DOCSLIB.ORG

ARYABHATA-I: an Early Indian Mathematician and His Work

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
ARYABHATA-I: an Early Indian Mathematician and His Work et International Journal on Emerging Technologies (Special Issue NCETST-2017) 8(1): 764-767(2017) (Published by Research Trend, Website: www.researchtrend.net ) ISSN No. (Print) : 0975-8364 ISSN No. (Online) : 2249-3255 ARYABHATA-I: An early Indian Mathematician and his work Govind Singh Assistant Professor-Amrapali Institute of Technology & Sciences, Haldwani (UK) INDIA ABSTRACT: The knowledge of History of early Indian Mathematics prior to 499 CE is not perfect and sufficient. In this paper we explore work done on mathematics by Arya Bhata-I and summarized in his book Arya Bhateeya. Key Words: Arya Bhata-I, Arya Bhateeya, summarized. These two parts are named as Dasageetika and I. INTRODUCTION Ary āshtashatam respectively. On the other hand Arya Kusuma Pura was the older name of Pataliputra Bhata himself gave only a single name to his complete situated in the Magadha state of ancient time, now a work, which was Arya Bhateeya . The second part i.e., place near the modern Patna in Bihar. Aryabhata-I was Ary āshtashatam again ramified as Ganita or one of the Mathematicians and Astronomers those mathematics, Kāla Kriya or the calculation of time and belonged to the Kusuma Pura School of learning, but it the Gōla or sphere. The last two parts deal with the is believed that he was a native of Kerala, a region of astronomical calculations and the first part, which south India. It is due the facts that, the calendar based consists of thirty three verses of the purely upon the systems given in the astronomical work of mathematical importance. Probably Aryabhata-I was Aryabhata-I have been followed to some extent in the first Indian astronomer who gave the value of Kerala even in the modern time. He mentioned in his obliquity as 24 0, however, explicitly in his astronomical work that he was twenty-three year old when the work [12 p. 107 and references therein]. The concept of Kaliyuga’s 3600 th year was progressing. This implies the place value system of numbers was, possibly, first that he was born in the year 476 CE and his work had introduced by Aryabhata-I in his work Arya Bhateeya been completed in 499 CE. Not much about his life is [13 p. 131 and references therein]. The mathematical known. He summarized his work in the book entitled contents dealt with mainly, the methods for extracting Arya Bhateeya however, it is a very small book. It is square roots and cube roots, some problems, the series, believed that the important work Arya Bhateeya had problems on geometry problems related to the quadratic been lost, fortunately, in 1864 an Indian scholar whose equations and the solution of the indeterminate name is Bhau Daji recovered a copy of this work [1 p. equations of the first degree. 40-41 and references therein].The Arya Bhateeya is Aryabhata-I was the first Indian mathematician who considered as the landmark work in history of described the method of solution of the indeterminate mathematical sciences of early India. Various equations. The problem of indeterminate equation was commentator like aroused in the following form. It is required to Bhaskara-I, Somesvera, Parmeswera, Nilkantha determine an integer ‘N’ which when divided ‘m’ Somayaji etc., wrote their valuable commentaries for leaves a remainder r 1, and when divided by ‘n’ leaves the clear understanding of mathematical terms in this the remainder r 2. work. Aryabhata’s one other work known as Therefore, N = mx + r 1 = ny + r 2 Aryabhata-siddhanta, which had been lost, but appeared Or, ny – mx = p through the references found in other works of Where, p = r 1 – r2 mathematical or astronomical interest written after the The equation mx + p = ny time of Aryabhata-I [10 p. 1 and references therein]. Where m, n, p are integers, positive or negative, then The small work, Arya Bhateeya, was divided into two the equation mx + p = ny is known as indeterminate parts by Brahmagupta in his own work Brahma Sphuta equation of the first degree. Clearly, the common factor Siddanta. The first part consists of ten verses from the of m and n should be also a factor common to p, introductory and concluding portions, while the second otherwise the equation is inconsistent. However, the part, which is the larger one, consists of one hundred latter Indian mathematicians called the method of eight verses. solution of indeterminate equation as kuttaka . Singh 764 Aryabhata was undoubtedly, a real mathematician who [11] about this important contribution of Aryabhata, discovered the method of solution of indeterminate that: equation of first degree. Probably, this is the most “Aryabhata-I’s value of π is a very close approximation