Parikarmāṣṭaka - Fundamental Operations

Introduction

Arithmetic deals with calculations using numbers. Pāṭīgaṇita is the Samskrit word for arithmetic and geometry .Pāṭīgaṇita is formed by combining the words Pāṭī (slate) and gaṇita (mathematics). Since gaṇita was done using a board of a slate , it was called Pāṭīgaṇita. For all transactions using numbers will require the basic operations of addition, subtraction, multiplication, division, squaring etc. Ancient Indian Mathematicians mentioned eight fundamental operations together called as Parikarmāṣṭaka.

Definition

Parikarma means arithmetic operations and aṣṭaka means group of eight. Parikarmāṣṭaka signifies eight basic operations.

The eight fundamental operations are  :

1. Saṅkalanam (addition)
2. Vyavakalanam (subtraction)
3. Guṇana (multiplication)
4. Bhājana (division)
5. Varga (squaring)
6. Varga-mūla (square root)
7. Ghana (cubing) and
8. Gana-mūla (cube root)

Addition and subtraction form the basis of all calculations. Bhāskara I mentions in the below shloka

संयोगभेदा गुणनागतानि शुद्धेश्च भागो गतमूलमुक्तम् ।

व्याप्तं समीक्ष्योपचयक्षयाभ्यां विद्यादिदं द्व्यात्मकमेव शास्त्रम् ॥ (Āryabhaṭīya-bhāṣya in Gaṇitapāda, p.43)

"All arithmetical operations resolve into two categories though usually considered to be four i.e, addition, subtraction, multiplication and division. The two main varieties are increase and decrease. Addition means to increase and subtraction means to decrease. These two varieties of operation penetrate the whole of mathematics. Multiplication and evolution ( square etc,) are particular kinds of addition and division and involution( square root etc) are particular kinds of subtraction. Every mathematical operations involves increase or decrease. Hence the whole of this science should be known as consisting truly of these two operations only."

Saṅkalana and Vyavakalana (Addition and Subtraction)

Addition is the first fundamental operation in mathematics. Subtraction is the reverse of addition.

Bhāskara II has mentioned about these operations in his work on Līlavatī.

कार्यः क्रमादुत्क्रमतोऽथवाऽङ्कयोगो यथास्थानकमन्तरं वा ॥ (Līlavatī , vs.12, p.12)

"The addition or subtraction (of digits in the given numbers) is to be done place wise right to left or left to right."

Write the given numbers one below the other so that the digits are aligned to their place value. Then starting with the units place add or subtract the digits, later move to tens and so on.

Samskrita names for Addition - yoga (addition), saṃyoga (sum), saṃyojana (joining together) , saṃyuti (sum), saṃyuti (sum), saṇkalana (making together).

Guṇana (Multiplication)

The multiplication of whole numbers is repeated addition. For example :

$2\quad X\quad 4=2+2+2+2=8$

Samskrita names for Multiplication - āhati (multiplication), ghāta (product), [guṇana , hanana, hati, vadha] (multiplication).

2	X	4	=	8
↑		↑		↑
guṇya (multiplicand)		guṇaka (multiplier)		guṇana-phala (Result of Multiplication)

Methods of Multiplication :

• Rūpa-guṇana - Direct Method
• Khaṇḍa-guṇana - Split Method
• Bhakta-guṇana - Factor Method
• Sthāna-vibhāga-guṇana - Place wise multiplication
• Iṣṭonayug-guṇana (Adding or subtracting a desired number)

Rūpa-guṇana - Direct method :

Here the tables of multiplier should be known. Multiplier is taken as a whole.Each digit of multiplicand is multiplied by the multiplier to get the product. In this method, the multiplier is taken a whole since it is small.

Example: 234 X 5 =

(1) (2)

2 3 4

x 5

1 1 7 0

Khaṇḍa-guṇana - Split Method :

Here the multiplier is split into sum of two numbers . This is represented as below

a X b = a X (c + d) = (a X c) + (a X d) where b = c + d.

This is the distributive property of multiplication over addition.

234 X 16 = 234 X (10 + 6 ) = (234 X 10) + (234 X 6) = 2340 + 1404 = 3744