Cubes in Līlāvatī: Difference between revisions

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[[Category:Mathematics]]
Here we will know how to find the cube of a number as mentioned in Līlāvatī.
[[Category: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.<ref>{{Cite book|last=|title=Līlāvatī Of Bhāskarācārya - A Treatise of Mathematics of Vedic Tradition|publisher=Motilal Banarsidass Publishers|year=2001|isbn=81-208-1420-7|location=New Delhi|pages=27-29}}</ref> If we want to find the cube of a number of two digits, say, 10a + b, write a<sup>3</sup> first. Below this, write 3a<sup>2</sup> b by shifting this result one place to the right. Below this write 3ab<sup>2</sup> after shifting it one place to the right. Below this write b<sup>3</sup> 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
{| class="wikitable" col2right
|+
|a<sup>3</sup> = 2<sup>3</sup>
|8
|
|
| rowspan="5" |
|
| rowspan="5" |
|
|8
|
|
|
|-
|3a<sup>2</sup>b = 3 X 2<sup>2</sup>X 7
|8
|4
|
|Shift this one place to the right
|
|8
|4
|
|
|-
|3ab<sup>2</sup> = 3 X 2 X 7<sup>2</sup>
|2
|9
|4
|Shift this one place to the right
|
|2
|9
|4
|
|-
|b<sup>3</sup> = 7<sup>3</sup>
|3
|4
|3
|
|
|
|3
|4
|3
|-
|
|
|
|
|
|'''1'''
|'''9'''
|'''6'''
|'''8'''
|'''3'''
|}'''Answer : 27<sup>3</sup> = 19683'''
==Example: Cube of 125==
125 = 10 X 12 + 5 which is the form 10a + b where a = 12 and b = 5
{| class="wikitable" col2right
|+
|a<sup>3</sup> = 12<sup>3</sup> (''Refer the calculation below'')
|1
|7
|2
|8
|
|1
|7
|2
|8
|
|
|
|-
|3a<sup>2</sup>b = 3 X 12<sup>2</sup>X 5
|2
|1
|6
|0
|Shift this one place to the right
|
|2
|1
|6
|0
|
|
|-
|3ab<sup>2</sup> = 3 X 12 X 5<sup>2</sup>
|9
|0
|0
|
|Shift this one place to the right
|
|
|
|9
|0
|0
|
|-
|b<sup>3</sup> = 5<sup>3</sup>
|1
|2
|5
|
|Shift this one place to the right
|
|
|
|
|1
|2
|5
|-
|
|
|
|
|
|
|'''1'''
|'''9'''
|'''5'''
|'''3'''
|'''1'''
|'''2'''
|'''5'''
|}'''Let us find 12<sup>3</sup>'''
 
12 = 10 X 1 + 2 which is the form 10a + b where a = 1 and b = 2
{| class="wikitable" col2right
|a<sup>3</sup> = 1<sup>3</sup>
|1
|
|
|1
|
|
|
|-
|3a<sup>2</sup>b = 3 X 1<sup>2</sup>X 2
|6
|
|Shift this one place to the right
|
|6
|
|
|-
|3ab<sup>2</sup> = 3 X 1 X 2<sup>2</sup>
|1
|2
|Shift this one place to the right
|
|1
|2
|
|-
|b<sup>3</sup> = 2<sup>3</sup>
|8
|
|Shift this one place to the right
|
|
|
|8
|-
|
|
|
|
|'''1'''
|'''7'''
|'''2'''
|'''8'''
|}12<sup>3</sup> = 1728
 
'''Answer : 125<sup>3</sup> = 1953125'''
==See Also==
[[लीलावती में 'घन']]
==References==
<references />
[[Category:Mathematics in Līlāvatī]]
[[Category:General]]

Latest revision as of 20:37, 30 August 2023

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

लीलावती में 'घन'

References

  1. ↑ Līlāvatī Of Bhāskarācārya - A Treatise of Mathematics of Vedic Tradition. New Delhi: Motilal Banarsidass Publishers. 2001. pp. 27–29. ISBN 81-208-1420-7.
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