Cubes by Bhārati Kṛṣṇa Tīrtha - Revision history

Ramamurthy: Headings modified

2023-08-30T14:16:27Z

Headings modified

← Older revision		Revision as of 19:46, 30 August 2023
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	~~== 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>

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	'''" Proportionally "'''		'''" Proportionally "'''

			== Steps ==
	The detailed steps are explained below .		The detailed steps are explained below .

Ramamurthy: Category updated

2023-08-16T12:26:05Z

Category updated

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	<references />		<references />
	[[Category:Mathematics by Bhārati Kṛṣṇa Tīrtha]]		[[Category:Mathematics by Bhārati Kṛṣṇa Tīrtha]]
			[[Category:General]]

Ramamurthy: Category updated

2023-06-22T10:28:31Z

Category updated

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	== References ==		== References ==
	<references />		<references />
	[[Category:Bhārati Kṛṣṇa Tīrtha]]		[[Category:Mathematics by Bhārati Kṛṣṇa Tīrtha]]

Ramamurthy: Category updated

2023-06-21T11:16:55Z

Category updated

← Older revision		Revision as of 16:46, 21 June 2023
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	== References ==		== References ==
	<references />		<references />
	~~[[Category:Mathematics]]~~
	[[Category:Bhārati Kṛṣṇa Tīrtha]]		[[Category:Bhārati Kṛṣṇa Tīrtha]]
	~~[[Category:Cubes]]~~

Ramamurthy: See Also added

2023-05-15T17:53:37Z

Sandeep at 12:28, 28 April 2023

2023-04-28T12:28:52Z

← Older revision		Revision as of 17:58, 28 April 2023
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	==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>

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	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>
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	\|		\|
	\|'''First term'''		\|'''First term'''
	\| colspan="2" \|'''Middle terms (second, third terms)'''		\| colspan="2" \| '''Middle terms (second, third terms)'''
	\|'''Fourth term'''		\|'''Fourth term'''
	\|-		\|-
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	\|'''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
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	\|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"
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	\|'''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"
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	\|'''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"
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	\|'''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]]

Ramamurthy: New category created

2023-04-28T08:19:34Z

New category created

← Older revision		Revision as of 13:49, 28 April 2023
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	==References==		==References==
	<references />		<references />
			[[Category:Mathematics]]
			[[Category:Bhārati Kṛṣṇa Tīrtha]]

Ramamurthy: New Page created

2023-04-28T08:14:30Z

New Page created

New page

==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>

'''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 <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>

third term will be <math>a^2b\ X\ \frac{b}{a} = a\ X\ b^2</math>

fourth term will be <math> ab^2\ X\ \frac{b}{a} = b^3</math>
{| class="wikitable"
|+
|
|'''First term'''
| colspan="2" |'''Middle terms (second, third terms)'''
|'''Fourth term'''
|-
|'''Step 1'''
|<math> a^3</math>
|<math>a^2\ X\ b</math>
|<math>a\ X\ b^2</math>
|<math>b^3</math>
|-
|'''Step 2'''
|<math> a^3</math>
|double of the middle
terms

<math>2(a^2\ X\ b)</math>
|double of the middle
terms

<math>2(a\ X\ b^2)</math>
|<math>b^3</math>
|-
|'''Step 3'''
|<math> a^3</math>
|add the two middle terms
<math>a^2\ X\ b + 2(a^2\ X\ b)</math>
|add the two middle terms
<math>a\ X\ b^2 + 2(a\ X\ b^2)</math>
|<math>b^3</math>
|-
|'''Final Step'''
|<math> a^3</math>
|'''https://alpha.indicwiki.in/index.php?title=Special:MathShowImage&hash=7526a2ceceb201826c5fffe6862f928f&mode=mathml'''
|'''https://alpha.indicwiki.in/index.php?title=Special:MathShowImage&hash=0df1c1fd92d7f7d0a8df80ed2d245d74&mode=mathml'''
|'''https://alpha.indicwiki.in/index.php?title=Special:MathShowImage&hash=2d275b176c3436e8981c70371f474c9c&mode=mathml'''
|}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"
|+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.
{| class="wikitable"
|-
|<math> a^3</math>
|<math>3(a^2\ X\ b) </math>
|<math>3(a\ X\ b^2)</math>
|<math>b^3</math>
|-
|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.
{| class="wikitable"
|-
|<math> a^3</math>
|<math>3(a^2\ X\ b) </math>
|<math>3(a\ X\ b^2)</math>
|<math>b^3</math>
|-
|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.
{| class="wikitable"
|-
|<math> a^3</math>
|<math>3(a^2\ X\ b) </math>
|<math>3(a\ X\ b^2)</math>
|<math>b^3</math>
|-
|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'''
==See Also==
==References==
<references />

← Older revision		Revision as of 23:23, 15 May 2023
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	\|}		\|}
	'''Answer : 25<sup>3</sup> = 15625'''		'''Answer : 25<sup>3</sup> = 15625'''

			== See Also ==
			[[घन - भारती कृष्ण तीर्थ]]

	== References ==		== References ==