PRESENTATION

RESMI.SB.Ed, MATHEMATICSNSS TRAINING COLLEGE,PANDALAMREG NUM:13304012

• Ancient Period

• Greek Period• Greek Period

• Hindu-Arabic Period

• Period of Transmission

• Early Modern Period

• Modern Period

A. Number Systems and Arithmetic• Development of numeration systems.• Creation of arithmetic techniques, lookup tables, the abacus and other

calculation tools.

B. Practical Measurement, Geometry and Astronomy• Measurement units devised to quantify distance, area, volume, and

time.• Geometric reasoning used to measure distances indirectly.

Ancient Period (3000 B.C. to 260 A.D.)

• Geometric reasoning used to measure distances indirectly.• Calendars invented to predict seasons, astronomical events. • Geometrical forms and patterns appear in art and architecture.

Practical Mathematics

As ancient civilizations developed, the need for practical mathematics increased. They required numeration systems and arithmetic techniques for trade, measurement strategies for construction, and astronomical calculations to track the seasons and cosmic cycles.

Babylonian Numerals

The Babylonian Tablet Plimpton 322

This mathematical tablet was recovered from an unknown place in the Iraqi desert. It was written originally sometime around 1800 BC. The tablet presents a list of Pythagorean triples written in Babylonian numerals. This numeration system uses only two symbols and a base of sixty.

Calculating Devices

Chinese Wooden Abacus

Roman Bronze “Pocket” Abacus

Babylonian Marble Counting Board

c. 300 B.C.

Greek Period (600 B.C. to 450 A.D.)

A. Greek Logic and Philosophy

� Greek philosophers promote logical, rational explanations of natural phenomena.

� Schools of logic, science and mathematics are established.

� Mathematics is viewed as more than a tool to solve practical problems; it is seen as a means to understand divine laws.

� Mathematicians achieve fame, are valued for their work.� Mathematicians achieve fame, are valued for their work.

B. Euclidean Geometry

� The first mathematical system based on postulates, theorems and proofs appears in Euclid's Elements.

Mathematics and Greek Philosophy

Greek philosophers viewed the universe in mathematical terms. Plato described five elements that form the world and related them to the five

regular polyhedra.

Euclid’s Elements

Greek, c. 800 Arabic, c. 1250 Latin, c. 1120

French, c. 1564 English, c. 1570 Chinese, c. 1607

Translations of Euclid’s Elements of Gemetry

Proposition 47, the Pythagorean Theorem

Archimedes and the Crown

Eureka!

Hindu-Arabian Period (200 B.C. to 1250 A.D. )

A. Development and Spread of Hindu-Arabic Numbers� A numeration system using base 10, positional notation, the zero symbol

and powerful arithmetic techniques is developed by the Hindus, approx. 150 B.C. to 800 A.D..

� The Hindu numeration system is adopted by the Arabs and spread throughout their sphere of influence (approx. 700 A.D. to 1250 A.D.).

B. Preservation of Greek Mathematics� Arab scholars copied and studied Greek mathematical works, principally in � Arab scholars copied and studied Greek mathematical works, principally in

Baghdad.

C. Development of Algebra and Trigonometry� Arab mathematicians find methods of solution for quadratic, cubic and

higher degree polynomial equations. The English word “algebra” is derived from the title of an Arabic book describing these methods.

� Hindu trigonometry, especially sine tables, is improved and advanced by Arab mathematicians

The Great Mosque of CordobaThe Great Mosque, Cordoba

During the Middle Ages Cordoba was the greatest center of learning in Europe, second only to Baghdad in the Islamic world. Islamic world.

Islamic Astronomy and Science

Many of the sciences developed from needs to fulfill the rituals and duties of Muslim worship. Performing formal prayers requires that a Muslim faces Mecca. To find Mecca from any part of the globe, Muslims invented the compass and developed the sciences of geography and geometry.

Prayer and fasting require knowing the times of each duty. Because these times are marked by astronomical phenomena, the science of astronomy underwent a major development.

