MATHS PROJECT

Greatness of Indian Mathematicians

Srinivasa Ramanujan

Srinivasa Ramanujan

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

Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below

Although there are numerous statements that could bear the name Ramanujan conjecture, there is one statement that was very influential on later work.  Ramanujan conjecture is an assertion on the size of the tau function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence ofPierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for his work on Weil conjectures.

The Ramanujan conjecture

Ramanujan–Hardy number 1729

• The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see Ramanujan. In Hardy's words.

• “I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."”

• The two different ways are these: 1729 = 13 + 123 = 93 + 103Generalizations of this idea have

created the notion of "taxicab numbers". Coincidentally, 1729 is also a Carmichael number.

AryabhataAryabhatta was born in 476A.D in

Kusumpur, India. Aryabhata is the first well known Indian mathematician. Born in Kerala, he completed his studies at the university of Nalanda. 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

HIS CONTRIBUTIONS

He made the fundamental advance in finding the lengths of chords of circles He gave the value of as 3.1416, claiming, for the first time, that it was an approximation. He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax -by = c, and also gave what later came to be known as the table of Sines. He also wrote a text book for astronomical calculations, Aryabhatasiddhanta. Even today, this data is used in preparing Hindu calendars (Panchangs). In recognition to his contributions to astronomy and mathematics, India's first satellite was named Aryabhata.

Bhaskracharya

Bhāskara  (also known as Bhāskara II and Bhāskarāchārya ("Bhāskara the teacher"), (1114–1185), was an Indian mathematician and astronomer. He was born near Vijjadavida (Bijāpur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. He lived in the Sahyadri region.

He was the first to give that any number divided by 0 gives infinity (00). He has written a lot about zero, surds, permutation and combination. He wrote, “The hundredth part of the circumference of a circle seems to be straight. Our earth is a big sphere and that’s why it appears to be flat.” He gave the formulae like sin(A ± B) = sinA.cosB ± cosA.sinB 

WHAT HE SAID?

. His famous book Siddhanta Siromani is divided into four sections -Leelavati , Bijaganita , Goladhayaya, and Grahaganita. Leelavati contains many interesting problems and was a very popular text book. Bhaskara introduced chakrawal, or the cyclic method, to solve algebraic equations.. Bhaskara can also be called the founder of differential calculus. He gave an example of what is now called "differential coefficient" and the basic idea of what is now called "Rolle's theorem". Unfortunately, later Indian mathematicians did not take any notice of this. Five centuries later, Newton and Leibniz developed this subject.

His Works

BRAHMAGUPTAHe was born in 598 AD

in Bhinmal city in the state of Rajasthan. He is renowned for introduction of negative numbers and operations on zero into arithmetic. He was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy.

Brahmagupta (598 A.D. -665 A.D.)

His Contributions

His main work was Brahmasphutasiddhanta, which was a corrected version of old astronomical treatise Brahmasiddhanta. This work was later translated into Arabic as Sind Hind. He formulated the rule of three and proposed rules for the solution of quadratic and simultaneous equations. He gave the formula for the area of a cyclic quadrilateral as where s is the semi perimeter. He gave the solution of the indeterminate equation Nx²+1 = y². He is also the founder of the branch of higher mathematics known as "Numerical Analysis".

Brahmagupta's theorem• Brahmagupta's theorem  states that if

a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.

• More specifically, let A, B, C and D be four points on a circle such that the lines AC and BD are perpendicular. Denote the intersection of AC and BD by M. Drop the perpendicular from M to the line BC, calling the intersection E. Let F be the intersection of the line EM and the edge AD. Then, the theorem states that F is the midpoint AD.

Mahavira• Mahavira was a 9th-century

Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. He gave the sum of a series whose terms are squares of an arithmetical progression and empirical rules for area and perimeter of an ellipse. He was patronised by the great Rashtrakuta king Amoghavarsha. Mahavira was the author of Ganit Saar Sangraha.

What he did?• He separated Astrology from

Mathematics. He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. He is highly respected among Indian Mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle. Mahavira’s eminence spread in all South India and his books proved inspirational to other Mathematicians in Southern India.

Varāhamihira

Varāhamihira (505–587 CE), also called Varaha or Mihira, was an Indian astronomer, mathematician, and astrologer who lived in Ujjain.[1] He is considered to be one of the nine jewels (Navaratnas) of the court of legendary ruler Vikramaditya

HIS CONTRIBUTIONS

Varahamihira (505-587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions:

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