Mathematics

There may be so many reasons for considering subject Mathematics as a difficult one. One of the challenges that Mathematics teaching faces are in the way it is defined. Usually, we observed that the visualization of Mathematics remains centered around numbers, complicated calculations, algorithms, definitions and memorization of facts, short-cuts and solutions including proofs. These all make Maths a little bit difficult in learning and practicing. We often observe in educational institutions at the school level, the engaging with exploration and new thoughts is discouraged. Teachers work on the common belief that there can be only one correct way to solve a problem and that Mathematics does not have possibilities of multiple solutions. If we promote the student’s way of thinking or solving the Maths problem, we can reduce the pressure of Maths from the mind of scholars.
Students learn the Mathematics in the following Classes in a systematic way:

We should emphasize the need for multiple ways of attempting problems, understanding that Mathematics is about exploring patterns, discovering relationships and building logic. We want all teachers to engage students in reading the book and help them in formulating and articulating their understanding of different concepts as well as finding a variety of solutions for each problem. We should make emphasis on allowing children to work with each other in groups and make an attempt to solve problems collectively.

We should appreciate them to talk to classmates about Mathematics and create problems based on the concepts that have learnt. We should convey a message everybody to recognize that Mathematics is not only about solving problems set by others or learning proofs and methods that are developed by others but is about exploration and building new ideas, arguments and facts. Doing and learning Mathematics is, therefore, about each person coming up with his/her methods and own rules.

The purpose of Mathematics has widened to include exploring mathematisation of experiences. This means that students can begin to relate the seemingly abstract ideas they learn in the classrooms to their own experiences and organise their experiences using these ideas. This requires them to have the opportunity to reflect and express both their new formulations as well as their hesitant attempt on mathematising events around them. We should always be emphasised the importance of language and Mathematics interplay. We should also be tried to keep the language simple and close to the language that the child uses typically.

The higher aim of mathematics education should be to develop the inner resources in school children. That is, development of mental ability such as logical and abstract thinking, reasoning, analysing, problem solving, etc. The mode of the transaction should be based on constructivism, which facilitates the learners to construct their own knowledge. Conceptual understanding of basic ideas and problem solving are the two main components of mathematics learning. It is well known that Mathematics is challenging and promising. Mathematics is one of the most critical subjects which not only decide the career of many young students but also enhances their ability of analytical and rational thinking and forms a base for Science and Technology.

The Classical Period (400 – 1200), which is often known as the golden age of Indian Mathematics. During this period a lots of Indian mathematicians such as Aryabhata, Brahmagupta, Bhaskara I and Bhaskara II, Varahamihira, Mahavira, etc. had set a remarkable landmark in history. They gave a broader and clearer shape to several branches of mathematics. Their contributions were not only for Indian Mathematics but it spread to Asia, Europe and the countries of Middle East also. They had shown a great achievement in Vedic mathematics and contributed in the various fields of astronomical and mathematical facts. In fact, mathematics of that period was included in the ‘astral science’ (jyotisha-shatra) and consisted of three sub-disciplines: mathematical sciences (ganita or tantra), horoscope astrology (hora or jataka) and divination (samhita). This tripartite division is seen in Varahamihira’s 6th century compilation—Pancasiddhantika (literally panca, “five,” siddhanta, “conclusion of deliberation”, dated 575 CE) of five earlier works, Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. The most of the texts were composed in Sanskrit verse, and were followed by prose commentaries.

Fields of Indian mathematics

The study of mathematics in ancient India was mainly in Arithmetic, Geometry, Algebra, Mathematical Logic and General Mathematics.

Sharadchandra Shankar Shrikhande

He was an Indian mathematician known for distinguished and well-recognized achievements in combinatorial mathematics. He had also share his experience along with R. C. Bose and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782. His specialty was combinatorics and statistical designs. In Mathematics, Shrikhande graph is used in statistical designs.
Shrikhande was born on October 19, 1917. He received a Ph.D. in the year 1950 from the University of North Carolina at Chapel Hill under the direction of R. C. Bose. He had worded as professor and taught at various universities in the USA and in India. He was a professor of mathematics at Banaras Hindu University. Shrikhande is a fellow of the Indian National Science Academy, the Indian Academy of Sciences and the Institute of Mathematical Institute, USA.

