GLIMPSES OF ANCIENT INDIAN MATHEMATICSPROF. MOHAN APTE CONTACT LANDLINE : 022 26140777 MOBILE : 9869487710 E-mail : [email protected] PROMINENT INDIAN MATHEMATICIANS Boudhana/Manava/ Apstamba/katyan Bakhashali Manuscript Aryabhat Brahmagupta Mhaviracharya Bhaskaracharya Narayan Pandit Madhav Nilakantha somayaji Shankar Variyar Jyeshtadeva Ganesh Daivadna Putman Somayaji Shankar Varman Ramanujan 1500 BC 1500 BC 200 BC /200 AD 499 AD 628 AD 850 AD 1150 AD 1356 AD 1400 AD 1500 AD 1500 AD 1550 AD 1520 AD 1732 AD 1823 AD 1910 AD Shulvasutra Shulvasutra Aryabhatiya Brahmasphutasiddhanta Ganitasarasangraha Lilavati/beejaganita Ganitakoumudi Tantrasangrah Kriyakarmakari Yuktibhasha Grahalaghav Karanapaddhati Sadratanamala WHAT INDIANS HAVE DISCOVERED IN MATHEMATICS DISCOVERY AND USE OF ZERO CALCULATIONS WITH ZERO DISCOVERY OF PLACE VALUES KNOWLEDGE ON INFINITY USE OF AVERAGE VALUES USE OF FRACTIONS USE OF RATIO AND PROPORTION PERMUTATIONS AND COMBINATION PARTNERSHIP AND SHARES LOANS AND INTERESTS INTEREST CALCULATION RULES OF CHARGING INTEREST RULES OF BODIES IN MOTION PROGRESSION OF THE TYPE PROGRESSION OF THE TYPE 1^3 + 2^3 + 3^3 + 4^3 + PROGRESSION OF THE TYPE n + n^2 + n^3 + n ^4 FIRST DEGREE INDETERMINATE EQUATION SECOND DEGREE INDETERMINATE EQUATION PYTHAGORUS THEOREM DISCOVERED BY BOUDHAYANA EXPLANATION OF BINOMIAL THEOREM CIRCLE – VERY VALUE OF Π AREA OF CIRCLE AND SPHERE ARC AND CHORD NEWTON’S INFINITE GP CONVERGENT SERIES DISCOVERED BY NILAKANTA SOMAYAJI SINE, COSINE, RADIUS AND ARC TAYLOR (1685 AD) SERIES OF SINE AND COSINE DISCOVERED BY NILAKANTA NEWTON GAUSS (1670) INTERPOLATION FORMULA DISCOVERED BY GOVINDASWAMI NEWTON’S (1660 AD) POWER SERIES DISCOVERED BY SOMAYAJI VOLUMES OF CONES GREGORY’S (1632 AD) SERIES FOR INVERSE TANGENT DISCOVERED BY MADHAVA CHARYA APPROXIMATE VALUES SURDS BIGNING OF CALCULUS : GANITA UKTIBHASHA PASCAL TRIANGLE : MERU PRASTAR FIBONASSCI SERIES : PINGALA : CHANDASHASTRA PERMUTATION AND COMBINATION BEEJAGANITA AND SO ON AND SO ON AND SO ON . mahapd\ma SaMkva: tsmaat\ jalaiQaMca An%yama\ maQyaM ParaQa-ma\ [it dSagauNaao<arM saM&a: saM#yaayaa: sqaanaanaaM vyavaharaqa-ma\ kRta: pUvaO-: Baaskracaaya- .Saas~aNaama\ gaiNatma\ maUQa-ina itYzit vaodaMga jyaaoitYa THE NUMBERS FROM ONE TO PARARDHA BHASKARACHARYA ek dSa Sat sahs~ Ayaut laxa p`yaut kaoTya: k`maSa: Aba-udma\ Abjama\ Kva.inaKva.IMPORTANCE OF MATHEMATICS yaqaa iSaKaa mayauraNaama\ naagaanaama\ maNayaao y tqaOva sava. puruYa: sahs~axa: sahs~pat\ sa BaUimama\ ivaSvatao vaR%vaa A%yat itYzt dSaaMgaulama\ saPtasyaasana\ pirQaya: i~: saPt saimaQa: kRta: ~IiNa Sata ~I sahs~aiNa i~MSacca dovaa: = 3309 gods Rik3.9 catuiBa-: saakM navaitma\ = 94 Rik 1.9.DECIMAL SYSTEM IN RIGVED puruYasaU> Rik x-90 Sahs~SaIYaa.6 AxaaOihNyaa p`saM#yaata rqaanaaM iWjasa<ama saaM#yaa gaiNat%va&O: sah Satanyaupir caOvaaYTaO tqaa BaUyaSca saPtit: gajaanaaM tu pirmaaNama\ ett\ &oyaM Satsahs~M tu sahs~aiNa navaOva tu naraNaama\ pMcaaSat\ Sataina ~Iia PaMcaYaYTI: sahs~aiNa tqaaEvaanaaM Satainaca dSaao<araiNa YaT\ p`ahu yaqa Chariats = Elephants = 21870 Foot soldiers = 109350 Horses = 65610 mahaBaart .155. 