Square root in Āryabhaṭīyam

Here we will be knowing how to find the square root as mentioned in Āryabhaṭīyam.

Verse

भागं हरेदवर्गान्नित्यं द्विगुणेन वर्गमूलेन ।

वर्गाद् वर्गे शुद्धे लब्धं स्थानान्तरे मूलम् ॥

Translation

Digits starting from the left till the square place to be subtracted from the maximum square value. Divide the non square place by twice the square root. Subtract the square of the quotient from the next square place. This process to be repeated till the last digit.

Example: Square root of 88209

Starting from unit place mark odd place as Varga (square) (V) and even place as Avarga (non-square) (A).

V	A	V	A	V
8	8	2	0	9
←

	V	A	V	A	V	Step details	Square Root
	8	8	2	0	9
- 22 = 4	4					Subtract the maximum possible square (4 = 22) from the digits till the left most-varga digit (8) . Now the square root is 2 and written in the Square root column	2
÷ 2 X 2 = 4)	4	8	(9			Bring down the digit of the next Avarga digit (8) and place it to the right of the remainder (4). Now the number is 48 and divide this by twice the square root that is currently in the square root column (2) = 2 X 2 = 4
	3	6				Subtract the above number from the maximum possible number 4 X 9 = 36 such that the quotient is less than 10. Here the quotient is 9.
	1	2	2			Bring down the the digit of the next varga digit (2) and place it to the right of the remainder (12), Now the number is 122
-92		8	1			Deduct square of the quotient (9) = 81 from it. Write the quotient (9) in the Square Root column next to the earlier number (2) . Now the square root is 29	2 9
÷ 2 X 29 = 58	58)	4	1	0	(7	Bring down the digit of the next Avarga digit (0) and place it to the right of the new remainder (41). Now the number is 410 and divide by twice the square root that is currently in the square root column (29) = 58
		4	0	6		Subtract the above number from the maximum possible number 58 X 7 = 406. Here the quotient is 7.
				4	9	Bring down the digit of the next varga digit (9) and place it to the right of the remainder (4), Now the number is 49
-72				4	9	Deduct square of the quotient (7) = 49 from it. Write this quotient (7) next to the square root obtained till now (29) in the square root column.	2 9 7
					0

Since the remainder is zero the given number is a perfect square.

Square root of 88209 = 297

Example: Square root of 117649

Starting from unit place mark odd place as Varga (square) (V) and even place as Avarga (non-square) (A).

A	V	A	V	A	V
1	1	7	6	4	9
←

	A	V	A	V	A	V	Step details	Square Root
	1	1	7	6	4	9
- 32 = 9		9					Subtract the maximum possible square (9 = 32) from the digits till the left most-varga digit (11) . Now the square root is 3 and written in the Square root column	3
÷ 2 X 3	6)	2	7	(4			Bring down the digit of the next Avarga digit (7) and place it to the right of the remainder (2). Now the number is 27 and divide this by twice the square root that is currently in the square root column (3) = 2 X 3 = 6
		2	4				Subtract the above number from the maximum possible number 6 X 4 = 24. Here the quotient is 4.
			3	6			Bring down the the digit of the next varga digit (6) and place it to the right of the remainder (3), Now the number is 36
- 42			1	6			Deduct square of the quotient (4) = 16 from it. Write the quotient (4) in the Square Root column next to the earlier number (3) . Now the square root is 34	3 4
÷ 2 X 34 = 68		68)	2	0	4	(3	Bring down the digit of the next Avarga digit (4) and place it to the right of the new remainder (20). Now the number is 204 and divide by twice the square root that is currently in the square root column (34) = 68
			2	0	4		Subtract the above number from the maximum possible number 68 X 3= 204. Here the quotient is 3.
					0	9	Bring down the digit of the next varga digit (9) and place it to the right of the remainder (0), Now the number is 9
-32						9	Deduct square of the quotient (3) = 9 from it. Write the quotient (3) in the Square Root column next to the earlier number (34) . Now the square root is 343	3 4 3
						0

Since the remainder is zero the given number is a perfect square.

Square root of 117649 = 343

See Also

आर्यभटीयम् में 'वर्गमूल'