excellent achievement of Aryabhata-I in the area of to the modern value and the most accurate among those pure mathematics [1 p. 95-96 and references therein]. of the ancients. There are reasons to believe that Some important identities, which were appeared first Aryabhata devised a particular method for finding this time in the Aryabhata’s work, are as the following [1 p. value. It is shown with sufficient grounds that 45 and references therein]: Aryabhata himself used it, and several later Indian ∑ n 2 = 1 2 + 2 2 + 3 2 + 4 2 + ……. + n 2 = [n(n+1)(2n+1)]/6 mathematicians and even the Arabs adopted it. The ∑ n 3 = 1 3 + 2 3 + 3 3 + 4 3 + ………. + n 3 = conjecture that Aryabhat’s value of π is of Greek origin (1+2+3+…..+n) 2 = [n 2(n+1) 2]/4 is critically examined and is found to be without 1+(1+2)+(1+2+3)+(1+2+3+4)+……. Up to n terms = foundation. Aryabhata discovered this value [n(n+1)(n+2)]/6 independently and also realised that π is an irrational Arya Bhateeya described rules of two important number. He had the Indian background, no doubt, but operations, known as Sa ṃkrama ṃa, i.e., the operation excelled all his predecessors in evaluating π. Thus the to find two unknown quantities when their sum and credit of discovering this exact value of π may be difference are given and these rules are as following: ascribed to the celebrated mathematician, Aryabhata-I.” “One should subtract the sum of the squares from the Thus Aryabhata was not only approximated a more square of the sum”. Implies that which is halved should accurate value of π but he was also aware of its be known as the product of two multipliers. In modern irrationality. However, a formal proof of the notations irrationality of π was given by a European [(m+n) 2-(m 2+n 2)]/2 = mn. mathematician only in 1761, whose name is Lambert [1 “The square root of the product multiplied by p. 153 and references therein]. four added to the square of the difference is increased The square root extracting method described in the or decreased by the difference, and halved”. In modern Aryabhata’s work was also found in the earlier Jainas notations canonical works in which, the square root of large [√{4mn+(m-n) 2}+(m-n)]/2 = m; [ √{4mn+(m- numbers for their cosmology was calculated. However, n) 2}-(m-n)]/2 = n. the method of root extraction was not mentioned clearly Bhaskara-I in his commentary on Arya Bhateeya of but it was explained clearly by Aryabhata in his Aryabhata-I entitled as, Aryabhatiya Bhashya, mathematical work. described the first of these as an alternative method to The cube root extraction method found in Aryabhata’s find the product of two quantities [15 p. 58 and work have not been appeared in the earlier references therein]. mathematical work so, most likely it was his original Aryabhata-I constructed a table of sines, to find the work. Algorithm for finding square root was also found approximate values by using a formula for {sin (n+1) θ in the works prior to Aryabhata’s work but, algorithm – sin n θ}, which is in terms of sin n θ and sin (n-1) θ [4 for cube root was first clearly enunciated by him in his p. 47 and references therein]. However, this sine table work Arya Bhateeya [14 p. 161 and references therein]. was included only the angles of multiples of (15/4) 0 and Aryabhata described his rule of cube extraction in the between 0 0 and 90 0 but, this is an important following verse: mathematical development in early India. aghanad bhajed dvit īī yat The approximation of π correct up to five significant trigu ṃena ghansya mulavarge ṃa | figures was first given by Aryabhata-I for to calculate vargastripurvaguita ṃṃ the circumference of a circle of diameter twenty śodhya ṃ prathamad ghana śca thousand units in the following verse: hganat ||5 ( Aryabhateeya, Ganitpada, verse 5) caturadhikam satamastagunam That is, “(Having subtracted the greatest possible cube dvasastistathasaharanam ayuta dvaya from the last cube place and then having written down viskambhasyasanno vrttav parinahah the cube root of the number subtracted in the line of the cube root), divide the second non cube place (standing That is, “add four to 100, multiply by eight, and then on the right of the last cube place) by thrice the square add 62,000. By this rule circumference of a circle with of the cube root (already obtained); (then) subtract from a diameter of 20,000 can be approached.” This gives the first non-cube place (standing on the right of the the ratio of the circumference to the diameter equal to second non-cube place) the square of the quotient {[(4+100)×8+62000]/20000} = 62832/20000 = 3.1416 multiplied by thrice the previous (cube root); and (then which is very close to the modern value of π = 3.14159 subtract) the cube (of the quotient) from the cube place correct up to five places.
