HISORY OF MATHEMATICS

HISORY OF MATHEMATICS

By RAHUL KARATHClass IX-C

ACKNOWLEDEMENTI greatfully thank the people who helped me in this project –

My parents My class teacher My school principal

TOPICS

• Babylonian mathematics• Golden ratio• Indian mathematics

Babylonian mathematics

The Babylonian civilisation in Mesopotamia replaced the Sumerian civilisation and the Acadian civilisation. The Babylonians inherited ideas from the Sumerians and from the Acadians. From the number systems of these earlier people came the base of 60, that is the sexagesimal system. Yet neither the Sumerian nor the Acadian system was a positional system and this advance by the Babylonians was undoubtedly their greatest achievement in terms of developing the number system.

The Sumerians had developed an abstract form of writing based on cuneiform (i.e. wedge-shaped) symbols. Their symbols were written on wet clay tablets which were baked in the hot sun and many thousands of these tablets have survived to this day. It was the use of a stylus on a clay medium that led to the use of cuneiform symbols since curved lines could not be drawn. The later Babylonians adopted the same style of cuneiform writing on clay tablets

A sample of one of their tablets

•

The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds, is just to write the sexagesimal fraction, 5 25/60 30/3600. We adopt the notation 5; 25, 30 for this sexagesimal number. As a base 10 fraction of the sexagesimal number 5; 25, 30 is 5 4/10 2/100 5/1000 which is written as 5.425 in decimal notation.

Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC. They give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 82 = 1,4 which stands for 82 = 1, 4 = 1 60 + 4 = 64

and so on up to 592 = 58, 1 (= 58

60 +1 = 3481). The Babylonians used the formula ab = [(a + b)2 - a2 - b2]/2

Division is a harder process. The Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a/b = a (1/b)

they constructed tables for n3 + n2 then, with the aid of these tables, certain cubic equations could be

solved. For example, consider the equation

ax3 + bx2 = c.

Some of the Babylonian mathematicians

Theon of Alexandria

Neugebaur

Moritz Cantor

Golden ratio

The golden ratio, also known as the divine proportion, golden mean, or golden section(Latin: sectio aurea) , is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron It’s an irrational mathematical constant, approximately 1.6180339887. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,golden number, and mean of Phidias

(φ)

The golden ratio is often denoted by the Greek letter phi, usually lower case-

The figure below illustrates the geometric relationship that defines this constant. Expressed algebraically:

This equation has as its unique positive solution the algebraic irrational number

Many artists and architects have proportioned their works to approximate the golden ratio

Indian mathematics

It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years.

We do know that the Harappans had adopted a uniform system of weights and measures. An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several scales for the measurement of length were also discovered during excavations

The main Sulbasutras were composed by Baudhayana (about 800 BC), Manava (about 750 BC), Apastamba (about 600 BC), and Katyayana (about 200 BC). These men were both priests and scholars but they were not mathematicians in the modern sense. Although we have no information on these men other than the texts they wrote, we have included them in our biographies of mathematicians.

There is another scholar, who again was not a mathematician in the usual sense, who lived around this period. That was Panini who achieved remarkable results in his studies of Sanskrit grammar. Now one might reasonably ask what Sanskrit grammar has to do with mathematics. It certainly has something to do with modern theoretical computer science, for a mathematician or computer scientist working with formal language theory will recognise just how modern some of Panini's ideas are.

INDIAN MATHEMATISIANS

Ramanujan

He was born on 22na of December 1887 in a small village of  Tanjore district, Madras.

He failed in English in Intermediate, so his formal studies were stopped but his self-study

of mathematics continued. He sent a set of 120 theorems to Professor Hardy of Cambridge. As a result he invited

Ramanujan to England. Ramanujan showed that any big number can be

written as sum of not more than four prime numbers.

He showed that how to divide the number into two or more squares or cubes.

when Mr Litlewood came to see Ramanujan in taxi number 1729, Ramanujan said that 1729 is the smallest number which can be written in the form of sum of cubes of two

numbers in two ways,i.e. 1729 = 93 + 103 = 13 + 123

since then the number 1729 is called Ramanujan’s number.

In the third century B.C, Archimedes noted that the ratio of circumference of a circle to

its diameter is constant. The ratio is now called ‘pi ( Π )’ (the 16th letter in the Greek

alphabet series) The largest numbers the Greeks and the Romans used were 106 whereas Hindus

used numbers as big as 1053 with specific names as early as 5000 B.C. during the

Vedic period.

Ramanujan

Aryabhatta

Aryabhatta was born in 476A.D in Kusumpur, India.

He was the first person to say that Earth is spherical and it revolves around the sun. He gave the formula (a + b)2 = a2 + b2 +

2ab

He taught the method of solving the following problems:

14 + 24 + 34 + 44 + 54 + …………+ n4 = n(n+1) (2n+1) (3n2+3n-1)/30

ARYABHATTA

Bhama Gupta

Brahma Gupta was born in 598A.D in Pakistan. He gave four methods of multiplication.

He gave the following formula, used in G.P

series a + ar + ar2 + ar3 +……….. + arn-1 = (arn-1)

÷ (r – 1)

He gave the following formulae : Area of a cyclic quadrilateral with side a, b, c, d=

√(s -a)(s- b)(s -c)(s- d)where 2s = a + b + c + dLength of its diagonals =

BRAHMA GUPTA

Shakuntala Devi

She was born in 1939 In 1980, she gave the product of two, thirteen

digit numbers within 28 seconds, many countries have invited her to demonstrate her

extraordinary talent

In Dallas she competed with a computer to see who give the cube root of 188138517 faster, she won. At university of USA she was asked to give

the 23rd root of 916748676920039158098660927585380162483106680144308622407126516427934657040867

09659327920576748080679002278301635492485238033574531693511190359657754734007568186

88305620821016129132845564895780158806771.

She answered in 50seconds. The answer is 546372891. It took a UNIVAC 1108 computer, full

one minute (10 seconds more) to confirm that she was right after it was fed with 13000

instructions.

SHAKUNTALA DEVI

CONCLUSIONThe article on history of mathematics

Is primarily into the origin of discoveriesin mathematics and to a lesser extentan investigation into the mathematical

methods and notations of the past

THANK YOU

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