61. Bineesha Project (Project Semester ii, 2000-2001). The Indian astronomer aryabhata is regardedas the founder of scientific astronomy in India and often called the http://courses.nus.edu.sg/course/phibalas/dialogue2001/Contribute/Bineesha proje

62. Aryabhata I (476-ca 550): Indian Astronomer And Mathematician. A thousand years before Copernicus, aryabhata propounds a heliocentric Aihole Inscription of King Pulakesin ii dating itself in the Kaliyuga Era side by http://o3.indiatimes.com/idea/archive/2005/08/05/208780.aspx

Browse A B C D ... Z document.write(''); CLICK HERE TO START BLOGGING Idea! All kinds of creative , adventurous etc. Ideas.... My Links Home Contact Login Blog Stats Posts - 20 Comments - 152 My Profile done...... BE(Mech) from Pune University.... Diploma In Advance Computing from ACTS-Pune University. ...... like.....Reading anything and everything. ..... Sports cricket,swimming,ping-pong . ......Yoga and jogging ................. Archives August, 2005 (2) July, 2005 (1) December, 2004 (2) November, 2004 (5) ... Aryabhata I (476-ca 550): Indian astronomer and mathematician. Aryabhata I (476-ca 550) Indian astronomer and mathematician U A thousand years before Copernicus, Aryabhata propounds a heliocentric universe with elliptically orbiting planets and a spherical Earth spinning on its axis, explaining the apparent rotation of the heavens. Writes Aryabhatiya, history's first exposition on plane and spherical trigonometry, algebra and arithmetic. Aryabhata wrote Aryabhatiya , finished in 499, which is a summary of Hindu mathematics up to that time, written in verse. Contribution to trigonometry The first actual appearance of the sine of an angle appears in the work of the Hindus.

63. SATELLITE TECHNOLOGY DEVELOPMENT aryabhata was the first Indian satellite launched into a near earth orbit on April 19 Bhaskara I and ii were the first two experimental remote sensing http://ceos.cnes.fr:8100/cdrom-00b2/ceos1/isro/abisro/old_sat.htm

S ATELLITE TECHNOLOGY DEVELOPMENT Aryabhata: Aryabhata was the first Indian satellite launched into a near earth orbit on April 19, 1975 by an Intercosmos rocket of the erstwhile USSR. Aryabhata carried three payloads - one each for X-ray, astronomy, solar physics and aeronomy. The mission was a success and provided the early experience to ISRO for planning, developing and managing a satellite mission. Bhaskara - I and II were the first two experimental remote sensing satellites launched on June 7, 1979 and November 20, 1981 respectively by Intercosmos rockets. Both the satellites carried two TV cameras, one in visible and the other in near-infrared band and having a 1 Km resolution and a 3-frequency Passive Microwave Radiometer. The Bhaskara mission saw the birth of a systematic ground segment and applications programme. Together with the capabilities developed in satellite and sensor development, a truly indigenous space-capability was the result of the Bhaskara mission. Rohini: A series of technological/scientific satellites, launched by India's own launch vehicle, SLV-3. First one was used to measure the performance of SLV-3, second and third carried a SMARTsensor - the first Charge Coupled Device (CCD) camera developed in-house. Rohini saw the entry of ISRO to the use of CCD technology and system design for sensors.

64. Science After Aryabhatta With aryabhata, we enter a new phase in which it becomes easier to trace the (ii) mathematics required for computations, (iii) division of time and http://www.hindunet.org/science_after_aryabhatta/

Sciences Yoga Ayurveda Indian Medicine ... Sciences Science After Aryabhatta Explore this topic in detail Recommend this page Comment on this page Print this page Create / Join Club Hindu Universe Links Articles Online Books Hindu Web Discussion Book- store Related sections In the earliest period of Indian science, it is exceptional when we know the authorship of a text or an idea. For example, although Yajnavalkya and Lagadha describe considerable astronomy, we do not know if this was developed by them or they merely summarized what was then well known. Likewise we are not sure of the individual contributions in the Shulba Sutras- of Baudhayana, Apastamba, and other authors- which describe geometry, or in Pingala's Chhandahsutra which shows how to count in a binary manner. The major exception to the anonymous nature of early Indian science is the grammatical tradition starting with Panini. This tradition is an application of the scientific method where the infinite variety of linguistic data is generated by means of a limited number of rules. With Aryabhata, we enter a new phase in which it becomes easier to trace the authorship of specific ideas. But even here there remain other aspects which are not so well understood. For example, the evolution of Indian medicine is not as well documented as that of Indian mathematics. Neither do we understand well the manner in which the philosophical basis underlying Indian science evolved.