Load more

Recommended publications
  • A Quartically Convergent Square Root Algorithm: an Exercise in Forensic Paleo-Mathematics
    A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics David H Bailey, Lawrence Berkeley National Lab, USA DHB’s website: http://crd.lbl.gov/~dhbailey! Collaborator: Jonathan M. Borwein, University of Newcastle, Australia 1 A quartically convergent algorithm for Pi: Jon and Peter Borwein’s first “big” result In 1985, Jonathan and Peter Borwein published a “quartically convergent” algorithm for π. “Quartically convergent” means that each iteration approximately quadruples the number of correct digits (provided all iterations are performed with full precision): Set a0 = 6 - sqrt[2], and y0 = sqrt[2] - 1. Then iterate: 1 (1 y4)1/4 y = − − k k+1 1+(1 y4)1/4 − k a = a (1 + y )4 22k+3y (1 + y + y2 ) k+1 k k+1 − k+1 k+1 k+1 Then ak, converge quartically to 1/π. This algorithm, together with the Salamin-Brent scheme, has been employed in numerous computations of π. Both this and the Salamin-Brent scheme are based on the arithmetic-geometric mean and some ideas due to Gauss, but evidently he (nor anyone else until 1976) ever saw the connection to computation. Perhaps no one in the pre-computer age was accustomed to an “iterative” algorithm? Ref: J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity}, John Wiley, New York, 1987. 2 A quartically convergent algorithm for square roots I have found a quartically convergent algorithm for square roots in a little-known manuscript: · To compute the square root of q, let x0 be the initial approximation.
    [Show full text]
  • Solving Cubic Polynomials
    Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. a 2. Then, given x2 + a x + a , substitute x = y − 1 to obtain an equation without the linear term. 1 0 2 (This is the \depressed" equation.) 3. Solve then for y as a square root. (Remember to use both signs of the square root.) a 4. Once this is done, recover x using the fact that x = y − 1 . 2 For example, let's solve 2x2 + 7x − 15 = 0: First, we divide both sides by 2 to create an equation with leading term equal to one: 7 15 x2 + x − = 0: 2 2 a 7 Then replace x by x = y − 1 = y − to obtain: 2 4 169 y2 = 16 Solve for y: 13 13 y = or − 4 4 Then, solving back for x, we have 3 x = or − 5: 2 This method is equivalent to \completing the square" and is the steps taken in developing the much- memorized quadratic formula. For example, if the original equation is our \high school quadratic" ax2 + bx + c = 0 then the first step creates the equation b c x2 + x + = 0: a a b We then write x = y − and obtain, after simplifying, 2a b2 − 4ac y2 − = 0 4a2 so that p b2 − 4ac y = ± 2a and so p b b2 − 4ac x = − ± : 2a 2a 1 The solutions to this quadratic depend heavily on the value of b2 − 4ac.
    [Show full text]
  • Chapter 5 Complex Numbers
    Chapter 5 Complex numbers Why be one-dimensional when you can be two-dimensional? ? 3 2 1 0 1 2 3 − − − ? We begin by returning to the familiar number line, where I have placed the question marks there appear to be no numbers. I shall rectify this by defining the complex numbers which give us a number plane rather than just a number line. Complex numbers play a fundamental rˆolein mathematics. In this chapter, I shall use them to show how e and π are connected and how certain primes can be factorized. They are also fundamental to physics where they are used in quantum mechanics. 5.1 Complex number arithmetic In the set of real numbers we can add, subtract, multiply and divide, but we cannot always extract square roots. For example, the real number 1 has 125 126 CHAPTER 5. COMPLEX NUMBERS the two real square roots 1 and 1, whereas the real number 1 has no real square roots, the reason being that− the square of any real non-zero− number is always positive. In this section, we shall repair this lack of square roots and, as we shall learn, we shall in fact have achieved much more than this. Com- plex numbers were first studied in the 1500’s but were only fully accepted and used in the 1800’s. Warning! If r is a positive real number then √r is usually interpreted to mean the positive square root. If I want to emphasize that both square roots need to be considered I shall write √r.
