Cubes by Bhārati Kṛṣṇa Tīrtha: Difference between revisions

From Vidyalayawiki

(New category created)
No edit summary
Line 1: Line 1:
==Introduction==
== Introduction ==
Here we will be learning how to find the cubes of two digit numbers. We will be using the sūtra<ref>{{Cite book|last=Singhal|first=Vandana|title=Vedic Mathematics For All Ages - A Beginners' Guide|publisher=Motilal Banarsidass|year=2007|isbn=978-81-208-3230-5|location=Delhi|pages=237-242}}</ref>
Here we will be learning how to find the cubes of two digit numbers. We will be using the sūtra<ref>{{Cite book|last=Singhal|first=Vandana|title=Vedic Mathematics For All Ages - A Beginners' Guide|publisher=Motilal Banarsidass|year=2007|isbn=978-81-208-3230-5|location=Delhi|pages=237-242}}</ref>


Line 12: Line 12:
Let a and b are the two digits .
Let a and b are the two digits .


Step 1 : We will be writing the four numbers in geometric ratio in exact proportion.
Step 1 : We will be writing the four numbers in geometric ratio in exact proportion.  


First term will be <math> a^3</math> , common ratio = <math>\frac{b}{a}</math>
First term will be <math> a^3</math> , common ratio = <math>\frac{b}{a}</math>


second term will be <math>a^3\ X\ \frac{b}{a} = a^2\ X\ b</math>
second term will be <math>a^3\ X\ \frac{b}{a} = a^2\ X\ b</math>
Line 25: Line 25:
|
|
|'''First term'''
|'''First term'''
| colspan="2" |'''Middle terms (second, third terms)'''
| colspan="2" |           '''Middle terms (second, third terms)'''
|'''Fourth term'''
|'''Fourth term'''
|-
|-
Line 56: Line 56:
|'''Final Step'''
|'''Final Step'''
|<math> a^3</math>
|<math> a^3</math>
|'''https://alpha.indicwiki.in/index.php?title=Special:MathShowImage&hash=7526a2ceceb201826c5fffe6862f928f&mode=mathml'''
|'''<math>3(a^2\ X\ b) </math>'''
|'''https://alpha.indicwiki.in/index.php?title=Special:MathShowImage&hash=0df1c1fd92d7f7d0a8df80ed2d245d74&mode=mathml'''
|'''<math>3(a\ X\ b^2)</math>'''
|'''https://alpha.indicwiki.in/index.php?title=Special:MathShowImage&hash=2d275b176c3436e8981c70371f474c9c&mode=mathml'''
|'''<math>b^3</math>'''
|}This is nothing but the expansion of the formula for <math> (a+b)^3</math> where a and b are two individual digits of the number. This will makes the calculation of cubes very fast and easy. This will be explained through the below examples.
|}
This is nothing but the expansion of the formula for <math> (a+b)^3</math> where a and b are two individual digits of the number. This will makes the calculation of cubes very fast and easy. This will be explained through the below examples.  
{| class="wikitable"
{| class="wikitable"
|+Cubes of natural numbers
|+Cubes of natural numbers
Line 82: Line 83:
|729
|729
|}
|}
===Example: 23<sup>3</sup>===
 
=== Example: 23<sup>3</sup> ===
Here a = 2 , b = 3 , Using the final step mentioned above.
Here a = 2 , b = 3 , Using the final step mentioned above.
{| class="wikitable"
{| class="wikitable"
Line 145: Line 147:
|'''6'''
|'''6'''
|'''7'''
|'''7'''
|}'''Answer : 23<sup>3</sup> = 12167'''
|}
===Example: 12<sup>3</sup>===
'''Answer : 23<sup>3</sup> = 12167'''
 
=== Example: 12<sup>3</sup> ===
Here a = 1 , b = 2 , Using the final step mentioned above.
Here a = 1 , b = 2 , Using the final step mentioned above.
{| class="wikitable"
{| class="wikitable"
Line 179: Line 183:
|'''2'''
|'''2'''
|'''8'''
|'''8'''
|}'''Answer : 12<sup>3</sup> = 1728'''
|}
===Example: 25<sup>3</sup>===
'''Answer : 12<sup>3</sup> = 1728'''
 
=== Example: 25<sup>3</sup> ===
Here a = 2, b = 5 , Using the final step mentioned above.
Here a = 2, b = 5 , Using the final step mentioned above.
{| class="wikitable"
{| class="wikitable"
Line 243: Line 249:
|'''2'''
|'''2'''
|'''5'''
|'''5'''
|}'''Answer : 25<sup>3</sup> = 15625'''
|}
==See Also==
'''Answer : 25<sup>3</sup> = 15625'''
==References==
 
== References ==
<references />
<references />
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Bhārati Kṛṣṇa Tīrtha]]
[[Category:Bhārati Kṛṣṇa Tīrtha]]
[[Category:Cubes]]

Revision as of 17:58, 28 April 2023

Introduction

Here we will be learning how to find the cubes of two digit numbers. We will be using the sūtra[1]

आनुरूप्येण

" Ānurūpyeṇa "

" Proportionally "

The detailed steps are explained below .

Let a and b are the two digits .

Step 1 : We will be writing the four numbers in geometric ratio in exact proportion.

First term will be , common ratio =

second term will be

third term will be

fourth term will be

First term Middle terms (second, third terms) Fourth term
Step 1
Step 2 double of the middle

terms

double of the middle

terms

Step 3 add the two middle terms

add the two middle terms

Final Step

This is nothing but the expansion of the formula for where a and b are two individual digits of the number. This will makes the calculation of cubes very fast and easy. This will be explained through the below examples.

Cubes of natural numbers
1 2 3 4 5 6 7 8 9
1 8 27 64 125 216 343 512 729

Example: 233

Here a = 2 , b = 3 , Using the final step mentioned above.

23 = 8 3 X ( 22 X 3) = 3 (4 X 3) 3 X (2 X 32) = 3 X (2 X 9) 33 = 27
8 36 54 27
8 36 54 Put 7 and carry over 2
8 36 54 + Carry over (2) 7
8 36 56 7
8 36 Put 6 and carry over 5 7
8 36 + Carry over (5) 6 7
8 41 6 7
8 Put 1 and carry over 4 6 7
8 + Carry over (4) 1 6 7
12 1 6 7

Answer : 233 = 12167

Example: 123

Here a = 1 , b = 2 , Using the final step mentioned above.

13 = 1 3 X ( 12 X 2) = 3 (1 X 2) 3 X (1 X 22) = 3 X (1 X 4) 23 = 8
1 6 12 8
1 6 Put 2 and carry over 1 8
1 6 + Carry over (1) 2 8
1 7 2 8

Answer : 123 = 1728

Example: 253

Here a = 2, b = 5 , Using the final step mentioned above.

23 = 8 3 X ( 22 X 5) = 3 (4 X 5) 3 X (2 X 52) = 3 X (2 X 25) 53 = 125
8 60 150 125
8 60 150 Put 5 and carry over 12
8 60 150 + Carry over (12) 5
8 60 162 5
8 60 Put 2 and carry over 16 5
8 60 + Carry over (16) 2 5
8 76 2 5
8 Put 6 and carry over 7 2 5
8 + Carry over (7) 6 2 5
15 6 2 5

Answer : 253 = 15625

References

  1. Singhal, Vandana (2007). Vedic Mathematics For All Ages - A Beginners' Guide. Delhi: Motilal Banarsidass. pp. 237–242. ISBN 978-81-208-3230-5.