PPP (complexity)

From formulasearchengine
Jump to navigation Jump to search

Template:IndicText

Madhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ..., and 90.00° (angles that are integral multiples of 3.75°, i.e. 1/24 of a right angle, beginning with 3.75 and ending with 90.00). The table is encoded in the letters of Devanagari using the Katapayadi system. This gives the entries in the table an appearance of the verses of a poem in Sanskrit.

Madhava's original work containing the sine table has not yet been traced. The table is seen reproduced in the Aryabhatiyabhashya of Nilakantha Somayaji (1444–1544) and also in the Yuktidipika/Laghuvivrti commentary of Tantrasamgraha by Sankara Variar (circa. 1500-1560). [1]

The table

Template:Trigonometry The image below gives Madhava's sine table in Devanagari as reproduced in Cultural foundations of mathematics by C.K. Raju.[1] The first twelve lines constitute the entries in the table. The last word in the thirteenth line indicates that these are "as told by Madhava".

Madhava's sine table in Devanagari

Values in Madhava's table

Diagram explaining the meaning of the values in Madhava's table

To understand the meaning of the values tabulated by Madhava, consider some angle whose measure is A. Consider a circle of unit radius and center O. Let the arc PQ of the circle subtend an angle A at the center O. Drop the perpendicular QR from Q to OP; then the length of the line segment RQ is the value of the trigonometric sine of the angle A. Let PS be an arc of the circle whose length is equal to the length of the segment RQ. For various angles A, Madhava's table gives the measures of the corresponding angles POS in arcminutes, arcseconds and sixtieths of an arcsecond.

As an example, let A be an angle whose measure is 22.50°. In Madhava's table, the entry corresponding to 22.50° is the measure in arcminutes, arcseconds and sixtieths of arcseconds of the angle whose radian measure is the modern value of sin 22.50°. The modern numerical value of sin 22.50° is 0.382683432363 and,

0.382683432363 radians = 180 / π × 0.382683432363 degrees = 21.926145564094 degrees.

and

21.926145564094 degrees = 1315 arcminutes 34 arcseconds 07 sixtieths of arcsecond.

In the Katapayadi system the digits are written in the reverse order. Thus in Madhava's table, the entry corresponding to 22.50° is 70435131.

Derivation of trigonometric sines from Madhava's table

For an angle whose measure is A, let

Then

Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be

Then according to the rules of the Katapayadi system of Kerala mathematicians we have

Madhava's value of pi

To complete the numerical computations one must have a knowledge of the value of π. It is appropriate that we use the value of π computed by Madhava himself. Nilakantha Somayaji has given this value of π in his Āryabhaṭīya-Bhashya as follows:[1]

Madhava's value of pi

A transliteration of the last two lines:

<poem>

    vibudha-netra-gaja-ahi-hutāśana
    tri-guṇa-veda-bha-vāraṇa-bāhavaḥ
    nava-nikharva-mite vr̥tivistare
    paridhi-mānam idaṁ jagadur budhāḥ

</poem>

The various words indicate certain numbers encoded in a scheme known as the bhūtasaṃkhyā system. The meaning of the words and the numbers encoded by them (beginning with the units place) are detailed in the following translation of the verse: "Gods (vibudha : 33), eyes (netra : 2), elephants (gaja : 8), snakes (ahi : 8), fires (hutāśana : 3), three (tri : 3), qualities (guṇa : 3), vedas (veda : 4), nakṣatras (bha : 27), elephants (vāraṇa : 8), and arms (bāhavaḥ : 2) - the wise say that this is the measure of the circumference when the diameter of a circle is nava-nikharva (900,000,000,000)."

So, the translation of the poem using the bhūtasaṃkhyā system will simply read "2827433388233 is, as the wise say, the circumference of a circle whose diameter is nava-nikharva (900,000,000,000)". That is, divide 2827433388233 (the number from the first two lines of the poem in reverse order) by nava-nikharva (900,000,000,000) to get the value of pi(π). This calculation yields the value π = 3.1415926535922. This is the value of π used by Madhava in his further calculations and is accurate to 11 decimal places.

Example

Madhava's table lists the following digits corresponding to the angle 45.00°:

This yields the angle with measure

The value of the trigonometric sine of 45.00° as given in Madhava's table is

Substituting the value of π computed by Madhava in the above expression, one gets sin 45° as 0.70710681.

This value may be compared with the modern exact value of sin 45.00°, namely, 0.70710678.

Comparison of Madhava's and modern sine values

In table below the first column contains the list of the twenty-four angles beginning with 3.75 and ending with 90.00. The second column contains the values tabulated by Madhava in Devanagari in the form in which it was given by Madhava. (These are taken from Malayalam Commentary of Karanapaddhati by P.K. Koru[2] and are slightly different from the table given in Cultural foundations of mathematics by C.K. Raju.[1]) The third column contains ISO 15919 transliterations of the lines given in the second column. The digits encoded by the lines in second column are given in Arabic numerals in the fourth column. The values of the trigonometric sines derived from the numbers specified in Madhava's table are listed in the fifth column. These values are computed using the approximate value 3.1415926535922 for π obtained by Madhava. For comparison, the exact values of the trigonometric sines of the angles are given in the sixth column.

