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In ancient time, mathematics was mainly used in an auxiliary or applied role. Thus, mathematical methods were used to solve problems in architecture and construction (as in the public works of the Harappan civilization) in astronomy and astrology (as in the words of the Jain mathematicians) and in the construction of Vedic altars (as in the case of the Shulba Sutras of Baudhayana and his successors). By the sixth or fifth century BCE, mathematics was being studied for its own sake, as well as for its applications in other fields of knowledge.

Supplementary to the Vedas are the Shulba Sutras. These texts are considered to date from 800 to 200 BCE. Four in number, they are named after their authors: Baudhayana (600 BCE), Manava (750 BCE), Apastamba (600 BCE), and Katyayana (200 BCE ). The sutras contain the famous theorem commonly attributed to Pythagoras. Some scholars (such as Seidenberg) feel that this theorem as opposed to the geometric proof that the Greeks, and possibly the Chinese, were aware of.

The Shulba Sutras introduce the concept of irrational numbers, numbers that are not the ratio of two whole numbers. For example, the square root of 2 is one such number. The sutras give a way of approximating the square root of number using rational numbers through a recursive procedure which in modern language would be a ‘series expansion’. This predates, by far, the European use of Taylor series.

Jain Mathematics (600 BCE to 500 CE)

Jain cosmology led to ideas of the infinite. This in turn, led to the development of the notion of orders of infinity as a mathematical concept. By orders of infinity, we mean a theory by which one set could be deemed to be ‘more infinite’ than another. In modern language, this corresponds to the notion of cardinality. For a finite set, its cardinality is the number of elements it contains. However, we need a more sophisticated notion to measure the size of an infinite set. In Europe, it was not until Cantors work in the nineteenth century that a proper concept of cardinality was established.

No account of Indian mathematics would be complete without a discussion of Indian numerals, the place-value system, and the concept of zero. The numerals that we use even today can be traced to the Brahmi numerals that seem to have made their appearance in 300 BCE. But Brahmi numerals were not part of a place value system. They evolved into the Gupta numerals around 400 CE and subsequently into the Devnagari numerals, which developed slowly between 600 and 1000 CE.

By 600 CE, a place-value decimal system was well in use in India. This means that when a number is written down, each symbol that is used has an absolute value, but also a value relative to its position. For example, the numbers 1 and 5 have a value on their own, but also have a value relative to their position in the number 15. The importance of a place-value system need hardly be emphasized. It would suffice to cite an often-quoted remark by La-place: ‘It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

The elevation of zero to the same status as other numbers involved difficulties that many brilliant mathematicians struggled with. The main problem was that the rules of arithmetic had to be formulated so as to include zero. While addition, subtraction, and multiplication with zero were mastered, division was a more subtle question. Today, we know that division by zero is not well-defined and so has to be excluded from the rules of arithmetic. But this understanding did not come all at once, and took the combined efforts of many minds. It is interesting to note that it was not until the seventeenth century that zero was being used in Europe, and the path of mathematics from India to Europe is the subject of much historical research.

The Classical Era of Indian Mathematics (500 to 1200 CE )

The most famous names of Indian mathematics belong to what is known as the classical era. This includes Aryabhata I (500 CE) Brahmagupta (700 CE), Bhaskara I (900 CE), Mahavira (900 CE), Aryabhatta II (1000 CE) and Bhaskarachrya or Bhaskara II (1200 CE)

One of Aryabhata’s discoveries was a method for solving linear equations of the form

ax + by = c. Here a, b, and c are whole numbers, and we seeking values of x and y in whole numbers satisfying the above equation. For example if a = 5, b =2, and c =8 then x =8 and y = -16 is a solution. In fact, there are infinitely many solutions:

x = 8 -2m

y = 5m -16

where m is any whole number, as can easily be verified. Aryabhata devised a general method for solving such equations, and he called it the kuttaka (or pulverizer) method. He called it the pulverizer because it proceeded by a series of steps, each of which required the solution of a similar problem, but with smaller numbers. Thus, a, b, and c were pulverized into smaller numbers.

Amongst other important contributions of Aryabhata is his approximation of Pie to four decimal places (3.14146). By comparison the Greeks were using the weaker approximation 3.1429. Also of importance is Aryabhata’s work on trigonometry, including his tables of values of the sine function as well as algebraic formulate for computing the sine of multiples of an angle.

Mathematics in the Modern Age

Ramanujan (1887- 1920) is perhaps the most famous of modern Indian mathematicians. Though he produced significant and beautiful results in many aspects of number theory, his most lasting discovery may be the arithmetic theory of modular forms. In an important paper published in 1916, he initiated the study of the Pie function. The values of this function are the Fourier coefficients of the unique normalized cusp form of weight 12 for the modular group SL2 (Z). Ramanujan proved some properties of the function and conjectured many more. As a result of his work, the modern arithmetic theory of modular forms, which occupies a central place in number theory and algebraic geometry, was developed by Hecke.

Present

“Education should be started with mathematics. For it forms well designed brains that are able to reason right. It is even admitted that those who have studied mathematics during their childhood should be trusted, for they have acquired solid bases for arguing which become to them a sort of second nature”. Mathematics is around us. It is present in diﬀerent forms whenever we pick up the phone, manage the money, travel to some place, play soccer, meet new friends; unintentionally in all these things mathematics is involved. There are huge illustrations that testify the presence of mathematics

in everything that we are doing. With some good understanding of simple and compound interest,

you can manage the way your money grows. The mathematical concept that deals with the chance of winning a lottery game is probability. If you want to calculate how much paint,wallpaper,flooring, carpeting or tile you have to buy to for your project then you must know the area of wall or floor.

Geometry in clothing Importance of Mathematics in everyday life Geometry in house decoration

Importance of Mathematics in everyday lifeGeometry in art: the Volswagen Logo Importance of Mathematics in everyday life Geometry in architecture: the Eiffel/Paris. Every area of mathematics has its own unique applications to the different career options. For example Algebra: computer sciences, cryptology, networking, study of symmetry in chemisty and physics Calculus (diﬀerential equations): Chemistry, biology, physics engineering, the motion of water, rocket science, molecular structure, option price modeling in business and economics models, etc...

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