1. a profound and importnat idea which appears so simple to us now that we ignore its true merit. to which we owe our modern system is all the in that it was the only one in all history to have Achieved this s triumph. called zephirum in Arabic. Ibn Musa Al Khuwarizmi . All the Numbers you may wish can be written. Pierre-Simon Laplace The measure of the genius of Indian civilization. Leonard Fibonacci We have decided to explain Indian calculating techniques using the Nine characters and to show how. We shall Appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius. Georges Ifrah : The modern number system The nine Indian figures are the following : 9.INDIAN DECIMAL SYSTEM OF NUMBERS OPINIONS OF MATHEMATICIANS It is India that gave us the ingenious method of expressing all numbers By means of ten symbols each symbol receiving a value of position as well as an absolute value. because of their simplicity and Conciseness.4.6.3.8..5. Two of the greatest men produced by antiquity.2.7. these characters are capable of expressing any number. This is why With these nine figures and the sign 0. SHULVA SUTRA (1500 BC) : SHYENA CHEETI . SHYEN CHEETI PLAN . . THE BAUDHAYAN THEOREM (1500 BC) THE SO CALLED PYTHAGORAS THEOREM 3 5 4 . PROOF OF BAUDHAYANA THEOREM P a C b c R b a S c Q c A c B X Y Area (ACPQYA) = square (ACRX) + square (XSQY) + triangle (CRP) + triangle (QSP) = SQUARE (BCPQ) + triangle (ABC) + triangle (BYQ) All triangles are congruent hence two triangles cancel out square (BCPQ) = square (ACRX) + square (XSQY) a ^2 = b^2 + c^2 . PYTHAGORUS TRIPLES IN SHULVA SUTRA . 414213/3 = 0.414213 GF = 0.069 Area of the square = 2 x2 = 4 C O D .13807 Area of circle = 3.13807 Hence OF = 1.13807^2 = 4.CIRCLEING A SQUARE : BAUDHAYAN METHOD E F A G B • With radius OA draw an arc AE • Select GF = GE/3 • Draw a circle with radius OF • Area ABCD = Area of circle of radius OF EXAMPLE Let AB = 2 hence AG = 1 and OA = √2 OA = OE = 1.414213 hence GE= .1416 x 1. CMdSaas~ ipMgala (500 BC) Mmaa~a#aMDmao$ Series) (Fibonacci yavda maa~avaR<aoYau k%yaokgaurva: kit ivdgaurva: [%yaaid &anaaqa-M #aMDmao$: kayaa-: . . 52/2 = 26) 26/2 divisible hence 1 13/2 not divisible hence 0 (13+1 =14. 8/2 = 4) 4/2 divisible hence 1 2/2 divisible hence 1 1/2 stop Decimal 51 = Pingal’s Binary. 010011 010011 0X2^0 + 1X2^1 + 0X2^2 + 0X2^3 + 1X2^4 +1X2^5 = 0 + 2 + 0 + 0 + 16 + 32 =50 (add 1 in the final answer) = 51 .BINARY NUMBERS IN PINGAL’S CHANDASHATRA The method 1> If divisible by 2 write one or write zero 2> If not divisible by two then write zero and add one EXAMPLE 51/2 not divisible hence 0 (51+1 = 52. 