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PI day - What Aryabhatta said about Pi
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PI day - What Aryabhatta said about Pi
Let us remember on Pi day Aryabhatta, the mathematician who mentioned what Pi is in a simple verse, First the verse in original Sanskrit followed by English transliteration.
चतुरधिकम् शतम्, अष्टगुणम् द्वाषष्टि
तथा सहस्राणाम् अयुत द्वय विष्कम्भस्य
आसन्न: वृत्त
चतुरधिकम् शतम् = 100+4 = 104; अष्टगुणम् = 104 * 8 = 832; द्वाषष्टि = 62; तथा = thus; सहस्राणाम् = 62 * 1000 = 62,000; अयुत = ten thousand fold;द्वय = by two; विष्कम्भस्य = of the diameter of a circle; आसन्न: = approximately; वृत्त = circle; परिणाह: = circumference
(((100+4) * 8 + (62 * 1000)) / 20000) * Diameter = Circumfrence of a circle, approximately.
In English:
Chaturadhikam shatam, ashTaguNam davaashashTi
Tathaa sahasraguNam ayuta davaya vishkambhasya
Aasannah vRtta
Chaturadhikam shatam = 100+4 = 104; ashTaguNam = 104 * 8 = 832; davashashTi = 62; tatha = thus / that is; sahasraguNam = 62 * 1000 = 62,000; ayuta = ten thousand fold; davaya = by two; vishkambhasya = of the diameter of a circle; aasannah = approximately; vRtta = circle; pariNaahah = circumference
(62,832 / 20,000) * Diameter = Circumfrence of a circle, approximately
or
3.1416 * D = Circumfrence of a circle, approximately
Aryabhatta didn't give a name for the approximate 3.1416 quantity however.
चतुरधिकम् शतम्, अष्टगुणम् द्वाषष्टि
तथा सहस्राणाम् अयुत द्वय विष्कम्भस्य
आसन्न: वृत्त
चतुरधिकम् शतम् = 100+4 = 104; अष्टगुणम् = 104 * 8 = 832; द्वाषष्टि = 62; तथा = thus; सहस्राणाम् = 62 * 1000 = 62,000; अयुत = ten thousand fold;द्वय = by two; विष्कम्भस्य = of the diameter of a circle; आसन्न: = approximately; वृत्त = circle; परिणाह: = circumference
(((100+4) * 8 + (62 * 1000)) / 20000) * Diameter = Circumfrence of a circle, approximately.
In English:
Chaturadhikam shatam, ashTaguNam davaashashTi
Tathaa sahasraguNam ayuta davaya vishkambhasya
Aasannah vRtta
Chaturadhikam shatam = 100+4 = 104; ashTaguNam = 104 * 8 = 832; davashashTi = 62; tatha = thus / that is; sahasraguNam = 62 * 1000 = 62,000; ayuta = ten thousand fold; davaya = by two; vishkambhasya = of the diameter of a circle; aasannah = approximately; vRtta = circle; pariNaahah = circumference
(62,832 / 20,000) * Diameter = Circumfrence of a circle, approximately
or
3.1416 * D = Circumfrence of a circle, approximately
Aryabhatta didn't give a name for the approximate 3.1416 quantity however.
indophile- Posts : 4338
Join date : 2011-04-29
Location : Glenn Dale, MD
Re: PI day - What Aryabhatta said about Pi
the real towering intellectual of ancient indian mathematics is madhava of sangamagrama (present day irinjilakudi in kerala).
read about the madhava-leibniz series formula for pi:
http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
that cirumference formulation useful as it is, is not sexy mathematics like the series solutions. this one converges, but converges slowly. ramanujan came up with much more astounding ones later.
read about the madhava-leibniz series formula for pi:
http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
that cirumference formulation useful as it is, is not sexy mathematics like the series solutions. this one converges, but converges slowly. ramanujan came up with much more astounding ones later.
MaxEntropy_Man- Posts : 14702
Join date : 2011-04-28
Re: PI day - What Aryabhatta said about Pi
here is ramanujan's:
computes eight additional digits for each extra term.
computes eight additional digits for each extra term.
MaxEntropy_Man- Posts : 14702
Join date : 2011-04-28
Re: PI day - What Aryabhatta said about Pi
Didn't know this one. The series goes to Infinity. To how many digits of Pi, has this Sum of the series been verified ?MaxEntropy_Man wrote:here is ramanujan's:
computes eight additional digits for each extra term.
rawemotions- Posts : 1690
Join date : 2011-05-03
Re: PI day - What Aryabhatta said about Pi
Totally agree. He predates newton by about 200 years and his work came pretty close to calculus. Anyway he is one of the few Indian mathematicians, that the West has grudgingly recognized. The newton and Leibnitz series is now called Madhava-Newton and Madhava-Leibnitz series. The sad thing is that, most kids studying today, still study older textbooks and do not know about this.MaxEntropy_Man wrote:the real towering intellectual of ancient indian mathematics is madhava of sangamagrama (present day irinjilakudi in kerala).
read about the madhava-leibniz series formula for pi:
http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
that cirumference formulation useful as it is, is not sexy mathematics like the series solutions. this one converges, but converges slowly. ramanujan came up with much more astounding ones later.
rawemotions- Posts : 1690
Join date : 2011-05-03
Re: PI day - What Aryabhatta said about Pi
Thanks for posting this!indophile wrote:Let us remember on Pi day Aryabhatta, the mathematician who mentioned what Pi is in a simple verse, First the verse in original Sanskrit followed by English transliteration.
चतुरधिकम् शतम्, अष्टगुणम् द्वाषष्टि
तथा सहस्राणाम् अयुत द्वय विष्कम्भस्य
आसन्न: वृत्त
चतुरधिकम् शतम् = 100+4 = 104; अष्टगुणम् = 104 * 8 = 832; द्वाषष्टि = 62; तथा = thus; सहस्राणाम् = 62 * 1000 = 62,000; अयुत = ten thousand fold;द्वय = by two; विष्कम्भस्य = of the diameter of a circle; आसन्न: = approximately; वृत्त = circle; परिणाह: = circumference
(((100+4) * 8 + (62 * 1000)) / 20000) * Diameter = Circumfrence of a circle, approximately.
In English:
Chaturadhikam shatam, ashTaguNam davaashashTi
Tathaa sahasraguNam ayuta davaya vishkambhasya
Aasannah vRtta
Chaturadhikam shatam = 100+4 = 104; ashTaguNam = 104 * 8 = 832; davashashTi = 62; tatha = thus / that is; sahasraguNam = 62 * 1000 = 62,000; ayuta = ten thousand fold; davaya = by two; vishkambhasya = of the diameter of a circle; aasannah = approximately; vRtta = circle; pariNaahah = circumference
(62,832 / 20,000) * Diameter = Circumfrence of a circle, approximately
or
3.1416 * D = Circumfrence of a circle, approximately
Aryabhatta didn't give a name for the approximate 3.1416 quantity however.
Idéfix- Posts : 8808
Join date : 2012-04-26
Location : Berkeley, CA
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