Indian Mathematics

"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

400 - 1200 CE

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

most Indian mathematicians were

astronomers

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

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"

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)

Brahmagupta's rule of three

ratios can be solved by cross multiplying