Angle A
in degrees
Madhava's numbers for specifying sin A Value of sin A
derived from
Madhava's table
Modern value
of sin A
in Devanagari script
using Katapayadi system
(as in Madhava's
original table)
in ISO 15919 transliteration
scheme
in Arabic numerals
(1)
(2)
(3)
(4)
(5)
(6)
03.75
श्रेष्ठो नाम वरिष्ठानां śreṣṭhō nāma variṣṭhānāṁ
22 05 4220
0.06540314 0.06540313
07.50
हिमाद्रिर्वेदभावनः himādrirvēdabhāvanaḥ
85 24 8440
0.13052623 0.13052619
11.25
तपनो भानु सूक्तज्ञो tapanō bhānu sūktajñō
61 04 0760
0.19509032 0.19509032
15.00
मध्यमं विद्धि दोहनं maddhyamaṁ viddhi dōhanaṁ
51 54 9880
0.25881900 0.25881905
18.75
धिगाज्यो नाशनं कष्टं dhigājyō nāśanaṁ kaṣṭaṁ
93 10 5011
0.32143947 0.32143947
22.50
छन्नभोगाशयांबिका channabhōgāśayāṁbikā
70 43 5131
0.38268340 0.38268343
26.25
मृगाहारो नरेशोयं mr̥gāhārō narēśōyaṁ
53 82 0251
0.44228865 0.44228869
30.00
वीरो रणजयोत्सुकः vīrō raṇajayōtsukaḥ
42 25 8171
0.49999998 0.50000000
33.75
मूलं विशुद्धं नाळस्य mūlaṁ viṣuddhaṁ nāḷasya
53 45 9091
0.55557022 0.55557023
37.50
गानेषु विरळा नराः gāneṣu viraḷā narāḥ
30 64 2902
0.60876139 0.60876143
41.25
अशुद्धिगुप्ता चोरश्रीः aśuddhiguptā cōraśrīḥ
05 93 6622
0.65934580 0.65934582
45.00
शम्कुकर्णो नगेश्वरः śaṃkukarṇō nageśvaraḥ
51 15 0342
0.70710681 0.70710678
48.75
तनूजो गर्भजो मित्रं tanūjō garbhajō mitraṃ
60 83 4852
0.75183985 0.75183981
52.50
श्रीमानत्र सुखी सखे śrīmānatra sukhī sakhē
25 02 7272
0.79335331 0.79335334
56.25
शशी रात्रौ हिमाहारौ śaśī rātrou himāhārou
55 22 8582
0.83146960 0.83146961
60.00
वेगज्ञः पथि सिन्धुरः vēgajñaḥ pathi sindhuraḥ
43 01 7792
0.86602543 0.86602540
63.25
छाया लयो गजो नीलो chāya layō gajō nīlō
71 31 3803
0.89687275 0.89687274
67.50
निर्मलो नास्ति सल्कुले nirmalō nāsti salkulē
05 30 6713
0.92387954 0.92387953
71.25
रात्रौ दर्पणमभ्रांगं rātrou darpaṇamabhrāṁgaṁ
22 81 5523
0.94693016 0.94693013
75.00
नागस्तुंग नखो बली nāgastuṁga nakhō balī
03 63 0233
0.96592581 0.96592583
78.75
धीरो युवा कथालोलः dhīrō yuvā kathālōlaḥ
92 14 1733
0.98078527 0.98078528
82.50
पूज्यो नारीजनैर्भगाः pūjyō nārījanairbhagāḥ
11 02 8043
0.99144487 0.99144486
86.25
कन्यागारे नागवल्ली kanyāgārē nāgavallī
11 32 0343
0.99785895 0.99785892
90.00
देवो विश्वस्थली भृगुः devō viśvasthalī bhr̥ guḥ
84 44 7343
0.99999997 1.00000000

Madhava's method of computation

No work of Madhava detailing the methods used by him for the computation of the sine table has survived. However from the writings of later Kerala mathematicians like Nilakantha Somayaji (Tantrasangraha) and Jyeshtadeva (Yuktibhāṣā) that give ample references to Madhava's accomplishments, it is conjectured that Madhava computed his sine table using the power series expansion of sin x.

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further references

  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • For an account of Madhava's computation of the sine table see : 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • For a thorough discussion of the computation of Madhava's sine table with historical references : 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Template:Indian mathematics

  1. 1.0 1.1 1.2 1.3 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Cite error: Invalid <ref> tag; name "Raju" defined multiple times with different content Cite error: Invalid <ref> tag; name "Raju" defined multiple times with different content Cite error: Invalid <ref> tag; name "Raju" defined multiple times with different content
  2. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (Published in 1953)