Ramamurthy: Headings modified

2023-09-01T08:31:25Z

Headings modified

← Older revision		Revision as of 14:01, 1 September 2023
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	~~==Introduction==~~
	Here we will be knowing a square root of a number as mentioned in Pāṭīgaṇitam.		Here we will be knowing a square root of a number as mentioned in Pāṭīgaṇitam.

	==Verse==		==Verse==
	विषमात् पदतस्त्यक्त्वा वर्गं स्थानच्युतेन मूलेन ।		विषमात् पदतस्त्यक्त्वा वर्गं स्थानच्युतेन मूलेन ।

Ramamurthy at 10:18, 24 August 2023

2023-08-24T10:18:48Z

← Older revision		Revision as of 15:48, 24 August 2023
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	'''Square root of 11943936 = 3456'''		'''Square root of 11943936 = 3456'''
	==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%E0%A4%AE%E0%A5%82%E0%A4%B2%27 पाटीगणितम् में 'वर्गमूल']		[[पाटीगणितम् में 'वर्गमूल']]
	==References==		==References==
	<references />		<references />
	[[Category:Mathematics in Pāṭīgaṇitam]]		[[Category:Mathematics in Pāṭīgaṇitam]]
	[[Category:General]]		[[Category:General]]

Ramamurthy: New Page created

2023-08-24T10:10:24Z

New Page created

New page

==Introduction==
Here we will be knowing a square root of a number as mentioned in Pāṭīgaṇitam.
==Verse==
विषमात् पदतस्त्यक्त्वा वर्गं स्थानच्युतेन मूलेन ।

द्विगुणेन भजेच्छेषं लब्धं विनिवेशयेत् पङ्क्तौ ॥ २५ ॥

तद्वर्गं संशोध्य द्विगुणं कुर्वीत पूर्ववल्लब्धम् ।

उत्सार्य ततो विभजेच्छेषं द्विगुणीकृतं दलयेत् ॥ २५ ॥
==Translation==
Having subtracted the (greatest possible) square from the (last) odd place (set down double the square root underneath the next place).<ref>{{Cite book|last=Shukla|first=Kripa Shankar|title=The Pāṭīgaṇita of Śrīdharācārya|publisher=Lucknow University|year=1959|location=Lucknow|pages=9-10}}</ref> By that double square root, that has left its place (i.e., which has been set down underneath the next place), divide the remainder; set down the quotient in the line (of double the square root), and, having subtracted the square of that (from the number above), make double of that (quotient). Then having moved the resulting quantity (in the line of double the square root) one place forward, divide by it as before. (Continue this process till all places are exhausted, and then) halve the doubled quantity (to get the square root).

This rule will be clear by the following examples:
===Example: Square root of 186624===
Starting from right denote the odd and even place by ''o'' and ''e'' respectively.
{| class="wikitable"
|+
!e
!o
!e
!o
!e
!o
|-
|1
|8
|6
|6
|2
| 4 |4
|-
| colspan="6" align="right" |←
|}Subtract the greatest possible square (16 = 42) from the last odd place (18) .18 -16 = 2. Write the double the square root of 16 ( which is 2 x 4 = 8) underneath the next place. We have
{| class="wikitable"
!o
!e
!o
!e
!o
!
|-
|2
|6
|6
|2
|4
|← remainder
|-
|
|8
|
|
|
|← line of double the square root
|}Divide 26 by 8 . Here the quotient will be 3 and remainder will be 26 - 24 = 2. Write the quotient 3 in the line of double the square root.
{| class="wikitable"
!e
!o
!e
!o
!
|-
|2
|6
|2
|4
|← remainder
|-
|8
|3
|
|
|← line of double the square root
|}Subtract the square of the quotient (i.e 32 = 9) 26 - 9 = 17. Write the double the square of the quotient (i.e 3 x 2 = 6) in the line of double the square root.
{| class="wikitable"
!e
!o
!e
!o
!
|-
|1
|7
|2
|4
|← remainder
|-
|8
|6
|
|
|← line of double the square root
|}At this stage one round of operation is over. Now move 86 one place forward.
{| class="wikitable"
!e
!o
!e
!o
!
|-
|1
|7
|2
|4
|← remainder
|-
|
|8
|6
|
|← line of double the square root
|}Divide 172 by 86 . Here quotient is 2 and remainder is 172 - 172 = 0. Write the quotient (2) in the line of double the square root.
{| class="wikitable"
!e
!o
!e
!o
!
|-
|
|
|
|4
|← remainder
|-
|
|8
|6
|2
|← line of double the square root
|}Finally subtracting the square of the quotient (22 = 4) from above (4 - 4 = 0) and doubling the quotient (2) 2 x 2 = 4 we get
{| class="wikitable"
!e
!o
!e
!o
!
|-
|
|
|
|0
|← remainder
|-
|
|8
|6
|4
|← line of double the square root
|}The process now ends. Hence we divide 864 by 2 = 432 is the required square root.