    [Show full text]
  • Rationale of the Chakravala Process of Jayadeva and Bhaskara Ii
    HISTORIA MATHEMATICA 2 (1975) , 167-184 RATIONALE OF THE CHAKRAVALA PROCESS OF JAYADEVA AND BHASKARA II BY CLAS-OLOF SELENIUS UNIVERSITY OF UPPSALA SUMMARIES The old Indian chakravala method for solving the Bhaskara-Pell equation or varga-prakrti x 2- Dy 2 = 1 is investigated and explained in detail. Previous mis- conceptions are corrected, for example that chakravgla, the short cut method bhavana included, corresponds to the quick-method of Fermat. The chakravala process corresponds to a half-regular best approximating algorithm of minimal length which has several deep minimization properties. The periodically appearing quantities (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, etc.) are correctly understood only with the new theory. Den fornindiska metoden cakravala att l~sa Bhaskara- Pell-ekvationen eller varga-prakrti x 2 - Dy 2 = 1 detaljunders~ks och f~rklaras h~r. Tidigare missuppfatt- 0 ningar r~ttas, sasom att cakravala, genv~gsmetoden bhavana inbegripen, motsvarade Fermats snabbmetod. Cakravalaprocessen motsvarar en halvregelbunden b~st- approximerande algoritm av minimal l~ngd med flera djupt liggande minimeringsegenskaper. De periodvis upptr~dande storheterna (jyestha-m~la, kanistha-mula, ksepaka, kuttakara, os~) blir forstaellga0. 0 . f~rst genom den nya teorin. Die alte indische Methode cakrav~la zur Lbsung der Bhaskara-Pell-Gleichung oder varga-prakrti x 2 - Dy 2 = 1 wird hier im einzelnen untersucht und erkl~rt. Fr~here Missverst~ndnisse werden aufgekl~rt, z.B. dass cakrav~la, einschliesslich der Richtwegmethode bhavana, der Fermat- schen Schnellmethode entspreche. Der cakravala-Prozess entspricht einem halbregelm~ssigen bestapproximierenden Algorithmus von minimaler L~nge und mit mehreren tief- liegenden Minimierungseigenschaften. Die periodisch auftretenden Quantit~ten (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, usw.) werden erst durch die neue Theorie verst~ndlich.
    [Show full text]
  • A Historical Survey of Methods of Solving Cubic Equations Minna Burgess Connor
    University of Richmond UR Scholarship Repository Master's Theses Student Research 7-1-1956 A historical survey of methods of solving cubic equations Minna Burgess Connor Follow this and additional works at: http://scholarship.richmond.edu/masters-theses Recommended Citation Connor, Minna Burgess, "A historical survey of methods of solving cubic equations" (1956). Master's Theses. Paper 114. This Thesis is brought to you for free and open access by the Student Research at UR Scholarship Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. A HISTORICAL SURVEY OF METHODS OF SOLVING CUBIC E<~UATIONS A Thesis Presented' to the Faculty or the Department of Mathematics University of Richmond In Partial Fulfillment ot the Requirements tor the Degree Master of Science by Minna Burgess Connor August 1956 LIBRARY UNIVERStTY OF RICHMOND VIRGlNIA 23173 - . TABLE Olf CONTENTS CHAPTER PAGE OUTLINE OF HISTORY INTRODUCTION' I. THE BABYLONIANS l) II. THE GREEKS 16 III. THE HINDUS 32 IV. THE CHINESE, lAPANESE AND 31 ARABS v. THE RENAISSANCE 47 VI. THE SEVEW.l'EEl'iTH AND S6 EIGHTEENTH CENTURIES VII. THE NINETEENTH AND 70 TWENTIETH C:BNTURIES VIII• CONCLUSION, BIBLIOGRAPHY 76 AND NOTES OUTLINE OF HISTORY OF SOLUTIONS I. The Babylonians (1800 B. c.) Solutions by use ot. :tables II. The Greeks·. cs·oo ·B.c,. - )00 A~D.) Hippocrates of Chios (~440) Hippias ot Elis (•420) (the quadratrix) Archytas (~400) _ .M~naeobmus J ""375) ,{,conic section~) Archimedes (-240) {conioisections) Nicomedea (-180) (the conchoid) Diophantus ot Alexander (75) (right-angled tr~angle) Pappus (300) · III.
    [Show full text]
  • The Evolution of Equation-Solving: Linear, Quadratic, and Cubic
    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work.