65. Biodata (i) Governing Council, Indian Institute of Science Bangalore since 2004 and (ii)Governing Council, aryabhata Institute of Observational Sciences since http://164.100.24.167:8080/members/website/Biodata.asp?no=1924

66. BANGLAPEDIA: Mathematics The great mathematicians, aryabhata (born in 476 AD), Brahmagupta (born in 598 AD) antiquity through aryabhata, Brahmagupta, Bhaskara ii down to Madhava http://banglapedia.search.com.bd/HT/M_0178.htm

Mathematics science of spatial and numerical relationships. The main divisions of pure mathematics include geometry, arithmetic, algebra, calculus, and trigonometry. Applied mathematics include statistics , numerical analysis, computing, mathematical theories of astronomy , electricity, optics, thermodynamics, and atomic studies. Prehistoric human beings probably learned to count at least up to ten on their fingers. The Chinese, Hindus, Babylonians and Egyptians all devised methods of counting and measuring that were of practical importance in their everyday lives. The first theoretical mathematician is believed to be Thales of Miletus (580 BC) who is believed to have proposed the first theorems in plane geometry. His disciple Pythagoras established geometry as a recognised science among the Greek. The later School of Alexander Geometers (4th and 3rd centuries BC) included Euclid and Archimedes. The present decimal numbers are based on a Hindu-Arabic system that reached Europe about AD 100 from Arab mathematicians of the Middle East such as Khwarizmi. The basic development of mathematics in India (including Bengal) took place between 500 BC and 500 AD, marked as Buddhist and Jaina period. Mathematics in Buddhist and Jaina period The topics of mathematics, according to the Sthananga-sutra (sutra 747) are ten in numbers: parikarma (four fundamental operations), vyavahara (subjects of treatment), rajju (geometry), rashi (mensuration of solid bodies), kalasavarna (fractions), yavat-tavat (simple equation), varga (quadratic equation), ghana (cubic equation), varga-varga (biquadratic equation) and vikalpa (permutation and combination). However, the historians of mathematics differ in explaining some of the terms from the commentator, Abhayadeva Suri (1050 AD).

67. Welcome To Society Of Indian Aerospace Technologies & Industries-SIATI Satellites(aryabhata,Bhaskara I ii,Rohini Series,IRS series,INSAT series).Launch vehicles(SLV,ASLV,PSLV,GSLV series) http://www.siatiaero.com/window.html

Home About Us Window Member ... President's Desk Core Competence – Cooperation Possibilities Evolution of Competence Indian Aerospace Industry Structure / Programmes Areas of Competence The Opportunities Possible Ways of Cooperation Evolution of Competence - Milestones :First Flight in Asia in Allahabad by Humber Motor Company Mono plane named Bleriot X1. :First Recorded Aviation Operation for Military Exercise in Aurangabad. :JRD Tata received the first Pilot’s license issued in India. :Setting up of INDIAN AIR FORCE Setting up the first national carrier - TATA Airlines *. :Hindustan Aircraft Limited started by Seth Walchand Hirachand. Starting of many Flying Clubs all over India. :Starting of 2 years’ Post Graduate Course in Aeronautical Engineering at Indian Institute of Science, Bangalore.

68. Space Programme The establishment of Equatorial Rocket Launching Station, conduct of SITE andSTEP, launches of aryabhata, Bhaskara I and ii, IRS1A and 1B and Apple, http://www.hcidhaka.org/indinfo/space.html

Space Programme In India, the space programme was formally launched in 1972 with the setting up of Space Commission and the Department of Space. The main objective of the programme is to provide space-based services in the spheres of communication, meteorology, resources survey and management, develop satellites, launch vehicles and associated ground systems. The past two decades witnessed the impressive progress made by the country's space programme. Space technology has not only enhanced India's communication capabilities, but has also contributed in meteorological forecasting, providing advanced disaster warning, search and rescue measures and distance education to remote areas. Similarly, space remote sensing provides vital inputs on agricultural, soil, forestry, land and water resources, environment, minerals, ocean development and management of drought and flood disasters, thus touching every facet of national development. An Integrated mission for sustainable development (IMSD), basically using IRS data, has been established at the recommendation of the Planning Commission for arriving at local specific prescriptions. INSAT System The Indian National Satellite System (INSAT) is a multipurpose operational satellite system for domestic telecommunications, meteorological observations and data relay nationwide direct satellite television broadcasting and radio, television programme distribution. The System is a joint venture of Department of Space (DOS), Department of Telecommunications (DOT), India Meteorological Department (IMD), All India Radio (AIR) and Doordarshan.