14/2 = 7) 7/2 not divisible hence 0 (7+1 = 8. 1622 3.14159265358979324 Formula for billion digits of PI naIlakMz 1500 AD SaMkr vaairyar 1600 AD puTmana saaomayaajaI 1738 AD SaMkr vama-na 1823 AD ramaanaujama\ 1915 AD .1415926536 3.1415926535897932 3.1416 3.141592653 3.088 3.INDIAN VALUES OF PI Saulva saU~ Aaya-BaT varahimaihr ba`(gauPt BaaskracaayamaaQava 1500 BC 499 AD 505 AD 3.1416 OR 22/7 3.14156 628 AD 1150 AD 1370 AD SQRT(10) = 3.141592654 3. 0818 PI 3 3.141674 Aasannaao (approximately) C .ARYABHAT (500 AD) : CORRECT VALUE OF PI caturiQakM Satma\ AYTgauNama\ WaSaiYTsqatasahs~aNaama\ AyautWya ivaYkMBasya Aasannaao vaR<apirNaah: B D A Let R = 5 O Polygon sides 6 12 384 side length 5 2.1058 3.588 0. AREA OF CIRCLE ARYABHAT METHOD A B C O If the circle is divided in to large number of small sections like ABO.etc.BCO. Each section can be treated as a triangle Area of triangle ABO = (1/2) X AB X BO Where AB = base and BO = radius ‘r’ Total area of the circle = (1/2) x r x SUM(AB) SUM (AB) = circumference of the circle = 2 x (PI) x r Area of circle = (1/2) x r x 2 x (PI) x r Area of circle = (PI) r^2 . SINES IN TERSE VERSE RODDAM NARASINGH : NATURE 20/27 2001 ARYABHAT’S SINE TABLE maiK BaiK fiK QaiK naiK HaiK =iK hsJa skik ikYga SGaik ikQva Glaik ikga` h@ya Qaik ikca sga JaSa =va @la Pt f C klaaQa-jyaa: . BHASKAR-1 (600 AD) SINE FORMULA . Lalla Varahamihir (505 AD) Bhaskaracharya ( 1114 AD) .BRAHMAGUPTA (628 AD) SINE FORMULA The Trigonometric relations Brahmagupta. BRAHMAGUPTA (628) EaI caapvaMSaitlako EaI vyaaGa`mauKo naRPao SaknaRpaNaama\ pHcaaSa%saMyau>Ov ba``a(: sfuTisaQdaMt: sajjanagaNagaiNatgaaolaiva%p`I%yaO i~MSaWYao-Na kRtao ijaYN First to give the formula for the area of a cyclic quadrilateral AREA OF CYCLIC QUADRILATERAL P= q= . which calculates the values of sine at different intervals.12 = 138.12/150 = 0.12 Sine(67) = 138. The formula was developed by Brahmagupta in 665.In trigonometry. The Brahmagupta interpolation formula is defined as: Angle Indian sine difference 0 0 15 39 30 75 45 106 60 130 75 145 90 150 First difference Second 39 36 31 24 15 5 -3 -5 -7 -9 -10 150Sine (67) = sine (60) + sine(7) = 130 + {(7/15)(15 + 24)/2 + (7/15)(15 – 24)/2} = 130 + 8. the Brahmagupta interpolation formula is a special case of the Newton-Stirling interpolation formula. which was later expanded by Newton and Stirling around a thousand years later to develop the more general Newton-Stirling interpolation formula.9208 (=9205 actual) . Brahmagupta's identity. also sometimes called Fibonacci's identity. each of which is a sum of two squares. says that the product of two numbers.In algebra. Specifically: . In other words the set of all sums of two squares is closed under multiplication. is itself a sum of two squares (and in two different ways). MAHAVIRACHARYA (815 – 878 AD) GANITASARSANGRAH POLINDROMS 139 X 109 = 15151 152207 X 73 = 11111111 14287143 X 7 = 100010001 12345679 X 9 = 111111111 142857143 X 7 = 1000000001 27994681 X 441 = 12345654321 333333666667 X 33 = 11000011000011 . SOME SIMPLE METHODS BY BHASKARACHRYA . 