1707 1783)

- important discoveries in calculusgraph 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

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

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.

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.

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

What does the future hold?

- Applications..
- Biology and Cybernetics

Clay Millenium Prizes

- Birch and Swinnerton-Dyer Conjectureif ?(1) is

equal to 0, then there are an infinite number of

rational points (solutions), and conversely, if

?(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 - ?(s) 0
- Yang-Mills and Mass Gap

P vs NP Problem