Square root in Pāṭīgaṇitam

Here we will be knowing a square root of a number as mentioned in Pāṭīgaṇitam.

Verse

विषमात् पदतस्त्यक्त्वा वर्गं स्थानच्युतेन मूलेन ।

द्विगुणेन भजेच्छेषं लब्धं विनिवेशयेत् पङ्क्तौ ॥ २५ ॥

तद्वर्गं संशोध्य द्विगुणं कुर्वीत पूर्ववल्लब्धम् ।

उत्सार्य ततो विभजेच्छेषं द्विगुणीकृतं दलयेत् ॥ २५ ॥

Translation

Having subtracted the (greatest possible) square from the (last) odd place (set down double the square root underneath the next place). By that double square root, that has left its place (i.e., which has been set down underneath the next place), divide the remainder; set down the quotient in the line (of double the square root), and, having subtracted the square of that (from the number above), make double of that (quotient). Then having moved the resulting quantity (in the line of double the square root) one place forward, divide by it as before. (Continue this process till all places are exhausted, and then) halve the doubled quantity (to get the square root).

This rule will be clear by the following examples:

Example: Square root of 186624

Starting from right denote the odd and even place by o and e respectively.

e	o	e	o	e	o
1	8	6	6	2	4
←

Subtract the greatest possible square (16 = 4²) from the last odd place (18) .18 -16 = 2. Write the double the square root of 16 ( which is 2 x 4 = 8) underneath the next place. We have

o	e	o	e	o
2	6	6	2	4	← remainder
	8				← line of double the square root

Divide 26 by 8 . Here the quotient will be 3 and remainder will be 26 - 24 = 2. Write the quotient 3 in the line of double the square root.

e	o	e	o
2	6	2	4	← remainder
8	3			← line of double the square root

Subtract the square of the quotient (i.e 3² = 9) 26 - 9 = 17. Write the double the square of the quotient (i.e 3 x 2 = 6) in the line of double the square root.

e	o	e	o
1	7	2	4	← remainder
8	6			← line of double the square root

At this stage one round of operation is over. Now move 86 one place forward.

e	o	e	o
1	7	2	4	← remainder
	8	6		← line of double the square root

Divide 172 by 86 . Here quotient is 2 and remainder is 172 - 172 = 0. Write the quotient (2) in the line of double the square root.

e	o	e	o
			4	← remainder
	8	6	2	← line of double the square root

Finally subtracting the square of the quotient (2² = 4) from above (4 - 4 = 0) and doubling the quotient (2) 2 x 2 = 4 we get

e	o	e	o
			0	← remainder
	8	6	4	← line of double the square root

The process now ends. Hence we divide 864 by 2 = 432 is the required square root.

As the remainder is zero . the square root is exact.

Square root of 186624 = 432

Example: Square root of 11943936

Starting from right denote the odd and even place by o and e respectively.

e	o	e	o	e	o	e	o
1	1	9	4	3	9	3	6
←

Subtract the greatest possible square (9 = 3²) from the last odd place (11).11 - 9 = 2. Write the double the square root of 9 (which is 2 x 3 = 6) underneath the next place. We have

o	e	o	e	o	e	o
2	9	4	3	9	3	6	← remainder
	6						← line of double the square root

Divide 29 by 6 . Here the quotient will be 4 and remainder will be 29 - 24 = 5. Write the quotient 4 in the line of double the square root.

e	o	e	o	e	o
5	4	3	9	3	6	← remainder
6	4					← line of double the square root

Subtract the square of the quotient (i.e 4² = 16) 54 - 16 = 38. Write the double the square of the quotient (i.e 4 x 2 = 8) in the line of double the square root.

e	o	e	o	e	o
3	8	3	9	3	6	← remainder
6	8					← line of double the square root

At this stage one round of operation is over. Now move 68 one place forward.

e	o	e	o	e	o
3	8	3	9	3	6	← remainder
	6	8				← line of double the square root

Divide 383 by 68 . Here quotient is 5 and remainder is 383 - 340 = 43 Write the quotient (5) in the line of double the square root.

o	e	o	e	o
4	3	9	3	6	← remainder
6	8	5			← line of double the square root

Finally subtracting the square of the quotient (5² = 25) from above (439 - 25 = 414), and doubling the quotient (5) 2 x 5 = 10 we get

o	e	o	e	o
4	1	4	3	6	← remainder
6	8	10			← line of double the square root
6	8+1	0
6	9	0

At this stage second round of operation is over. Now move 690 one place forward.

o	e	o	e	o
4	1	4	3	6	← remainder
	6	9	0		← line of double the square root

Divide 4143 by 690 . Here quotient is 6 and remainder is 4143 - 4140 = 3 Write the quotient (6) in the line of double the square root.

		e	o
		3	6	← remainder
6	9	0	6	← line of double the square root

Finally subtracting the square of the quotient (6² = 36) from above (36 - 36 = 0), and doubling the quotient (6) 2 x 6 = 12 we get

		e	o
			0	← remainder
6	9	0	12	← line of double the square root
6	9	0+1	2
6	9	1	2

The process now ends. Hence we divide 6912 by 2 = 3456 is the required square root.

As the remainder is zero . the square root is exact.

Square root of 11943936 = 3456

See Also

पाटीगणितम् में 'वर्गमूल'