Squares in Līlāvatī

Introduction

Here we will know how to find the square of a number as mentioned in Līlāvatī.

Verse 19 :

समद्विघातः कृतिरुच्यतेऽथ

स्थाप्योऽन्त्यवर्गो द्विगुणान्त्यनिघ्नाः ।

स्वस्वोपरिष्टाच्च तथाऽपरेऽङ्काः

त्यक्त्वान्त्युमत्सार्य पुनश्च राशिम् ॥ 19 ॥

Translation

The product of a number with itself is called its square. To square a number use the following procedure: First write the square of the extreme left-hand digit on its top. Then multiply the next [i.e., second] digit by the double of the first digit and write the result on the top. Next, multiply the third digit by the double of the first digit and write the result on the top. In this way arrive at the unit's place. Next cross the first digit and shift the number so formed one place to the right. Then repeat the same procedure. Finally add all the products written at the top and the sum is the required square.

Example: Square of 297

	8	8	2	0	9	Steps: In row A write 297. We start from left side. Write square of 2 = 4 above 2. Twice of 2 is 4 . Multiply the second digit (9) by 4 = 36 . Write this number (36) above 4 with 6 is above 9. Then multiply the third digit (7) by 4 = 28. Write this number (28) above 36 with 8 above 7. Shift 297 to the right and cross out 2 as in the row X. Now the number is 97 The above procedure is repeated. Square of 9 = 81 is written above the line B with 1 is above 9. Twice of 9 is 18 . Multiply the last digit (7) by 18 = 126 . Write this number (126) above 81 with 6 is above 7. Again Shift 297 to the right and cross out 2 and 9 as in the row Y. Square of 7 = 49 is written above the line C with 9 is above 7. Finally add all the values above A , column wise from the right side.
D				4	9
C		1	2	6
		8	1
B		2	8
	3	6
	4
A	2	9	7
X		2	9	7
Y			2	9	7

Answer: Square of 297 = 88209

Example: Square of 425

	18	0	6	2	5	Steps: In row A write 425. We start from left side. Write square of 4 = 16 above 4. Twice of 4 is 8 . Multiply the second digit (2) by 8 = 16 . Write this number (16) above 16 with 6 is above 2. Then multiply the third digit (5) by 8 = 40. Write this number (40) above 16 with 0 above 5. Shift 425 to the right and cross out 4 as in the row X. Now the number is 25 The above procedure is repeated. Square of 2 = 4 is written above the line B with 4 is above 2. Twice of 2 is 4 . Multiply the last digit (5) by 4 = 20 . Write this number (20) above 4 with 0 is above 5. Again Shift 425 to the right and cross out 4 and 2 as in the row Y. Square of 5 = 25 is written above the line C with 5 is above 5. Finally add all the values above A , column wise from the right side.
D				2	5
C			2	0
			4
B		4	0
	1	6
	16
A	4	2	5
X		4	2	5
Y			4	2	5

Answer: Square of 425 = 180625

Example: Square of 12345

	1	5	2	3	9	9	0	2	5
F								2	5
E							4	0
						1	6
D						3	0
					2	4
					9
C					2	0
				1	6
			1	2
			4
B				1	0
				8
			6
		4
	1
A	1	2	3	4	5
P		1	2	3	4	5
Q			1	2	3	4	5
R				1	2	3	4	5
S					1	2	3	4	5

Answer: Square of 12345 = 152399025

See Also

लीलावती में 'वर्ग'