Agenda

Aryabhatta - 476 - 520 AD

Varahamihira - 505 – 587 AD

Brahmagupta - 598 – 655 AD

Bhaskra–I - 962 -1019 AD

Aryabhatta476 - 520 AD

Aryabhatta

The place-value system

Knowledge of zero was implicit in Aryabhata's place -value system as a place holder for the powers of ten with

null coefficients.

• Place value system and zero

Aryabhatta

Area of a triangle as “tribhujasya phalashariram samadalakoti bhujardhasamvargah”.

That translates to: "for a triangle, the result of a perpendicular with the half-side is the area.“

• Mensuration and Trigonometry

Aryabhatta

In “Aryabhatiya”, Aryabhata provided elegant results for the summation of series of squares and

cubes and

•Algebra

Varahamihira505 AD - 587 AD

Varahamihira• Trigonometry

Some important trigonometric results attributed to Varahamihira.

Embellished findings in attractive poetic and metrical styles.

The usage of a large variety of meters is especially evident in his Brihat Jataka and Brihat-Samhita.

Brahmagupta598 – 655 AD

Brahmagupta

• Zero

Brahmagupta's Brahmasphutasiddhanta is the very first book that mentions zero as a number, hence Brahmagupta is considered as the man who found zero.

He gave rules of using zero with other numbers.

Zero plus a positive number is the positive number etc.

Many cultures knew four fundamental operations.

The way we do nowbased on Hindu arabic number system first appeared in Brahmasputa siddhanta. Contrary to popular opinion, the four fundamental operations (addition, subtraction, multiplication and division) did not appear first in Brahmasputha Siddhanta, but they were already known by the Sumerians at least 2500 BC.

• Arithmetic

Brahmagupta

Formula for cyclic quadrilaterals in geometry.

Given the lengths of the sides of any cyclic quadrilateral, he gave an approximate and an exact formula for the figure's area.

The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral.

Brahmagupta

• Geometry

Brahmagupta

In some of the verses before verse 40, Brahmagupta gives constructions of various figures with arbitrary sides.

He essentially manipulated right triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and a scalene cyclic quadrilateral.

• Measurements and constructions

Bhaskra–I 962 -1019 AD

Bhaskra–I

The representation of numbers in a positional system.

Numbers were not written in figures, but in words and were organized in verses.

Eg. The number 1 was given as moon, since it exists only once; the number 2 was represented by wings, twins, or eyes, since they always occur in pairs; the number 5 was given by the (5) senses.

• Representation of numbers

Bhaskra–I

Bhaskara stated theorems about the solutions of today so called Pell equations.

In modern notation, he asked for the solutions of the Pell equation 8x2 + 1 = y2.

It has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, e.g., (x,y) = (6,17).

• Further contributions