As the remainder is zero . the square root is exact.

'''Square root of 186624 = 432'''
===Example: Square root of 11943936===
Starting from right denote the odd and even place by ''o'' and ''e'' respectively.
{| class="wikitable"
|+
!e
!o
!e
!o
!e
!o
!e
!o
|-
|1
|1
|9
|4
|3
|9
|3
| 4 |6
|-
| colspan="8" align="right" |←
|}Subtract the greatest possible square (9 = 32) from the last odd place (11).11 - 9 = 2. Write the double the square root of 9 (which is 2 x 3 = 6) underneath the next place. We have
{| class="wikitable"
!o
!e
!o
!e
!o
!e
!o
!
|-
|2
|9
|4
|3
|9
|3
|6
|← remainder
|-
|
|6
|
|
|
|
|
|← line of double the square root
|}Divide 29 by 6 . Here the quotient will be 4 and remainder will be 29 - 24 = 5. Write the quotient 4 in the line of double the square root.
{| class="wikitable"
!e
!o
!e
!o
!e
!o
!
|-
|5
|4
|3
|9
|3
|6
|← remainder
|-
|6
|4
|
|
|
|
|← line of double the square root
|}Subtract the square of the quotient (i.e 42 = 16) 54 - 16 = 38. Write the double the square of the quotient (i.e 4 x 2 = 8) in the line of double the square root.
{| class="wikitable"
!e
!o
!e
!o
!e
!o
!
|-
|3
|8
|3
|9
|3
|6
|← remainder
|-
|6
|8
|
|
|
|
|← line of double the square root
|}At this stage one round of operation is over. Now move 68 one place forward.
{| class="wikitable"
!e
!o
!e
!o
!e
!o
!
|-
|3
|8
|3
|9
|3
|6
|← remainder
|-
|
|6
|8
|
|
|
|← line of double the square root
|}Divide 383 by 68 . Here quotient is 5 and remainder is 383 - 340 = 43 Write the quotient (5) in the line of double the square root.
{| class="wikitable"
!o
!e
!o
!e
!o
!
|-
|4
|3
|9
|3
|6
|← remainder
|-
|6
|8
|5
|
|
|← line of double the square root
|}Finally subtracting the square of the quotient (52 = 25) from above (439 - 25 = 414), and doubling the quotient (5) 2 x 5 = 10 we get
{| class="wikitable"
!o
!e
!o
!e
!o
!
|-
|4
|1
|4
|3
|6
|← remainder
|-
|6
|8
|10
|
|
| rowspan="3" valign="top" |← line of double the square root
|-
|6
|8+1
|0
|
|
|-
|6
|9
|0
|
|
|}At this stage second round of operation is over. Now move 690 one place forward.
{| class="wikitable"
!o
!e
!o
!e
!o
!
|-
|4
|1
|4
|3
|6
|← remainder
|-
|
|6
|9
|0
|
|← line of double the square root
|}Divide 4143 by 690 . Here quotient is 6 and remainder is 4143 - 4140 = 3 Write the quotient (6) in the line of double the square root.
{| class="wikitable"
!
!
!e
!o
!
|-
|
|
|3
|6
|← remainder
|-
|6
|9
|0
|6
|← line of double the square root
|}Finally subtracting the square of the quotient (62 = 36) from above (36 - 36 = 0), and doubling the quotient (6) 2 x 6 = 12 we get
{| class="wikitable"
!
!
!e
!o
!
|-
|
|
|
|0
|← remainder
|-
|6
|9
|0
|12
| rowspan="3" valign="top" |← line of double the square root
|-
|6
|9
|0+1
|2
|-
|6
|9
|1
|2
|}The process now ends. Hence we divide 6912 by 2 = 3456 is the required square root.

As the remainder is zero . the square root is exact.

'''Square root of 11943936 = 3456'''
==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%E0%A4%AE%E0%A5%82%E0%A4%B2%27 पाटीगणितम् में 'वर्गमूल']
==References==
<references />
[[Category:Mathematics in Pāṭīgaṇitam]]
[[Category:General]]

Square root in Pāṭīgaṇitam - Revision history

Ramamurthy: Headings modified

Ramamurthy at 10:18, 24 August 2023

Ramamurthy: New Page created