Ramanujan

Ramanujan was born on 22nd of December 1887 in a small village of Tanjore district, Madras. He was good in Mathematics and logical subjects but failed in English in Intermediate, so his formal studies were stopped but his self-study of mathematics continued. Once he sent a set of 120 theorems to Professor Hardy of Cambridge. Professor Hardy get impressed and invited Ramanujan to England.
Do you know taxi number 1729 is now Ramanujan Number? Why? When Mr. Littlewood came to see Ramanujan. He hired a taxi numbered 1729. Mr. Littlewood had shown his curiosity about that number. Ramanujan declared 1729 as 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. Ramanujan proved that a big number can be written as sum of not more than four prime numbers. He also showed that how to divide the number into two or more squares or cubes.

Bhaskara I

Bhaskara had keen interest in astronomy. He wrote three astronomical contributions. In 629 he created the Aryabhatiya, about mathematical astronomy which was written in verses. The comments referred exactly to the 33 verses dealing with mathematics. The consideration of variable equations and trigonometric formulae were there. His work Mahabhaskariya about mathematical astronomy is divided into eight chapters. A remarkable approximation formula for sin x can be seen in chapter 7.

Bhaskara II

Bhaskara II (1114 -1185) or Bhaskaracharya (“Bhaskara the teacher”) was born in a village of Mysore district. He was the first Mathematician who describe that any number divided by 0 gives infinity. He has written a lot about 20, sequences, permutation, combination and surds. Bhaskara was an Indian mathematician and astronomer. Bhaskaracharya has been called the greatest mathematician of medieval India. He is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. In 12th century, Bhaskara and his works represent a significant contribution to mathematical and astronomical knowledge. His main work the Siddhanta Shiromani, Sanskrit for “Crown of treatises,” is divided into four parts called Lilavati, Bijaganita, Grahaganita and Goladhyaya. These four sections deal with arithmetic, algebra, mathematics of the planets and spheres respectively.
Some of Bhaskara’s contributions to mathematics include the following:

• A proof of the Pythagorean theorem by calculating the same area in two different ways and then cancelling out terms to get a² + b² = c².
• In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained. Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
• A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the Chakravala method.
• The first general method for finding the solutions of the problem x² – ny² = 1 (so-called “Pell’s equation”) was given by Bhaskara II.

Baudhayana

He, (fl. c. 800 BCE) was most likely also a priest but he was an Indian mathematician also. He was the author of the earliest Sulba Sutra – appendices to the Vedas giving rules for the construction of altars called the Baudhayana Sulbasútra, which contained several important mathematical results. He is much older than the famous mathematician Apastambha. He belongs to the Yajurveda school. He is accredited with calculating the value of pi to some degree of precision and with discovering a fact what is now known as the Pythagorean theorem.

Who is the father of Indian mathematics?

Aryabhata is the father of Indian mathematics. He was the mathematician who gave an accurate approximation for π.He correctly explains the causes of eclipses of the Sun and the Moon. His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.

He wrote in the Aryabhatiya that: Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given. This gives π = 62832/2000 = 3.1416 which is an almost accurate value.

Actually, π = 3.14159265 correct to 8 places. He also gave a table of sines to calculate the approximate values at intervals of 3 degree and 45 minutes. For this purpose, he used a formula for sin(n + 1)x — sin nx in terms of sin nx and sin(n − 1)x. He introduced the versine (versin = 1 – cosine) into trigonometry.

Aryabhata relates the radius of the planetary orbits with the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the orbits of the planets are ellipses later on it was found true.

