Square in Pāṭīgaṇitam - Revision history

Ramamurthy: /* Translation 24 */

2023-09-05T12:03:35Z

Translation 24

← Older revision		Revision as of 17:33, 5 September 2023
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	The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number), or the sum of as many terms of the series whose first term is 1 and common difference 2, or the product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.		The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number), or the sum of as many terms of the series whose first term is 1 and common difference 2, or the product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.

	These rules are illustrated with examples below.		These rules are illustrated with examples below.

	=== The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number) ===		=== The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number) ===

Ramamurthy: Internal link updated.

2023-09-05T12:03:00Z

Internal link updated.

← Older revision		Revision as of 17:33, 5 September 2023
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	\|}		\|}
	==See Also==		==See Also==
	[https://alpha.indicwiki.in/index.php?title=%E0%A4%AA%E0%A4%BE%E0%A4%9F%E0%A5%80%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%AE%E0%A5%8D_%E0%A4%AE%E0%A5%87%E0%A4%82_%27%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%27 पाटीगणितम् में 'वर्ग']		[[पाटीगणितम् में 'वर्ग']]
	==References==		==References==
	<references />		<references />
	[[Category:Mathematics in Pāṭīgaṇitam]]		[[Category:Mathematics in Pāṭīgaṇitam]]
	[[Category:General]]		[[Category:General]]

Ramamurthy: Headings modified

2023-09-05T12:01:32Z

Headings modified

← Older revision		Revision as of 17:31, 5 September 2023
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	=== The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number ===		=== The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number ===
	'''<math>n^2=(n-a)(n+a)+a^2</math>'''		'''<math>n^2=(n-a)(n+a)+a^2</math>'''


	Let us assume the number <math>a=1</math>		Let us assume the number <math>a=1</math>

Ramamurthy: /* Example: Square of 125 */

2023-09-05T11:59:16Z

Example: Square of 125

← Older revision		Revision as of 17:29, 5 September 2023
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	The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number), or the sum of as many terms of the series whose first term is 1 and common difference 2, or the product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.		The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number), or the sum of as many terms of the series whose first term is 1 and common difference 2, or the product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.

	These rules are illustrated with examples below.		These rules are illustrated with examples below.

	=== The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number). ===		=== The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number) ===
	'''<math>n^2=n \times n</math>'''		'''<math>n^2=n \times n</math>'''
	{\| class="wikitable"		{\| class="wikitable"
	\|+
	!Number		!Number
	!Square		!Square
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	\|9 X 9 = 81		\|9 X 9 = 81
	\|}		\|}

			=== The sum of as many terms of the series whose first term is 1 and common difference 2 ===
	'''<math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math>'''		'''<math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math>'''

	~~The sum of as many terms of the series whose first term is 1 and common difference 2.~~
	{\| class="wikitable"		{\| class="wikitable"
	!Number		!Number
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	\|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81		\|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
	\|}		\|}

			=== The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number ===
	'''<math>n^2=(n-a)(n+a)+a^2</math>'''		'''<math>n^2=(n-a)(n+a)+a^2</math>'''

	~~The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.~~

	Let us assume the number <math>a=1</math>		Let us assume the number <math>a=1</math>

Ramamurthy: /* Verse 24 */

2023-09-05T11:54:39Z

Verse 24

← Older revision		Revision as of 17:24, 5 September 2023
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	These rules are illustrated with examples below.		These rules are illustrated with examples below.

			=== The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number). ===
	'''<math>n^2=n \times n</math>'''		'''<math>n^2=n \times n</math>'''

	~~The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number).~~
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+

Ramamurthy: /* Translation 24 */

2023-09-05T11:53:26Z

Translation 24

← Older revision		Revision as of 17:23, 5 September 2023
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	These rules are illustrated with examples below.		These rules are illustrated with examples below.

	<math>n^2=n \times n</math>		'''<math>n^2=n \times n</math>'''

	The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number).		The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number).
Line 206:		Line 206:
	\|9 X 9 = 81		\|9 X 9 = 81
	\|}		\|}
	<math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math>		'''<math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math>'''

	The sum of as many terms of the series whose first term is 1 and common difference 2.		The sum of as many terms of the series whose first term is 1 and common difference 2.
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	\|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81		\|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
	\|}		\|}
	<math>n^2=(n-a)(n+a)+a^2</math>		'''<math>n^2=(n-a)(n+a)+a^2</math>'''

	The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.		The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.