Painting of astronomers at work in the observatory of Istanbul

Al-Khwarizmi Abu Abdullah Muhammad bin Musa al-Khwarizmi, c. 800 A.D. was a Persian mathematician, scientist, and author. He worked in Baghdad and wrote all his works in Arabic.

He developed the concept of an algorithm in mathematics. The words "algorithm" and "algorism" derive ultimately from his name. His ultimately from his name. His systematic and logical approach to solving linear and quadratic equations gave shape to the discipline of algebra, a word that is derived from the name of his book on the subject, Hisab al-jabr wa al-muqabala (“al-jabr” became “algebra”).

He was also instrumental in promoting the Hindu-arabic numeration system.

Evolution of Hindu-Arabic

Numerals

A. Discovery of Greek and Hindu-Arab mathematics• Greek mathematics texts are translated from Arabic into Latin;

Greek ideas about logic, geometrical reasoning, and a rational view of the world are re-discovered.

• Arab works on algebra and trigonometry are also translated into Latin and disseminated throughout Europe.

Period of Transmission (1000 AD – 1500 AD)

into Latin and disseminated throughout Europe.

B. Spread of the Hindu-Arabic numeration system• Hindu-Arabic numerals slowly spread over Europe• Pen and paper arithmetic algorithms based on Hindu-Arabic

numerals replace the use the abacus.

Leonardo of Pisa

From Leonardo of Pisa’s famous book Liber Abaci (1202 A.D.):

"These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated."

The Abacists and Algorists

CompeteThis woodblock engraving of a competition between arithmetic techniques is from from Margarita Philosphica by Gregorius Reich, (Freiburg, 1503).

Lady Arithmetic, standing Lady Arithmetic, standing in the center, gives her judgment by smiling on the arithmetician working with Arabic numerals and the zero.

Rediscovery of Greek GeometryLuca Pacioli (1445 - 1514), a Franciscan friar and mathematician, stands at a table filled with geometrical tools (slate, chalk, compass, dodecahedron model, etc.), dodecahedron model, etc.), illustrating a theorem from Euclid, while examining a beautiful glass rhombicuboctahedron half-filled with water.

Pacioli and Leonardo Da Vinci

Luca Pacioli's 1509 book The Divine Proportion was illustrated by Leonardo Da Vinci.

Shown here is a drawing of an icosidodecahedron and an "elevated" form of it. For the elevated forms, each face is augmented with a pyramid composed of equilateral triangles.

Early Modern Period (1450 A.D. – 1800 A.D.)

A. Trigonometry and Logarithms• Publication of precise trigonometry tables, improvement of surveying

methods using trigonometry, and mathematical analysis of trigonometric relationships. (approx. 1530 – 1600)

• Logarithms introduced by Napier in 1614 as a calculation aid. This advances science in a manner similar to the introduction of the computer.

B. Symbolic Algebra and Analytic Geometry• Development of symbolic algebra, principally by the French • Development of symbolic algebra, principally by the French

mathematicians Viete and Descartes • The cartesian coordinate system and analytic geometry developed by

Rene Descartes and Pierre Fermat (1630 – 1640)

C. Creation of the Calculus• Calculus co-invented by Isaac Newton and Gottfried Leibniz. Major

ideas of the calculus expanded and refined by others, especially the Bernoulli family and Leonhard Euler. (approx. 1660 – 1750).

• A powerful tool to solve scientific and engineering problems, it opened the door to a scientific and mathematical revolution.

Viète and Symbolic AlgebraIn his influential treatise In Artem Analyticam Isagoge (Introduction to the Analytic Art, published in1591) Viète demonstrated the value of symbols. He suggested using letters as symbols for quantities, both known and unknown. unknown.

François Viète1540-1603

Napier’s Logarithms

John Napier

In his Mirifici Logarithmorum Canonis descriptio (1614) the Scottish nobleman John Napier introduced the concept of logarithms as an aid to calculation.

