Page 18 - Guidelines for Training/ Orientation of Faculty on IKS

6. Development of Place Value System

Earliest evidence of the use of place value system. Numerals found in the inscriptions (Brāhmi &
Kharosṭhi). Use of Zero as a symbol in Piṅgala’s Chandaḥśāstra. References to use of decimal place
value system in the commentary Vyāsabhāṣya on Yogasūtra and in Southeast Asian Inscriptions.
Different systems of numeration employing place value system. Bhūtasaṅkhyā system. Āryabhaṭan
system. Kaṭapayādi system. Algorithms for arithmetical operations based on decimal place value
system.

7. Āryabhaṭīya of Āryabhaṭa
Āryabhaṭa, his period and his work Āryabhaṭīya. Names of the notational places. Square and Squaring.
Algorithm for finding the square root. Cube and cubing. Algorithm for finding the cube root. Formula
for the area of a triangle. Bhāskara I on altitude and area of a triangle. Numerical examples

Area of a circle, trapezium and other planar figures. Approximate value of π. Computation of tabular

Rsines (geometric and difference equation methods). Approximate formula for Rsine (as given by
Bhāskara I). Problems related to gnomonic shadow. Bhujā-koṭi-karṇa-nyāya, jyā-śara-nyāya and their
applications. Arithmetic progressions. Finding sum of natural numbers, sum of sums, and so on.

Some algebraic identities. Rule of three. Problems on interest calculation. Ekavarṇa-samikaraṇa and
anekavarṇa-samikaraṇa. The Kuṭṭaka problem (sāgra and niragra-kuṭṭaka). Illustrative examples.

8. Brāhmasphuṭasiddhānta of Brahmagupta

Introduction. Twenty logistics. Cube root. Rule of Three, Five Seven, etc. Mixtures. Interest calculations,
etc. Progressions: Arithmetic and Geometric. Plane figures. Triangles, right triangles and quadrilaterals.

Diagonals of a cyclic quadrilateral. Rational triangles and quadrilaterals. Chords of a circle. Volumes
with uniform and tapering cross-sections. Pyramids and frustum. Shadow problems.

Mathematical operations with plus, minus and zero. Rules in handling surds (karaṇī) Operations with
unknowns (avyakta-ṣaḍvidha). Equations with single unknowns (ekavarṇa-samīkaraṇa). Equations
with multiple unknowns (anekavarṇa-samīkaraṇa). Equations with products of unknowns (bhāvita).
Brahmagupta on kuṭṭaka. The Second order indeterminate equation (Vargaprakṛti). Bhāvanā principle
and its applications.

9. Bakṣālī Manuscript

The discovery of Bakṣālī Manuscript. Its antiquity and uniqueness. Use of symbols. Symbol for
negative sign (kṣaya). Symbol for denoting unknown quantities (yāvatāvat). Solution of indeterminate
equations. Formula for approximate value of surds. Some interesting problems involving
simultaneous equations.

10. Gaṇitasārasaṅgraha of Mahāvīra
Introduction. Arithmetical operations, operations with zero. Squares, cubes, square roots, cube
roots. Arithmetical and Geometric progressions, Citi (summation). Manipulations with fractions and
solutions of equations. Mixed problems including interest calculations.

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