1/1/1/2/2/90/90 1/1/1/2/2x90+90/90 ------1/1/1/2/270/90 1/1/1/2x270+90/270-------1/1/1/630/270 1/1/1x630+270/630 -------1/1/900/630 1/1x900+630/900 ----------1/1530/900 1x1530+900/1530 ----------2430/1530 2430/100 ----.Baaskracaaya.reminder : 18 X= 18 and y = 30 100x18 + 90 = 63x30 = 1890 .reminder : 30 and 1530/63 ----.: kuT\Tk (pulveriser) Solve : (100x + 90)/63 = y Solve : 100x – 63y + 90 = 0 Do continuous division as follows num/deno reminder division 100/63 37 1 63/37 26 1 37/26 11 1 26/11 4 2 11/4 3 2 4/3 1 1 Write as follows 1/1/1/2/2/1/90/0 ‘Multiply last but one to last but third and add last and remove it’ 1/1/1/2/2/1x90+0/90 ---. : cak`vaala 8x^2 + 1 = y^2 By observation x =1 and y = 3 x = 6 and y = 17 Multiple 8 short long 1 3 6 17 6x6-1=35 17x6-3=99 35x6-6=204 99x6-17=577 204x6-35=1189 577x6-99=3363 .Baaskracaaya. he could not solve it. The values of x and y are as follows.32.61. . X = 226153980 y = 1766319049 In connection with the Pell’s equation Bhaskara gave solutions for five cases .11. History of mathematics : Boyer ‘Bhaskara’s Chakrawala method is beyond all praise is certainly The finest thing achieved in the theory of numbers before Lagrange’ Herman Hankel (1839-1873) : Famous German mathematician.vaga-p`kRit cak`vaala pQdt INDITERMINATE EQUATION OF SECOND ORDER French mathematician Fermat (1604-1664)asked his friend Bessy to solve This equation. Bhaskaracharya solved it five Century before Lagrange. 8. The equation was solved By Lagrange in 18th century. and 67 His solutions for 61 is an impressive feet in calculations And its verifications alone will tax the efforts of the reader. The result is then multiplied by 3 and divided by zero.k: KgauNaao inajaaQa-yau@t: i~iBa: ca gauiNat: K)t: i~YaiYT: Addition of a number and its half is multiplied by zero. Idea of Limit in Bhaskaracharya’s Lilawati : 45 If we take K as d tends to zero . If the answer is 63 find the number. SQUARE OF A NUMBER BRAHMAGUPTA BHASKARACHARYA 7^2 2X9X7 9^2 2X2X7 2X2X9 2^2 (6x2)x73 3^2 (3x2)x7 7^2 . 7413 SIN(25. 180/7 = 25.7413) = 0.DRAWING REGULAR POLYGONS OF SIDES 3 TO 9 : BHASKARACHARYA’S METHOD AB = AC + BC = 2AC =2 x R x SIN(Ѳ) n X 2 Ѳ = 360 hence Ѳ = 180/n One side of polygon = 2 x R x SIN(180/n) A R Ѳ O Ѳ C has given following values of SIN(180/n) Triangle : 103923/120000 Square : 84853/120000 Pentagon : 70534/120000 Hexagon : 60000/120000 Heptagon : 52055/120000 Octagon : 45922/120000 Nonagon : 41031/120000 EXAMPLE Heptagon : n = 7.4339 = 4339/10000 = (4339 x 12)/(10000 x 12) = 52068/120000 One side of regular Heptagon = 2 x R x 52068/120000 B . MDaMSaucaMd`aQamakuMiBapala: Aanaunanaunnanananaunnaina%yama\ PI = 31415926536/10 000 000 000 PI = 3. ca C ja Ja Ha T z D Z Na t qa d Qa na Pa f ba Ba ma ya r la va Sa Ya sa h xa 0 cca.1415926536 .kTpyaaid pQQat 1 2 3 4 5 6 7 8 9 0 k K ga Ga [. . 