Mathematics In India

Aryabhatta: The Great Mathematician

Aryabhatta wrote the Aryabhatiya, a 121-verse manuscript (476-529 A.D.). He invented methods for arithmetic, geometry, algebra, and trigonometry in addition to astronomy. Also calculated the value of the 'pi' to be 3.1416 and talked about mathematical concepts like numerical squares and cube roots.

Through sine functions, Aryabhata is credited with developing trigonometry.

Background

Aryabhatiya was particularly well-known in South India, where it influenced a number of mathematicians in the subsequent era. This poem, written in poetry couplets, is on mathematics and astronomy.

Following an introduction that includes astronomical tables and Aryabhata's vowel number notation system, the book is divided into three sections.

• Ganita (mathematics)

• Kala-kriya (time calculations)

• gola (philosophy, sphere).

Ganita: Aryabhata defines the first ten decimal places in ganita and explains how to use the decimal number system to calculate square and cubic roots. He then discusses geometric measures in addition to the properties of two intersecting circles and identical right-angled triangles.

Aryabhata also discovered a method for constructing a sin table using Pythagoras' theorem.

Additionally, mathematical series, quadratic equations, compound interest ratios and proportions, and solutions to various linear equations are all included.

Bhaskara I named Aryabhata's general solution for linear indeterminate equations kuttakara ("pulveriser") because it involved decomposing the problem into smaller and smaller problems with decreasing coefficients, a process analogous to the Euclidean algorithm and also related to the method of continued fractions.

Kala-kriya: Aryabhata then studied astronomy, with an emphasis on the planets' motions along the ecliptic.

Among the subjects covered are time definitions, eccentric and epicyclic models of planetary motion, planetary longitude corrections for various terrestrial locations, and a theory of "lords of the hours and days" (an astrological concept used for determining propitious times for action).

Gola: Aryabhatiya concludes gola with spherical astronomy, in which he demonstrates how to apply plane trigonometry to spherical geometry by projecting points and lines on the surface of a sphere onto appropriate planes.

• Solar and lunar eclipses are predicted, and the apparent westward movement of the stars is explained explicitly as a result of the spherical Earth rotating about its axis.

Aryabhata also correctly identified reflected sunlight as the source of illumination for the Moon and planets.

During Iran's Sasanian period, Aryabhatasiddhanta had a significant influence on the development of Islamic astronomy. Its contents were preserved in part through the writings of Varahamihira, Bhaskara I, and Brahmagupta.

It is one of the earliest works of astronomy, establishing midnight as the start of each day.

Varahamihira

Varahamihira lived in Ujjain and wrote the Panchasiddhan Tika, Brihat Samhita, and Brihat Jataka, among other works. Five early astronomical systems, including the Surya Siddhanta, are summarised in the first chapter. Additionally, zero addition and subtraction are referred to as panchasiddhantika.

The Paitamaha Siddhanta, another system he describes, appears to have several similarities to Lagadha's ancient Vedanga Jyotisha. Brihat Samhita is a fascinating collection of themes that shed light on the ideas that were prevalent at the time. The Brihat Jataka appears to be an astrological text influenced heavily by Greek astrology.

Brahmagupta

Brahmagupta was born in Bhilamala, Rajasthan, in the year 598. He published his renowned work, Brahmasphuta Siddhanta, in 628 A.D. In western and northern India, his rivalry with Aryabhata's school had a huge influence.

Throughout 771 A.D., Brahmagupta's work was translated into Arabic in Baghdad and became known as Sindhind in the Arabic world.

• The solution of a critical second-order indeterminate problem in number theory is one of Brahmagupta's most important contributions. For decades, the Khandakhadyaka, another of his works, was a well-known reference manual for astronomical computations.

Bhaskara

Bhaskara was a brilliant mathematician and astronomer from Karnataka. One of his mathematical accomplishments is the 'concept of differentials'. He wrote the four-part novel Siddhanta Shiromani.

• Lilavati on arithmetic

• Bijaganita on algebra, 

• Ganitadhyaya

• Goladhyaya teaches astronomy.

The book includes examples of Pythagoras theorem applications, trirasika and kuttaka procedures, and questions on permutations and combinations, in addition to various introductory topics in arithmetic, the geometry of triangles, and quadrilaterals.

The Bijaganita is an advanced algebra book, and it is the first of its kind in Indian culture. Operations with unknowns, as well as the kuttaka and chakravala techniques for solving indeterminate equations, are among the topics covered.

• His planetary movements theories are more refined than the older Siddhanta's. Kerala had a thriving legacy of mathematics and astronomy following Bhaskara, which considered itself a successor to the Aryabhata school.