    [Show full text]
  • Crowdsourcing
    CROWDSOURCING The establishment of the ZerOrigIndia Foundation is predicated on a single premise, namely, that our decades-long studies indicate that there are sound reasons to assume that facilitating further independent scientific research into the origin of the zero digit as numeral may lead to theoretical insights and practical innovations equal to or perhaps even exceeding the revolutionary progress to which the historic emergence of the zero digit in India somewhere between 200 BCE and 500 CE has led across the planet, in the fields of mathematics, science and technology since its first emergence. No one to date can doubt the astounding utility of the tenth and last digit to complete the decimal system, yet the origin of the zero digit is shrouded in mystery to this day. It is high time, therefore, that a systematic and concerted effort is undertaken by a multidisciplinary team of experts to unearth any extant evidence bearing on the origin of the zero digit in India. The ZerOrigIndia Foundation is intended to serve as instrument to collect the requisite funds to finance said independent scientific research in a timely and effective manner. Research Academics and researchers worldwide are invited to join our efforts to unearth any extant evidence of the zero digit in India. The ZerOrigIndia Foundation will facilitate the research in various ways, chief among which is to engage in fundraising to finance projects related to our objective. Academics and researchers associated with reputed institutions of higher learning are invited to monitor progress reported by ZerOrigIndia Foundation, make suggestions and/or propose their own research projects to achieve the avowed aim.
    [Show full text]
  • Real Numbers 6.3B
    6.3b Real Numbers Lesson Tutorials A cube root of a number is a number that when multiplied by itself, and then multiplied by itself again, equals the given number. A perfect cube is a number that can be written as the cube of an integer. EXAMPLE 1 Finding Cube Roots Find the cube root of each number. a. 8 b. −27 2 ⋅ 2 ⋅ 2 = 8 −3 ⋅ (−3) ⋅ (−3) = −27 So, the cube root of 8 is 2. So, the cube root of − 27 is − 3. 3 — The symbol √ is used to represent a cube root. Cubing a number and fi nding a cube root are inverse operations. Use this relationship to solve equations involving cubes. EXAMPLE 2 Solving an Equation Find the surface area of the cube. Volume ä 125 ft3 Use the formula for the volume of a cube to fi nd the side length s. Remember s V = s 3 Write formula for volume. The volume V of a cube = 3 with side length s is 125 s Substitute 125 for V. 3 — — given by V = s . The 3 3 3 s √ 125 = √ s Take the cube root of each side. surface area S is given s by S = 6s 2. 5 = s Simplify. The side length is 5 feet. Use a formula to fi nd the surface area of the cube. S = 6s 2 Write formula for surface area. = 6(5)2 Substitute 5 for s. = 150 Simplify. The surface area of the cube is 150 square feet. Find the cube root of the number. 1.
    [Show full text]
  • And Cubic Equations by the Odd Number Method 1
    Calculating Machine Solution of Quadratic and Cubic Equations by the Odd Number Method 1. Introduction. Eastlack1 has published a method for the solution of quadratic equations by means of a calculating machine. His process is ex- tended here to the solution of cubic equations. In the ordinary manual operation of calculating machines, the use of the method of solving cubic equations presented here will not be found to be as convenient as the use of certain other methods, such as that of Newton. The method is described here, however, in the belief that it may find application in large scale, automatic computing machines (such as the IBM Sequence Controlled Calculator or the ENIAC) where a large number of operations is not objec- tionable, provided that the operations are repetitive and sufficiently simple. We limit our discussion to real roots. Eastlack's method of solving quadratic equations is first reviewed so that the extension of the method to cubic equa- tions may be clearer. 2. Extraction of square roots by the odd number method. The ordinary method for finding on the calculating machine the square root of a number of 3 or 4 digits of the form (10* + y)2 = 100*2 + 20*y + y1 consists in removing the 100*2 term by subtracting the x odd numbers, 1, 3, 5, • • -, (2x — 1), from the first one or two digits of (10* + y)2, and then removing the remainder of (10* + y)2 by subtracting the y numbers, (20* + 1), (20* + 3), • • -, (20* + 2y - 1). This process of square root extraction may evidently be extended to the evaluation of roots having more than two digits.