69. Read About Aryabhata At WorldVillage Encyclopedia. Research Aryabhata And Learn aryabhata. Everything you wanted to know about aryabhata but had no clue how the sine table (ii) mathematics required for computations (iii) division of http://encyclopedia.worldvillage.com/s/b/Aryabhata

Culture Geography History Life ... WorldVillage Aryabhata From Wikipedia, the free encyclopedia. astronomers of the classical age of India . He was born in AD in Ashmaka but later lived in Kusumapura , which his commentator AD) identifies with Patilputra (modern Patna Aryabhata gave the world the digit "0" (zero) for which he became immortal. His book, the , presented astronomical and mathematical theories in which the Earth was taken to be spinning on its axis and the periods of the planets were given with respect to the sun (in other words, it was heliocentric ). This book is divided into four chapters: (i) the astronomical constants and the sine table (ii) mathematics required for computations (iii) division of time and rules for computing the longitudes of planets using eccentrics and epicycles (iv) the armillary sphere, rules relating to problems of trigonometry and the computation of eclipses. In this book, the day was reckoned from one sunrise to the next, whereas in his he took the day from one midnight to another. There was also difference in some astronomical parameters. Aryabhata also gave an accurate approximation for edit See also Indian mathematicians edit External links http://www.jqjacobs.net/astro/aryabhata.html

70. Indian National Science Academy Publication Part ii English translation of the astronomical works of Lalla (1981) 340 pp . Aryabhatiya of aryabhata (Critically edited with translation and notes) http://www.insaindia.org/Publication/historypubli.htm

Home FAQ's Site Map Feedback ... Contact Us Publications of History Of Science In India * Caraka Samhita : A Scientific Synopsis P Ray and H N Gupta (Rev. Ed.) (1980) 124 pp. A scientifically classified accounts of the contents of Caraka Samhita, the medical treatise of ancient India. Rs.30/= US $ 10/= * A Bibliography of Sanskrit Works in Astronomy and Mathematics S N Sen with the research assistance of A K Bag and S R Sarma (1966) 258 pp. A bibliography of source materials indispensable for the study of history of astronomy and mathematics in ancient and medieval India. Rs.20/= US $ 5/= Jahangir : The Naturalist M A Alvi and A Rahman (1967) 140 pp. A study of Jahangir's scientific interests in the fields of animals and plants and also in chemical technology, medicine and astronomy. Rs.60/= US $ 20/= (PB) Rs.75/= US $ 25/= (HB) * Fatullah Shirazi M A Alvi and A Rahman (1968) 37 pp. A study of a sixteenth century Indian scientist of remarkable varsatility. Rs.2.50 US $ 0.33 Susruta Samhita : A Scientific Synopsis P Ray, H N Gupta and M Ray (1980) 459pp. The work gives a classified account in two parts, the first dealing with theories and practices of surgical treatment, and the second a classified account on living creatures, medicinal plant and animal substances of vegetable and mineral origin, medicinal and surgical terms, appliances and intruments, weights and measures etc.

71. Indian National Science Academy History Of Science KV Sarma, Aryabhatiya of aryabhata (critical edition with English (ii) “TheMethod for finding out the Number of Moons and their Families in the http://www.insaindia.org/History/compro66.htm