Srinivasa Ramanujan

Srinivasa Ramanujan (1887-1920) was one of India’s geniuses and greatest mathematician. He had worked in the field of the analytical theory of numbers. He also performed remarkable works on elliptical functions, infinite series and continued fractions. In 1990 he started to work on mathematics summing in geometric and arithmetic series deriving his own methods. He also provided a method to solve cubic equations in 1902. In 1904 Ramanujan had focused his study towards the deep research. He developed the series ∑(1/n) and calculated Euler’s constant to 15 decimal places. In 1908, Ramanujan studied continued fractions and divergent series.

Sridhara (c. 870, India – c. 930 India)

He was a great Indian mathematician and wrote on practical applications of algebra which separates algebra from arithmetic. He was one of the first Mathematician who derive a formula for solving quadratic equations. He was broadly known for two treatises: Trisatika (sometimes called the Patiganitasara) and the Patiganita. His major work was Patiganitasara. It was witten in three hundred slokas that is why it was named Trisatika. This book contains the discussion of counting of numbers, natural number, division, zero, multiplication, squares, cubes, fraction, interest-calculation, joint business or partnership and mensuration.
He has written, “If O(zero) is added to any number, the sum is the same number; If 0(zero) is subtracted from any number, the number remains unchanged; If 0(zero) is multiplied by any number, the product is 0(zero)”. He has said nothing about division of any number by 0(zero). In the case of dividing a fraction he has found out the method of multiplying the fraction by the reciprocal of the divisor.

Mahavira

He was one of the great Indian Mathematician of 9th-century. He mainly belongs from Gulbarga. He had asserted that the square root of a negative number did not exist. He also gave the sum of a series whose terms are squares of an AP 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. The separation of Astrology from Mathematics is basically done by him. He is highly respected among all the Indian Mathematicians. Mahavira’s eminence spread in all South India and his books proved inspirational to other Mathematicians in Southern India.

Brahmagupta

He was the first mathematician who uses zero as a number. He set many rules to compute with zero. Only positive numbers appear in Brahmaphuta Siddhanta. Brahmagupta’s most famous work is his creation Brahmasphutasiddhanta. Chapter 18 of this book provides the solution of the general linear equation. It is important to note that Brahmagupta found the result in terms of the sum of the first n integers. He has given the formula of the sum of the squares of the first n natural numbers.
Brahmagupta’s Formula: His most famous result in geometrical Mathematics is his formula for cyclic quadrilaterals. Brahmagupta gave two formulae, one for an approximate and the other for an exact formula for the figure’s area. The approximate area is the product of the halves of the sums of the opposite sides of a quadrilateral. The accurate area is the square root the product of the half of the sum of the sides diminished by each side of the quadrilateral.

Varahamihira

Varahamihira (Devanagari) (505–587) was not only a mathematician but he was an astronomer and astrologer also. He was also called Varaha or Mihira. He lived in Ujjain and was considered to be one of the nine jewels of the court of legendary ruler Vikramaditya. Varahamihira’s main work was the book Pañcasiddhantika dated ca. 575 CE. This book contains all the information about older Indian texts which are now lost. It is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, like the Surya Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta, Romaka Siddhanta and Paitamaha Siddhantas. It is a summary of Vedanga Jyotisha as well as Hellenistic astronomy. Varahamihira was the first mathematician who mentioned the work of Pancha Siddhantika that the ayanamsa, or the shifting of the equinox is 50.32 seconds.

The Five Siddhantas

The 11th century Arabian scholar Alberuni also described the details of “The Five Astronomical Canons”: “They [the Indians] have 5 Siddhantas:

1. Surya-Siddhanta, ie. the Siddhanta of the Sun, composed by Latadeva,
2. Vasishtha-siddhanta, so called from one of the stars of the Great Bear, composed by Vishnucandra,
3. Pulisa-siddhanta, so called from Paulisa, the Greek, from the city of Saintra, which is supposed to be Alexandria, composed by Pulisa.
4. Romaka-siddhanta, so called from the Rum, ie. the subjects of the Roman Empire, composed by Srishena.
5. Brahma-siddhanta, so called from Brahman, composed by Brahmagupta, the son of Jishnu, from the town of Bhillamala between Multan and Anhilwara, 16 yojanas from the latter place.”