000 PI = CIRCUMFERENCE/DIAMETER = 2872433388233/900.= 900.OOO.imato vaR<aivastro piriGa maanama\ [dma\ jaga ivabauQa = 33 nao~ =2 gaja =8 Aih =8 hutaSana =3 i~ = 3 gauNa =3 vaod =4 Ba =27 vaarNa = 8 baahvaa: = 2 nava inaKva.000 = 3.maaQava (1350 to 1435) : VALUE OF PI ivabauQa nao~ gaja Aih hutaSana i~ gauNa vaod Ba vaarNa b nava inaKva.000.1415926535922 .000.000. Mathematician POWER SERIES Madhav Gregory Series Madhav Newton Series Series for PI .MADHAV OF SANGAMGRAMAM (1350 – 1435) Greatest of the Indian medieval Astronomer. 16228 four digits accuracy Select x1 = 6 y1 = 19 and x2 = 228 y2 = 721 X = 6x721 +228x19 = 8658 and y = 19x721 + 10x6x228 = 27379 SQRT(10) = 27379/8658 = 3.NARAYANA PANDIT (1340 -1400) SQUARE ROOT OF A NUMBER GANITA KAUMUDI Nx^2 + 1 = y^2 N + 1/x^2 = y^2 If x is very small then N = y^2/x^2 Solve 10x^2 +1 = y^2 By observation x1 = 6 and y1 = 19 Let x2 = 6 and y2 = 19 X = x1y2 +x2y1 = 6x19 +6x19 = 228 Y = y1y2 +Nx1x2 =19x19 + 10x6x6 = 721 SQRT(10) = y/x = 721/228 = 3.162277662 eight digit accuracy . Still calculations are correct. Accepted 11 year cycle to reduce number of days from the epoch. Simplified the calculations. Most brilliant Astronomer of Medieval India. Removed the use of trigonometry.ga`hlaaGava Written by Ganesh Daivadna in 1520 AD. 11 year cycle consists of only 4016 days. Accepted all over India form almanac preparations. jyaacaapkma-rihtM saulaGaup`karma\ ktR-ma\ ga`hp`krNaM sfuTma\ ]Vtaosisma . Epoch Chaitra Shukla Pratipada Shake 1442 (1520 AD) Ganesh wrote more than 15 Astronomical books. MAGIC SQUARES 30 16 18 22 20 36 24 34 Total = 100 10 Nagarjun 1st century AD 10 44 32 14 28 26 40 6 2 Varahamihir 505 AD 5 4 7 3 8 1 6 5 2 7 4 8 3 6 1 Total = 18 7 Khajuraho 1000 AD 2 16 9 12 7 1 13 3 6 8 10 15 14 11 5 4 Total = 34 . . 8.vaaHCa kRtaQaa. 5 42 49 2 7 100 100 6 48 4 100 100 3 43 5 100 46 8 44 100 45 1 47 100 100 100 100 . 1.kRtmaokhInama\ iWyao ga`ho YaaoDSa saPtma itqyaavataro p`qamao ca iSaYTma\ iWsaPt YaT\ HyaYT ku vaod b vaaHCaExpected Number kRtaQaa-Expected number/2 kRtmaokhInama\ by 1 iWyao ga`ho YaaoDSa -Reduce -9 -16 -2 saPtmao AYTmao itqaI -7 Avatar . 6.10 p`qamao -8 -1 -15 nd th th th th th th st Following numbers are to be entered in sequence – 2. 7. 4. 3. PRINCE OF PI RAMANUJAM (1887 – 1820) Proposed several formulas for pi including the formula to find billion digits of pi Accurate up to 8 digits Accurate up to 15 digits Accurate up to 9 digits Accurate up to 8 digits Accurate up to 18 digits .