Home FAQ's Site Map Feedback ... Proforma Indian National Commission for History of Science Research Project IJHS Conferences Publications Projects Approved By The Indian National Commission For History Of Science For The Financial Year - 1966-2000 Ancient Period Medieval Period Modern Period Ancient Period Caraka Samhita: A Scientific Synopsis, 1961-63. Published by INSA, 1965. Reprinted 1980. Bibliography of Sanskrit Works on Astronomy and Mathematics, 1961-64 . Published by INSA, 1966. H.D. Sankalia, Some Aspects of Prehistoric Technology in India, 1968-69 . Published by INSA, New Delhi, 1970. Susruta Samhita: A Scientific Synopsis, 1969-71. Published by INSA, 1980. Reprinted 1993. Bina Chatterjee, Sisyadhivrddhidatantra of Lalla (critical edition with Commentary of Mallikarjuna Suri), 1969-71. Published by INSA, 1981. Bina Chatterjee, Sisyadhivrddhidatantra of Lalla (English translation with mathematical notes.) 1969-71. Published by INSA, 1981. The Sulba-Sutras of Baudhayana, Apastamba, Katyayana and Manava (with English translation and commentary), 1969-71. Published by INSA, 1983.

72. Indiaclub.com Search Results - ProductID: 6403 aryabhata Life and Contributions by DS Hooda , JN Kapur (Author ALERT) PART SEVEN Concluding Remarks CHAPTER ii LIFE HISTORY OF aryabhata PART ONE http://www.indiaclub.com/shop/SearchResults.asp?prodstock=6403

73. Historia Matematica Mailing List Archive: Re: [HM] Indian Astro The opening stanzas of Part I and part ii are quoted in English translation. As a side note, I wonder if aryabhata s 476550AD astronomy http://sunsite.utk.edu/math_archives/.http/hypermail/historia/feb00/0043.html

Re: [HM] Indian astronomy and mathematics Subject: Re: [HM] Indian astronomy and mathematics From: K. Subramaniam ( subra@hbcse.tifr.res.in Date: Fri Feb 04 2000 - 01:01:45 EST Next message: Colin McLarty: "Re: [HM] Emmy Noether" Previous message: William C Waterhouse: "Re: [HM] Emmy Noether" In reply to: Kim Plofker: "Re: [HM] Indian astronomy and mathematics" Next in thread: Kim Plofker: "Re: [HM] Indian astronomy and mathematics" Reply: K. Subramaniam: "Re: [HM] Indian astronomy and mathematics" Reply: Kim Plofker: "Re: [HM] Indian astronomy and mathematics" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] I am seeking a clarification on Kim Plofker's post. I too looked up Shukla and Sarma's critical edition of Aryabhatiya mentioned by K.P. The discussion of the lost work Aryabhata-siddhanta appears on pgs.lxiii- lxviii. Shukla and Sarma say that Brahmagupta's Khanda-khadyaka was planned in two parts. (p.lxvii) The first part summarized the teachings of Aryabhata- siddhanta without making any alteration, modification or addition (except one or two rules). Part II set forth Brahmagupta's corrections and

74. Historia Matematica Mailing List Archive: Re: [HM] Indian Astro The discussion of the lost work aryabhatasiddhanta appears on pgs. lxiii-lxviii . The opening stanzas of Part I and part ii are quoted in English http://sunsite.utk.edu/math_archives/.http/hypermail/historia/feb00/0049.html

Re: [HM] Indian astronomy and mathematics Subject: Re: [HM] Indian astronomy and mathematics From: Kim Plofker ( Kim_Plofker@Brown.edu Date: Sun Feb 06 2000 - 01:07:12 EST Next message: Milo Gardner: "[HM] base 60, Indian, Babylonian or Sumerian?" Previous message: owner-historia-matematica@chasque.apc.org: "(no subject)" In reply to: K. Subramaniam: "Re: [HM] Indian astronomy and mathematics" Next in thread: Dinesh Maheshwari: "Re: [HM] Indian astronomy and mathematics" Reply: Kim Plofker: "Re: [HM] Indian astronomy and mathematics" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... subra@hbcse.tifr.res.in Well, I swore I'd give up this thread, but this is such a thoughtful and have been edited with an English translation by Bina Chatterjee (2 vols: New Delhi, 1970). The eight chapters of the Purva-khandakhadyaka deal in the usual fashion with the main questions of classical Indian astronomy: mean motions of the planets (including the sun), corrections for true motions of the planets, Three Questions (on direction, place, and time), lunar eclipse, solar eclipse, planetary risings and settings, size and

75. Kuttaka aryabhata s solution of the equation axby=c was known as the Kuttaka (pulverizer) in Bhaskara ii later made a modification to aryabhata s Kuttaka. http://www.math.sfu.ca/histmath/India/5thCenturyAD/Kuttaka.html