Ramamurthy: /* Verse 23 */

2023-09-05T11:52:40Z

Verse 23

← Older revision		Revision as of 17:22, 5 September 2023
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	These rules are illustrated with examples below.		These rules are illustrated with examples below.
	~~===~~<math>n^2=n \times n</math>~~===~~
			<math>n^2=n \times n</math>

	The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number).		The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number).
	{\| class="wikitable"		{\| class="wikitable"
Line 204:		Line 206:
	\|9 X 9 = 81		\|9 X 9 = 81
	\|}		\|}
			<math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math>

	~~== <math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math> ==~~
	The sum of as many terms of the series whose first term is 1 and common difference 2.		The sum of as many terms of the series whose first term is 1 and common difference 2.
	{\| class="wikitable"		{\| class="wikitable"
Line 238:		Line 240:
	\|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81		\|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
	\|}		\|}
	~~===~~<math>n^2=(n-a)(n+a)+a^2</math>~~===~~		<math>n^2=(n-a)(n+a)+a^2</math>

	The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.		The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.

Ramamurthy: /* n2=1+3+5+7+9+11+13+15+17........+ t⁢o n t⁢e⁢r⁢m⁢s{\displaystyle n^{2}=1+3+5+7+9+11+13+15+17........+\ to\ n\ terms} */

2023-09-05T11:51:45Z

n2=1+3+5+7+9+11+13+15+17........+ t⁢o n t⁢e⁢r⁢m⁢s{\displaystyle n^{2}=1+3+5+7+9+11+13+15+17........+\ to\ n\ terms}

← Older revision		Revision as of 17:21, 5 September 2023
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	\|9 X 9 = 81		\|9 X 9 = 81
	\|}		\|}
	===<math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math>===
			== <math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math> ==
	The sum of as many terms of the series whose first term is 1 and common difference 2.		The sum of as many terms of the series whose first term is 1 and common difference 2.
	{\| class="wikitable"		{\| class="wikitable"

Ramamurthy: New Page created

2023-09-05T11:51:16Z

New Page created

New page

Here we will be knowing how to find square of a number as per Pāṭīgaṇitam.
==Verse 23==
कृत्वाऽन्त्यपदस्य कृतिं शेषपदैर्द्विगुणमन्त्यमभिहन्यात् ।

उत्सार्य्योत्सार्य पदाच्छेषं चोत्सारयेत् कृतये ॥ २३ ॥
==Translation 23==
To obtain the square of a number (proceed successively as follows)<ref>{{Cite book|last=Shukla|first=Kripa Shankar|title=The Pāṭīgaṇita of Śrīdharācārya|publisher=Lucknow University|year=1959|location=Lucknow|pages=8-9}}</ref>: Having squared the last digit (''antya-pada'') (i.e., having written the square of the last digit over the last digit), multiply the remaining digits by twice the last, moving it from place to place (towards the right, and set down the resulting products over the respective digits); then (rub out the last digit and) move the remaining digits (one place to the right).

This rule is based on the formula <math>(a+b)^2=a^2+2ab+b^2</math>.

