History of mathematics - Pedagogy of Mathematics

It includes Prehistory: from primitive counting to Numeral systems, Archaic mathematics in Mesopotamia and egypt, Birth of mathematics as a deductive science in Greece: Thales and Pythagoras and Role of Aryabhatta in Indian Mathematics.

1. History of Mathematics with Special Reference to Aaryabhatta Jemima Sultana Department of Education Aligarh Muslim University

2. Introduction Modern Mathematics having roots in ancient Egypt and Babylonia, really flourished in ancient Greece. It is remarkable in Arithmetic (Number theory) and DeductiveGeometry. Mathematics written in ancient Greek was translated into Arabic, together with some mathematics of India. Mathematicians of Islamic Middle East significantly developed Algebra. Later some of this mathematics was translated into Latin and became the mathematics of Western Europe. Over a period of several hundred years, it became the mathematics of the world.

3. 1. Prehistory: from primitive counting to Numeral systems Numeral Systems The origin of the earliest civilizations such as Sumer (in Mesopotamia), Egypt and Minoan (in Crete) goes back to 3500-4000BC. Needs of trade, city management, measurement of size, weight and time required a unified system to make calculations and represent the results. The earliest Sumerian Systems of Measures and Calendars are dated by 4000BC. Special clay tokens were invented to count sheep, days and other objects. In 3000BC in the city Uruk there were more than a dozen of different counting systems in use. About this time, Abacus as a tool of calculation was invented. Later, as a writing system was developed (pressing cuneiform signs on clay tablets with a reed stylus), the Sumer sexagesimal numeral system based on powers of 60 was elaborated (do not confuse with hexadecimal system based on 16). Nowadays Sumerian system is used for time (hour, minutes, seconds) and angle measurements (360°).

4. Mesopotamian numerals

5. 2. Archaic Mathematics in Mesopotamia (Babylonia) and Egypt Babylonian Mathematics: not much of geometry, but amazing arithmetic and algebra. Babylonian mathematics used pre-calculated clay tablets in cuneiform script to assist with arithmetic. For example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. Babylonian algebra was not symbolic, but it was rhetoric: instead of symbols for unknown and signs just words were used, for example, an equation x+1=2 was expressed as “a thing plus one equals two”.

6. The Plimpton 322 tablet (1800 BC) in Plimpton collection at Columbia University. It contains a list of Pythagorean triples, i.e., integers (a,b,c) such that a^2+b^2=c^2. It seems that a general formula for such triples was known, although no direct evidence of this was ever found.

7. Egyptian mathematics: unit fractions, more geometry, but less algebra Rhind (or Ahmes) mathematical papyrus (1650 BC) in British Museum, 6m length. It was found during illegal excavation and sold in Egypt to Scottish antiquarian Rhind in 1858. It is a problem book that was copied by scribe Ahmes from an older papyrus dated by 1800-2000BC.There are 87 problems with solutions in arithmetic, algebra and geometry. The most of arithmetical problems are related to the unit Egyptian fractions and involve in particular finding least common multiples of denominators and decomposition of 2/n into unit fractions.

8. 3. Birth of mathematics as a deductive science in Greece: Thales and Pythagoras Thales of Miletus (624-546 BC) the first philosopher and mathematician in Greek tradition, one of seven Sages of Greece, founder of Milesian natural philosophy school.

9. Archimedes Archimedes(287-212BC) mathematician, physicist, engineer, astronomer, inventor regarded as one of the leading (in fact, the greatest) scientists in classical antiquity.

10. 4. Mathematics of late Hellenistic Period

11. Indian Mathematics Originates in Vedic mathematics in Sanskrit sutras with multiplication rules and formulas (like areas of geometric figures, may be Pythagoras Th) hidden between the Vedic hymns.

12. Aaryabhatta Aryabhatta was a fifth century mathematician, astronomer, astrologer and physicist. He was a pioneer in the field of mathematics. At the age of 23, he wrote Aryabhattiya, which is a summary of mathematics of his time. There are four sections in this scholarly work. In the first section he describes the method of denoting big decimal numbers by alphabets. In the second section, we find difficult questions from topics of modern day Mathematics such as number theory, geometry, trigonometry and Beejganita (algebra). The remaining two sections are on astronomy.

13. Aaryabhatta Aryabhatta showed that zero was not a numeral only but also a symbol and a concept. Discovery of zero enabled Aryabhatta to find out the exact distance between the earth and the moon. The discovery of zero also opened up a new dimension of negative numerals. As we have seen, the last two sections of Aryabhattiya were on Astronomy. Evidently, Aryabhatta contributed greatly to the field of science, too, particularly Astronomy. In ancient India, the science of astronomy was well advanced. It was called Khagolshastra.Khagol was the famous astronomical observatory at Nalanda, where Aryabhatta studied. Infact science of astronomy was highly advanced and our ancestors were proud of it.

14. Aaryabhatta The aim behind the development of the science of astronomy was the need to have accurate calendars, a better understanding of climate and rainfall patterns for timely sowing and choice of crops, fixing the dates of seasons and festivals, navigation, calculation of time and casting of horoscopes for use in astrology. Knowledge of astronomy, particularly knowledge of the tides and the stars, was of great importance in trade, because of the requirement of crossing the oceans and deserts during night time.

15. Aaryabhatta Disregarding the popular view that our planet earth is “Achala” (immovable), Aryabhatta stated his theory that,earth is round and rotates on its own axis. He explained that the appearance of the sun moving from east to west is false by giving examples. One such example was: When a person travels in a boat, the trees on the shore appear to move in the opposite direction. He also correctly stated that the moon and the planets shined by reflected sunlight. He also gave a scientific explanation for solar and lunar eclipse clarifying that the eclipse were not because of Rahhu and/or Ketu or some other rakshasa (demon,). Do you realize now, why the first satellite sent into orbit by India has been named after Aryabhatta?

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