Aryabhata's Kuttaka Aryabhata 's solution of the equation ax-by=c was known as the Kuttaka (pulverizer) in Hindu mathematics. The equation ax-by=c, arose as a result of the following: It is required to determine an integer N which when divided by a leaves a remainder r' and when divided by b leaves a remainder r''. From this we get: N = xa + r', N = yb + r''. By equating the first equation with the second we get xa + r' = yb + r'' or ax-by = c for c = r''-r'. Aryabhata noted that any factor common to a and b should be a factor of c, otherwise the equation has no solution. Dividing a, b and c by the greatest common factor of (a,b) we can reduce the equation to the form where a and b are relatively prime. In the discussion that follows we can assume that (a,b)=1 and without loss of generality c>0. Aryabhata's method of solution to the equation ax-by=c is given in stanza 32 and 33 of his Aryabhatiya. The following translation of his method was by Bhaskara I, who was a pupil of Aryabhata's teachings. Bhaskara I added some steps in the translation if it was missing in the original.

76. Rent Receipt spanning as it did the times from aryabhata to Bhaskara ii aryabhata, aryabhata’s Theory Rotation of Earth, Indian Journal or History of Science, http://www.vidyaonline.net/arvindgupta/jvn.htm

The culture of science C.D. Deshmukh Memorial Lecture at the India International Centre, Delhi By Jayant Vishnu Narlikar Emeritus Professor, Inter-University Centre for Astronomy and Astrophysics, Pune Ladies and Gentlemen, It is a great honour to be invited to deliver this prestigious lecture in the memory of the late C.D. Deshmukh, especially at the India international Centre that was C.D.’s creation and one of his most beloved projects. I feel somewhat inhibited to fulfil my assignment today, since from my childhood I had looked upon C.D. Deshmukh as a towering personality. I have memories of his visit to our house back in early nineteen fifties, when I was in secondary school. My parents asked me on that occasion to recite some Sanskrit shlokas . I did so with some trepidation since I had heard about how great a scholar C.D. was in Sanskrit. I also recall feeling very happy when the 'Guest of Honour' complimented me on my Sanskrit pronunciation. On that occasion he had come from a function at the Women's College of the Banaras Hindu University. The university is often referred to as B.H.U. and C.D. used this fact in one of his typical witty remarks. He told the women students - as students of the B.H.U. were 'Bhu-kanyas' and as such they should emulate the ideal of Bhukanya -Sita.

77. Astronomia Vedica Astronomia Vedica Part ii. Cosmological Time Cycles in Indian Astronomy In Sanskrit Astronomy is known as Khagola Sasthra and aryabhata worked at an http://www.eastrovedica.com/html/astronomiavedica2.htm

Astronomia Vedica - Part II Cosmological Time Cycles in Indian Astronomy In Sanskrit Astronomy is known as Khagola Sasthra and Aryabhata worked at an astronomical observatory called Khagola. He studied at the University of Nalanda which housed more than 9 million books. The Sexagesimal Division of a Day ( Sixtieth Division ) One Cosmological Cycle is 4.32 Billion years, known as a Brahma day. The Life span of Brahma is 100 sidereal years or 2*4.32*360*100 = 3.1104*10^14 years ! Indian Astronomy is graced by such gigantic calculations starting from 1/21600th of a day to 3.1104*10^14 years ! The Ursa Major Cycle The constellation of Ursa Major ( The Saptha Rishies ) move backwards along the Zodiac, staying in a constellation for 100 years. To make a circuit of the Zodiac, they take 27*100 = 2700 years. This is known as an Ursa Major Cycle. Remarks Prof Drayson in "Asiatic Researches ", " The Indians thought proper to connect their mythology with an astronomical period of a strange nature. It is that of the Seven Rishies, moving along the Zodiac in a retrograde motion of 2700 years." Ursa Major was in Regulus at the start of the Mahabharatha War. The first astronomical calender was erected by the Indian emperor Vaivaswatha Manu ( circa 8736 BC ) and it was based on the Ursa Major Cycle. D or Lunar Day ( Thidhi ) When we deduct the longitude of the Sun from the longitude of the Moon, we get the Thidhi or Lunation