This rule is illustrated with an example below.
===Example: Square of 125===
Here the last digit it 1 . Its square (12 = 1) is placed over itself.
{| class="wikitable"
|+
|1
|
|
|-
|1
|2
|5
|}Twice the last digit 2 X 1 = 2 placing it below the rest of the digits (2 or 5) and rubbing out the last digit (1). Now the work on the pāṭī appears as
{| class="wikitable"
|1
|
|
|-
|
|2
|5
|-
|
|
|2
|}Multiply 25 by 2 (below) 25 X 2 = 50 and place the result (50) over the respective figures (25), we get
{| class="wikitable"
|1
|5
|0
|-
|
|2
|5
|}One round of operation is completed. Move the remaining digits (25) one place forward to the right. we get
{| class="wikitable"
|1
|5
|0
|
|-
|
|
|2
|5
|}Now the process is repeated . In the remaining digits (25) the last digit is 2. Square of the last digit (22 = 4) is placed above 2
{| class="wikitable"
|1
|5
|0 + 4
|
|-
|
|
|2
|5
|}
{| class="wikitable"
|1
|5
|4
|
|-
|
|
|2
|5
|}Twice the last digit 2 X 2 = 4 placing it below the rest of the digit (5) and rubbing out the last digit (2). Now the work on the pāṭī appears as
{| class="wikitable"
|1
|5
|4
|
|-
|
|
|
|5
|-
|
|
|
|4
|}Multiply 5 by 4 (below) 5 X 4 = 20 and place the result (20) over the respective figure (5), we get
{| class="wikitable"
|1
|5
|4 + 2
|0
|-
|
|
|
|5
|}
{| class="wikitable"
|1
|5
|6
|0
|-
|
|
|
|5
|}Second round of operation is completed. Move the remaining digit (5) one place forward to the right. we get
{| class="wikitable"
|1
|5
|6
|0
|
|-
|
|
|
|
|5
|}Now the process is repeated . In the remaining digits (5) the last digit is 5. Square of the last digit (52 = 25) is placed above 5
{| class="wikitable"
|1
|5
|6
|0 + 2
|5
|-
|
|
|
|
|5
|}
{| class="wikitable"
|1
|5
|6
|2
|5
|-
|
|
|
|
|5
|}As there are no remaining digits the process ends. 5 being rubbed out. Now the work on the pāṭī appears as
{| class="wikitable"
|1
|5
|6
|2
|5
|}'''Square of 125 = 15625'''
==Verse 24==
सदृशद्विराशिघातो रूपादिद्विचयपदसमासो (वा) ।

इष्टोनयुतपदवधो वा तदिष्टवर्गान्वितो वर्गः ॥ २४ ॥
==Translation 24==
The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number), or the sum of as many terms of the series whose first term is 1 and common difference 2, or the product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.

These rules are illustrated with examples below.
===<math>n^2=n \times n</math>===
The square (of a given number) is also equal to the product of two equal numbers (each equal to the given number).
{| class="wikitable"
|+
!Number
!Square
! rowspan="4" |
!Number
!Square
! rowspan="4" |
!Number
!Square
|-
|1
|1 X 1 = 1
|4
|4 X 4 = 16
|7
|7 X 7 = 49
|-
|2
|2 X 2 = 4
|5
|5 X 5 = 25
|8
|8 X 8 = 64
|-
|3
|3 X 3 = 9
|6
|6 X 6 = 36
|9
|9 X 9 = 81
|}
===<math>n^2=1+3+5+7+9+11+13+15+17........+ \ to \ n \ terms</math>===
The sum of as many terms of the series whose first term is 1 and common difference 2.
{| class="wikitable"
!Number
!Square
! rowspan="4" |
!Number
!Square
! rowspan="4" |
!Number
!Square
|-
|1
|1
|4
|1 + 3 + 5 + 7 = 16
|7
|1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
|-
|2
|1 + 3 = 4
|5
|1 + 3 + 5 + 7 + 9 = 25
|8
|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64
|-
|3
|1 + 3 + 5 = 9
|6
|1 + 3 + 5 + 7 + 9 + 11 = 36
|9
|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
|}
===<math>n^2=(n-a)(n+a)+a^2</math>===
The product of the difference and the sum of the given number and an assumed number plus the square of the assumed number.

Let us assume the number <math>a=1</math>
{| class="wikitable"
!Number
!Square
! rowspan="4" |
!Number
!Square
! rowspan="4" |
!Number
!Square
|-
|1
|(1 - 1)(1 + 1) + 12 = 1
|4
|(4 - 1)(4 + 1) + 12 = 16
|7
|(7 - 1)(7 + 1) + 12 = 49
|-
|2
|(2 - 1)(2 + 1) + 12 = 4
|5
|(5 - 1)(5 + 1) + 12 = 25
|8
|(8 - 1)(8 + 1) + 12 = 64
|-
|3
|(3 - 1)(3 + 1) + 12 = 9
|6
|(6 - 1)(6 + 1) + 12 = 36
|9
|(9 - 1)(9 + 1) + 12 = 81
|}
==See Also==
[https://alpha.indicwiki.in/index.php?title=%E0%A4%AA%E0%A4%BE%E0%A4%9F%E0%A5%80%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%AE%E0%A5%8D_%E0%A4%AE%E0%A5%87%E0%A4%82_%27%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%27 पाटीगणितम् में 'वर्ग']
==References==
<references />
[[Category:Mathematics in Pāṭīgaṇitam]]
[[Category:General]]