• Sankara's Yuktidipika and Kriyakramakari, as well as Jyeshthadeva's Ganitayuktibhasha in Malayalam, are among the many other exponents of the school who have written expositions and comments. 

• Kerala mathematician's work foreshadowed the subsequent development of calculus in Europe in many ways.

Madhava

Madhava (1340-1425 A.D.) devised a method for determining the moon's location every 36 minutes. He also developed methods for calculating the planets' motions.

• He expanded trigonometric functions and 'pi' to eleven decimal digits using power series expansions.

Madhava developed his own method of calculus-based on his trigonometry abilities. He was a self-taught mathematician from Kerala who lived a century before Newton and Liebnitz.

Nilakantha Somayaji

Nilakantha (1444-1545 A.D.) was a renowned astronomer who authored a number of works.

Nilakantha appears to have discovered the correct formulation for the planets' centres of mass equation, and his solar system model must be considered authentically heliocentric. He also improved Madhava's power series methods.

Mathematicians in Kerala developed methods light years ahead of those used in nineteenth-century European mathematics.

The Decimal System and the Origin of Zero

Zero was known to the ancient Indians, and this knowledge spread beyond India. Both the decimal and binary systems have their origins in India.

• In 825 and 1025 A.D., Al Khawarizmi and Al-Nasavi refer to it as ta-rikh ai Hindi and al-amal al-Hindi, respectively.

• Brahmagupta and Bhaskar devised several rules for performing mathematical operations on zero, the latter having a good understanding of number systems and the ability to solve equations.

• The symbols for nine digits and a zero had been established in their entirety by the fifth century A.D.

Binomial Numbers

According to Barend van Nooten, binary numbers were known at the time of Pingala's Chhandahshastra. Pingala, who lived in the early first century B.C., used binary digits to classify Vedic metres. Knowledge of binary numbers demonstrates a strong understanding of mathematics.

The Pramanas' Triad: There are three main elements to acquiring mathematical knowledge and expertise that are essential for mastering any field.

• Pratyaksha (perception)

• Anumana, (inference)

• Agama or Sabda, or textual or traditional knowledge.

Pramanas are the collective name for these three elements.

India's geometry

In India, geometry arose as a method of constructing altars for Vedic sacrifices. Early Hindu geometric studies are known as sulvas.

• The Sulvas, also known as the Sulva Sutras, are instructions for constructing altars of worship.

• The Sulva geometers must have implicitly accepted a number of postulates in order to perform geometric procedures.

• The Sulva postulates deal with how geometric forms like straight lines, rectangles, circles, and triangles are divided.

The Sulvasutra geometers were aware of the Pythagoras theorem over two hundred years before Pythagoras (all four major Sulvasutras contain an official declaration of the theorem) and addressed issues such as finding a circle with the same area as a square and vice versa during their studies.

Mathematical Tradition of the Jain

The mathematical development of the country has been aided by the Jain tradition. The Jain scholars, unlike the Vedic people, got their inspiration for mathematics from contemplating the universe rather than religious acts. Even primarily philosophical Jain texts contain mathematical ideas.

The Jains developed a complex cosmography that included mathematics.

• Circle geometry, the arithmetic of huge powers of 10, permutations and combinations, and classifications of infinities were among the topics covered in the early Jain texts, which date from around the fifth century B.C. to the second century A.D. (whose plurality had been recognised by the Jains).

• Furthermore, Jain scriptures provide unique methods for calculating circular arc lengths in terms of the length of the corresponding chord and the height above the chord, as well as the area of areas subtended by circular arcs and their chords.

Vedic Mathematics: A Misnomer

The term "Vedic mathematics" refers to Shri Bharati Krishna Tirthaji's 1965 book of the same name.

• The book describes shortcut methods for certain types of arithmetic or algebraic calculations, as well as short Sanskrit phrases that serve as memory aids for the individual exercises; sixteen of these phrases are referred to as sutras,' and another group of similar phrases as'sub-sutras.'

• These sentences have been proven to have no origin in Vedic literature and are external to the Vedas in terms of language, style, and mathematical substance.

• For the most part, the mathematics detailed in the book is a twentieth-century concept known as Vedic mathematics.

Key takeaways

• The Indus Valley Civilization and the Vedic Sulba Sutras demonstrate early, sophisticated uses of geometry and standardized measurements for urban planning and rituals.
• Indian mathematicians transformed zero from a philosophical concept ('shunya') into a foundational numerical value and formalized the base-10 decimal place-value system.
• Brahmagupta established the fundamental rules for arithmetic operations involving zero and introduced the concept of negative numbers.
• The Kerala School of Mathematics discovered infinite series for trigonometric functions centuries before the European development of calculus.
• Srinivasa Ramanujan's intuitive 20th-century discoveries in number theory and infinite series continue to influence modern advanced mathematics and physics.

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