    [Show full text]
  • 31 Indian Mathematicians
    Indian Mathematician 1. Baudhayana (800BC) Baudhayana was the first great geometrician of the Vedic altars. The science of geometry originated in India in connection with the construction of the altars of the Vedic sacrifices. These sacrifices were performed at certain precalculated time, and were of particular sizes and shapes. The expert of sacrifices needed knowledge of astronomy to calculate the time, and the knowledge of geometry to measure distance, area and volume to make altars. Strict texts and scriptures in the form of manuals known as Sulba Sutras were followed for performing such sacrifices. Bandhayana's Sulba Sutra was the biggest and oldest among many Sulbas followed during olden times. Which gave proof of many geometrical formulae including Pythagorean theorem 2. Āryabhaṭa(476CE-550 CE) Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra (present day Patna, Bihar). Notabl e Āryabhaṭīya, Arya-siddhanta works Explanation of lunar eclipse and solar eclipse, rotation of Earth on its axis, Notabl reflection of light by moon, sinusoidal functions, solution of single e variable quadratic equation, value of π correct to 4 decimal places, ideas circumference of Earth to 99.8% accuracy, calculation of the length of sidereal year 3. Varahamihira (505-587AD) Varaha or Mihir, was an Indian astronomer, mathematician, and astrologer who lived in Ujjain. He was born in Avanti (India) region, roughly corresponding to modern-day Malwa, to Adityadasa, who was himself an astronomer.
    [Show full text]
  • How Cubic Equations (And Not Quadratic) Led to Complex Numbers
    How Cubic Equations (and Not Quadratic) Led to Complex Numbers J. B. Thoo Yuba College 2013 AMATYC Conference, Anaheim, California This presentation was produced usingLATEX with C. Campani’s BeamerLATEX class and saved as a PDF file: <http://bitbucket.org/rivanvx/beamer>. See Norm Matloff’s web page <http://heather.cs.ucdavis.edu/~matloff/beamer.html> for a quick tutorial. Disclaimer: Our slides here won’t show off what Beamer can do. Sorry. :-) Are you sitting in the right room? This talk is about the solution of the general cubic equation, and how the solution of the general cubic equation led mathematicians to study complex numbers seriously. This, in turn, encouraged mathematicians to forge ahead in algebra to arrive at the modern theory of groups and rings. Hence, the solution of the general cubic equation marks a watershed in the history of algebra. We shall spend a fair amount of time on Cardano’s solution of the general cubic equation as it appears in his seminal work, Ars magna (The Great Art). Outline of the talk Motivation: square roots of negative numbers The quadratic formula: long known The cubic formula: what is it? Brief history of the solution of the cubic equation Examples from Cardano’s Ars magna Bombelli’s famous example Brief history of complex numbers The fundamental theorem of algebra References Girolamo Cardano, The Rules of Algebra (Ars Magna), translated by T. Richard Witmer, Dover Publications, Inc. (1968). Roger Cooke, The History of Mathematics: A Brief Course, second edition, Wiley Interscience (2005). Victor J. Katz, A History of Mathematics: An Introduction, third edition, Addison-Wesley (2009).
    [Show full text]
  • An Analysis on Extracting Square and Cube Roots by Aryabhata's Methods
    ADBU-Journal of Engineering Technology An Analysis on Extracting Square and Cube Roots by Aryabhata’s Methods. Yumnam Kirani Singh, C-DAC, IIPC Building, NIT Campus, Silchar, Assam. Email:yumnam.singh[@]cdac.in Abstract— Finding accurate root of a number is still considered as challenging in computer science community. The only popularly known method to compute root is the long division method of finding square root. Aryabhata contributed two methods similar to long division method to compute square root and cube root. In recent years, his methods have also been studied, explained and tried to implement as computer algorithms. The explained methods and proposed algorithms fail to give correct results. Some analyses have been made on these methods in order to ascertain why the algorithms fail. Improved algorithms have been provided to give correct result while computing square root or cube root using Aryabhata’s methods. Keywords—cube root, square root, Bino’s Model of multiplication, Large number manipulation, Long division method, Aryabhata’s methods. (Article history: Received 12 November 2016 and accepted 30 December 2016) In this paper, the algorithms for extraction of square root and I. INTRODUCTION cube root using Aryabhata’s methods are analyzed and Aryabhatta is one of the most revered ancient Indian improved so that it gives for correct results while computing mathematicians who made significant contributions in various the roots. The paper is divided into four sections. Section II fields of mathematics and astronomy [1, 2]. His contribution discusses algorithm for extraction of square root and section in mathematics has also been studied for possible applications III discusses the algorithm to compute cube root.
    [Show full text]