Indian Mathematics

Bhaskara

(114-1185 CE) - most well-known Indian mathematician - first person to declare that any number divided by zero is infinity - also the first to declare that the sum of any number and infinity is infinity - Famous for his book "Siddhanta Siromani" - introduced "chakrawal" (cyclic method) to solve algebraic equations - can be called the founder of differential calculus - gave an example of what is now called "differential coefficients" - renowned for his concept of instantaneous motion

Aryabhata

(475-550 CE) - first well-known Indian mathematician - wrote "Aryabhatiya" (499 CE) - wrote a test for astronomical calculations titled: "Aryabhatasiddhata" which is still used today for preparing Hindu calendars - India's first satellite was named "Aryabhata" in recognition of his contributions to astronomy and mathematics - along with another Indian mathematician, he found the solution of linear Diophantine equations of the form ax + by = c, where a, b, and c are integers and one is looking for all integer solutions x,y

Brahmagupta

(598-665 CE) - introduction of negative numbers - arithmetic operations with zero - main work was "Brahmsphutasiddhanta" - formulated the "rule of three" - proposed rules for the solutions of quadratic and simultaneous equations - gave the formula for the area of a quadrilateral inscribed in a circle in terms of a semiperimeter "s" - was the first mathematician to treat algebra and arithmetic as two different branches of mathematics - gave solutions to the indeterminate equation Nx^2 + 1 = y^2 (a special case of this came to be known as Pell's equation) - considered to be the founder of Numerical Analysis

"Siddhanta Siromani"

- 1150 CE - written by Bhaskara 1) Leelavati (arithmetic) 2) Bijaganita (algebra) 3) Goladhaya (sphere-celestial globe) 4) Grahaganita (mathematics and the planets)

"Granita Sara Sangraha"

- 850 CE - first textbook on arithmetic in present-day form

Chord/half-chord of a circle

- Greek trigonometric calculations revolved around the chord of an angle - Indians learned the theory from Hipparchus, a predecessor of Ptolemy - Mathematicians realized the importance of starting with a central angle (alpha), doubling it (2 alpha), then taking half the chord of 2 alpha => called this "jya"

400 - 1200 CE

- Indian mathematics flourished - most fundamental contribution => invention of the decimal system and zeros - introduced and worked with negative numbers

Sulba Sutras

- dates back to the 8th century BCE - lists several simple Pythagorean Triples - simple Pythagorean theorem for the sides of a square - geometric solutions for linear and quadratic equations in a single unknown - close figure of sqrt(2) => 1 + (1/3) + (1/(3)(4)) + (1/34)

differential coefficients

- example of this was given by Bhaskara - version of the derivative and basic idea of what is known as Rolle's Theorem

Kerala School of Astronomy and Mathematics

- founded by Madhava of Sangamagrama in Kerala, South India - flourished between the 14th and 16th centuries - most important mathematical concept => series expansions for trigonometric functions - able to develop Taylor Series expansions, differentiation, term-by-term integration, convergence tests, and iterative methods for solutions of non-linear equations - developed the theory that the area under a curve is its integral, but not a full blown calculus of the integral and derivative

What does "jya" have to do with sine?

- jya = half of the chord of 2alpha - the length of jya = sin(alpha)

Jya

- means "half-chord" - mistranslated into Latin as "sinus" which is where we get the word sine

"Aryabhatiya"

- written by Aryabhata in 499 CE - astronomical treatise section titled "Granita" (calculations) - found the lengths of chords of circles by using the half-chord rather than the full chord method - gave a value of pi to be 3.1416, claiming for the first time that it was an approximation - gave methods for extracting square roots, summing arithmetic, and geometric series - provided (what would later be known as) the table of sines

Mahavira

- wrote "Granita Sara Sangraha" - only Indian mathematician to refer to an ellipse (the Greeks, by contrast, studied conic sections in great detail)

most Indian mathematicians were

astronomers

Brahmagupta's rule of three

ratios can be solved by cross multiplying