History of Hindu Mathematics: A Source Book v1.c8.s3: 3. Subtraction

History of Hindu Mathematics: A Source Book v1.c8.s3

Language: English

3. Subtraction

v1.c8.s3.1

**Terminology.** Āryabhaṭa II (950) gives the following definition of subtraction:

v1.c8.s3.2

"The taking out (of some number) from the *sarvadhana* (total) is subtraction; what remains is called *śeṣa* (remainder)."³

The terms *vyutkalita* (made apart), *vyutkalana* (making apart), *śodhana* (clearing), *pâtana* (causing to fall), *viyoga* (separation), etc., have been used for subtraction. The terms *śeṣa* (residue) and *antara* (difference) have been used for the remainder. The minuend has been called *sarvadhana* or *viyojya* and the subtrahend *viyojaka*.

v1.c8.s3.5

**The Operation.** Bhâskara II gives the method of subtraction thus:

v1.c8.s3.6

"Subtract the numbers according to their places in the direct or inverse order."⁴

v1.c8.s3.7

⁴ *L*, p. 2.

v1.c8.s3.8

| Sum of units | 2,5,2,3,8,0,0 | 20 |
| Sum of tens | 3,9,1,1,0 | 14 |
| Sum of hundreds | 1,0,0,1 | 2 |
| Sum of sums | | 360 |

v1.c8.s3.9

The horizontal process has been adopted by the commentator so that both the 'direct' and 'inverse' processes may be exhibited by a single illustration. It was never used in practice.

v1.c8.s3.10

**Direct Process.** Sûryadâsa¹ explains the process of subtraction with reference to the example. 1000 — 360 thus:

v1.c8.s3.11

¹ In his commentary on the *Lîlâvatî*.

v1.c8.s3.12

“Hence making the subtraction as directed, six cannot be subtracted from the zero standing in the tens place, so taking ten and subtracting six from it, the remainder (four) is placed above (six), and this ten is to be subtracted from the next place. For, as the places of unit, etc., are multiples of ten, so the figure of the subtrahend that cannot be subtracted from the corresponding figure of the minuend is subtracted from ten, the remainder is taken and this ten is deducted from the next place. In this way this ten is taken to the last place until it is exhausted with the last figure. In other words, numbers upto nine occupy one place, the differentiation of places begins from ten, so it is known ‘how many tens there are in a given number’ and, therefore, the number that cannot be subtracted from its own place is subtracted from the next ten, and the remainder taken.”

v1.c8.s3.13

The above refers to the direct process, in which subtraction begins from the units place.

v1.c8.s3.14

**Inverse Process.** The inverse process is similar, the only difference being that it begins from the last place of the minuend, and the previously obtained partial differences are corrected, if required. The process is suitable for working on a *pâṭî* (board) where figures can be easily rubbed out and corrected. This process seems to have been in general use in India, and was considered to be simpler than the direct process.²

v1.c8.s3.15

² According to Gaṅgâdhara, the inverse process of working is easier in the case of subtraction, and the direct in the case of addition.

Linked texts (2)

Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Mahāsiddhānta (Āryasiddhānta) — Āryabhaṭa II, Sanskrit