Cubes in Līlāvatī
Here we will know how to find the cube of a number as mentioned in Līlāvatī.
Verse 24 :
समत्रिघातश्च घनः प्रदिष्टः
स्थाप्यो घनोऽन्त्यस्य ततोऽन्त्यवर्गः ।
आदित्रिनिघ्नस्तत आदिवर्ग:
त्र्यन्त्याहतोऽथादिघनश्च सर्वे ॥ 24 ॥
Translation
Cube of a given number is its product with itself thrice over.[1] If we want to find the cube of a number of two digits, say, 10a + b, write a3 first. Below this, write 3a2 b by shifting this result one place to the right. Below this write 3ab2 after shifting it one place to the right. Below this write b3 after shifting it one place to the right. Add all the results, and the result is the cube. This procedure can be modified by starting from b but then each time the shifting should be made to the left. If there are more than two digits, then find the cube of the two digits at the extreme left and continue with the procedure given above.
Example: Cube of 27
27 = 10 X 2 + 7 which is the form 10a + b where a = 2 and b = 7
| a3 = 23 | 8 | 8 | |||||||||
| 3a2b = 3 X 22X 7 | 8 | 4 | Shift this one place to the right | 8 | 4 | ||||||
| 3ab2 = 3 X 2 X 72 | 2 | 9 | 4 | Shift this one place to the right | 2 | 9 | 4 | ||||
| b3 = 73 | 3 | 4 | 3 | 3 | 4 | 3 | |||||
| 1 | 9 | 6 | 8 | 3 |
Answer : 273 = 19683
Example: Cube of 125
125 = 10 X 12 + 5 which is the form 10a + b where a = 12 and b = 5
| a3 = 123 (Refer the calculation below) | 1 | 7 | 2 | 8 | 1 | 7 | 2 | 8 | ||||
| 3a2b = 3 X 122X 5 | 2 | 1 | 6 | 0 | Shift this one place to the right | 2 | 1 | 6 | 0 | |||
| 3ab2 = 3 X 12 X 52 | 9 | 0 | 0 | Shift this one place to the right | 9 | 0 | 0 | |||||
| b3 = 53 | 1 | 2 | 5 | Shift this one place to the right | 1 | 2 | 5 | |||||
| 1 | 9 | 5 | 3 | 1 | 2 | 5 |
Let us find 123
12 = 10 X 1 + 2 which is the form 10a + b where a = 1 and b = 2
| a3 = 13 | 1 | 1 | |||||
| 3a2b = 3 X 12X 2 | 6 | Shift this one place to the right | 6 | ||||
| 3ab2 = 3 X 1 X 22 | 1 | 2 | Shift this one place to the right | 1 | 2 | ||
| b3 = 23 | 8 | Shift this one place to the right | 8 | ||||
| 1 | 7 | 2 | 8 |
123 = 1728
Answer : 1253 = 1953125
See Also
References
- ↑ Līlāvatī Of Bhāskarācārya - A Treatise of Mathematics of Vedic Tradition. New Delhi: Motilal Banarsidass Publishers. 2001. pp. 27–29. ISBN 81-208-1420-7.