John Napier1550-1617

Kepler and the Platonic Solids

Johannes Kepler1571-1630

Kepler’s first attempt to describe planetary orbits used a model of nested regular polyhedra (Platonic solids).

Newton’s Principia – Kepler’s Laws

“Proved”

Isaac Newton

1642 - 1727

Newton’s Principia Mathematica (1687) presented, in the style of Euclid’s Elements, a mathematical theory for celestial motions due to the force of gravity. The laws of Kepler were “proved” in the sense that they followed logically from a set of basic postulates.

Newton’s CalculusNewton developed the main ideas of his calculus in private as a young man. This research was closely connected to his studies in physics. Many years later he published his results to establish priority for himself as establish priority for himself as inventor the calculus.

Newton’s Analysis Per Quantitatum Series, Fluxiones, Ac Differentias, 1711, describes his calculus.

Leibniz’s Calculus

Gottfied Leibniz

Leibniz and Newton independently developed the calculus during the same time period. Although Newton’s version of the calculus led him to his great discoveries, Leibniz’s concepts and his style of notation form the basis of modern calculus.

1646 - 1716

A diagram from Leibniz's famous 1684 article in the journal Acta eruditorum.

Leonhard Euler

Leonhard Euler was of the generation that followed Newton and Leibniz. He made contributions to almost every field of mathematics and was the most prolific mathematics writer of all time.

His trilogy, Introductio in analysin infinitorum, Institutiones calculi differentialis, and Institutiones Institutiones calculi differentialis, and Institutiones calculi integralis made the function a central part of calculus. Through these works, Euler had a deep influence on the teaching of mathematics. It has been said that all calculus textbooks since 1748 are essentially copies of Euler or copies of copies of Euler.

Euler’s writing standardized modern mathematics notation with symbols such as:

f(x), e, π, i and ∑ .

Leonhard Euler1707 - 1783

Modern Period (1800 A.D. – Present)

A. Non-Euclidean Geometry• Gauss, Lobachevsky, Riemann and others develop alternatives to Euclidean geometry

in the 19th century.• The new geometries inspire modern theories of higher dimensional spaces, gravitation,

space curvature and nuclear physics.

B. Set Theory• Cantor studies infinite sets and defines transfinite numbers • Set theory used as a theoretical foundation for all of mathematics

C. Statistics and ProbabilityC. Statistics and Probability• Theories of probability and statistics are developed to solve numerous practical

applications, such as weather prediction, polls, medical studies etc.; they are also used as a basis for nuclear physics

D. Computers• Development of electronic computer hardware and software solves many previously

unsolvable problems; opens new fields of mathematical research.

E. Mathematics as a World-Wide Language• The Hindu-Arabic numeration system and a common set of mathematical symbols are

used and understood throughout the world.• Mathematics expands into many branches and is created and shared world-wide at an

ever-expanding pace; it is now too large to be mastered by a single mathematician

Current Branches of Mathematics1. Foundations

• Logic & Model Theory • Computability Theory & Recursion Theory • Set Theory • Category Theory

2. Algebra• Group Theory • Ring Theory

(includes elementary algebra)

4. Geometry & Topology• Euclidean Geometry • Non-Euclidean Geometry • Absolute Geometry • Metric Geometry • Projective Geometry • Affine Geometry • Discrete Geometry & Graph Theory • Differential Geometry

(includes elementary algebra) • Field Theory • Module Theory • Galois Theory • Number Theory • Combinatorics • Algebraic Geometry

3. Mathematical Analysis • Real Analysis & Measure Theory

(includes elementary Calculus) • Complex Analysis • Tensor & Vector Analysis • Differential & Integral Equations • Numerical Analysis • Functional Analysis & Theory of Functions

• Differential Geometry • General Topology • Algebraic Topology

5. Applied Mathematics• Probability Theory • Statistics • Computer Science • Mathematical Physics • Game Theory • Systems & Control Theory