78. Salmonsens Konversationsleksikon / Anden Udgave / Bind II: Arbejderhaver - Benzo Arthur af Bretagne, Søn af Henrik ii Plantagenet s tredieældste Søn Gotfred, f . aryabhata, ind. Astronom og Matematiker, f. 476 e. Kr. 190, 191 http://runeberg.org/salmonsen/2/2/

Salmonsens konversationsleksikon Anden Udgave Bind II: Arbejderhaver - Benzol Project Runeberg Catalog Recent Changes Comments? ... Anden Udgave redigeret af Chr. Blangstrup Bind II: Arbejderhaver - Benzol K¸benhavn A/S J. H. Schultz Forlagsboghandel Trykt hos J. H. Schultz A/S MCMXV F¶rord till den digitala utg¥van Det h¤r ¤r andra bandet av tjugosex i andra utg¥van . Se v¥rt f¶rord till Salmonsens konverssationsleksikon och andra utg¥van The above contents can be inspected in scanned images: smudstitel titelside Korrstapel Proof bar for this volume Indhold / Table of Contents Titel og indhold smudstitel titelside Accentbetegnelse og Lydskrift ... Rettelser og Trykfejl A Arbejderhaver, se Kolonihaver Arbejderhygiejne Arbejderkammer, se Arbejdsraad Arbejderkommissioner ... Azymitter B B Ba B¥¥th, Albert Ulrik, sv. Digter (1853-1912) B¥¥th-Holmberg, Cecilia, sv. Forfatterinde, f. 1857 ... Benzol (Benzin, Fenylbrinte) Project Runeberg, Fri Dec 17 04:42:59 2004 (aronsson) (diff) (history) (download) http://runeberg.org/salmonsen/2/2/

79. Bhutas : Some Reflections On Modern Scientific Concepts And Traditional Indian T Why the astronomical tradition from aryabhata to Bhaskara ii did not continuebeyond the twelfth century? For, during those seven centuries (5001200 ad) http://ignca.nic.in/ps_04003.htm

THE NATURE OF MATTER Previous Page Contents of the Book Next Page Bhutas Some Reflections on Modern Scientific Concepts and Traditional Indian Thoughts Jayant V. Narlikar Against the background of some elementary knowledge about the many different streams of ancient thoughts, I have ventured to ask some questions visa-vis scientific ideas of modern times where I do know a little, perhaps just enough to gauge the extent of my ignorance. About our ancient traditions I do not even know the extent of my ignorance! So I had better begin with the modern scientific end. What does science say about the nature of matter on different physical scales? Scales of Structure A few years ago Professor Phillip Morrison had made a beautiful but short film entitled ‘Powers of Ten’. It started with a scene very common in the Western world, that of a couple picnicking in a city park. Then the camera zooms out showing the larger scale of the park, then zooms farther to show the city, then the state and so on. Each scene is followed by another with ten times larger scale. How far does this go on? Taking 1 metre as the scale we measure our local distances in (— the picnicking couple occupied an area of a few square metres), the tenfold increases in length go on till about 1026 metres — that is, about a hundred million million million million metres! A few significant steps enroute to this gigantic scale are given in Table I below.

80. 80.07.11: A Chronological History Of ¹ With Developmental Activities In Problem During the sixth century, there lived a Hindu mathematician, aryabhata, whosebest known I and ii, New York Dover Publications, Inc., 1958. Vol. I, pp. http://www.yale.edu/ynhti/curriculum/units/1980/7/80.07.11.x.html

Yale-New Haven Teachers Institute Home A Chronological History of ¹ with Developmental Activities in Problem Solving by Anthony P. Solli Contents of Curriculum Unit 80.07.11: Introduction Prerequisites Level Part I: Chronological History ... To Guide Entry Introduction : The unit begins with a historical development of ¹ and proceeds with examples of activities to help students develop a deeper appreciation of the mathematical beauty and values of ¹. Prerequisites : Students should have at least a working knowledge of fractions and decimals, plus an intuitive understanding of the nomenclature, terminology, vocabulary, and formulae of basic geometry. Level: The historical part of this unit may be used in any grade level, preferably middle school. The activities part could be used in any middle school level, although it is recommended for average and above average groups working in small groups or independently. It also may be used as a motivational or summary part of a geometry unit or pre algebra course. The length of time needed could be a week or two, depending on the amount of time you would spend on the history of ~r and doing all or some of the suggested activities.

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