Indian Mathematics PDF

Mathametics Vinayak Rajat Bhat Associate Professor Centre for Indian Knowledge Systems Chanakya University Bengaluru  Indian Mathematics: Unique aspects Mathematical works should not be dried ones Indian mathematics is a seamless blend...

Mathametics Vinayak Rajat Bhat Associate Professor Centre for Indian Knowledge Systems Chanakya University Bengaluru  Indian Mathematics: Unique aspects Mathematical works should not be dried ones Indian mathematics is a seamless blend of poetry, literature, logic, and mathematical thinking weaved into a single work. Therefore, there is no fear or stress in learning Mathematics. Mathematics was considered as a part of life Mathematics could be found in temple inscriptions, literary work addressing issues of life, and in a discussion on religion or spirituality. Bhāskarācārya in his Līlāvatī, for example, brings interesting mathematical concepts by posing interesting riddles to a student and solving them. Vyāsa-bhāṣya on Yoga-sūtra refers to the decimal place value system as an example while discussing a philosophical issue.  Indian Mathematics: Unique aspects Uninterrupted tradition of mathematical thinking; it has widely spread across the length & breadth of India Mathematical concepts were developed by those from Gāndhāra (modern- day Afghanistan) to those in Bengal, as well as by those from Kashmir to Kerala The use of sūtras is prevalent in Indian mathematics वृत्तक्षेत्रे परिधिगुणितव्यासपादः फलं तत् (Līlāvatī ) In a circle, the circumference multiplied by one-fourth the diameter is the area. Indian mathematicians adopted the constructive approach where the emphasis is on finding a procedure to solve a problem rather than merely seeking proofs of existence of a solution  Ancient Indian’s tryst with Mathematics Geometry is an ancient Science in India Just with a pole anchored on the ground and a thread attached to it, Indians were able to generate complex geometrical shapes What we see here is a procedure for construction of a square mentioned in Baudhāyana-śulba-sūtra, an ancient mathematical text It is taught in the Department of Mathematics in some Universities in the West as “Rope Geometry” https://www.youtube.com/watch?v=mFAxkauprw8 Ancient Indian’s tryst with Mathematics: Construction of Square 1. Place a rope equal to the measure of the side (X) of the square to be constructed on East-West line. 2. Mark the ends of the rope as A and B and its centre as C. 3. Mark the midpoints of AC and BC as D and E. 4. Fix the ends of the rope on the points D and E and pull it towards the South to the maximum. 5. Place one end of the rope on C and stretch it along CF. 6. Mark point G such that CG = X/2. 7. Fix the ends of the rope on points A and G and pull it towards South-East to the maximum and fix point H (AH=GH=X/2) 8. Fix the ends of the rope on points B and G and pull it towards South-West to the maximum and fix point I (BI=GI=X/2) 9. Repeat steps 4 to 8 in the North direction to get the other two vertices J and K.  Mathematics: Contributions of Ancient Indians (3000 BCE to 600 CE) Sl. Details of the No. Period, Location Salient Contributions Work/Mathematician 1 Vedic Texts 3000 BCE or earlier The earliest recorded mathematical knowledge, number system, Pythagorean type triplets; Decimal system of naming numbers, the concept of infinity; 2 Lagadha – Vedāṅga- ~ 1300 BCE Astronomical concepts; a mathematical model for sun-moon jyotiṣa movement in time; equinoxes & solstices; 3 Śulba-sūtras 800 - 600 BCE Earliest Texts of Geometry; Approximate value of the square root (Baudhāyana, Āpasthamba, of 2, and π. Exact procedures for the construction and Kātyāyana and Mānava) transformations of squares, rectangles, trapezia , etc. 4 Pāṇini –Aṣṭādhyāyī 500 BCE; Śalātura (in Algorithmic approaches; Originator of the Backus–Naur Form Khyber province in (BNF) used in programming languages today); Context-sensitive Pakistan) rules, Arrays, inheritance, polymorphism. 5 Piṅgala - Chandaḥ- 300 BCE Binary sequences; Conversion of Binary to Decimal system and śāstra vice versa; 'Meru Prastara' (Pascal's triangle); Optimal Algorithms to calculate powers; Zero as a Symbol 6 Bauddha about 500 BCE to 500 Multi-valued logic, Discussion of indeterminate and infinite Mathematical works CE numbers;  Mathematics: Contributions of Ancient Indians (3000 BCE to 600 CE) Sl. Details of the Period, Location Salient Contributions No. Work/Mathematician 7 Jaina Mathematical works –Sūrya- 200 BCE to 300 Concept of logarithms, large numbers; algorithms for raising a Prajñapti, Jambūdvīpa-prajñapti, CE number by a power; the arc of a circle; combinatorics; mensuration; Bhagavatī-sūtra, Sthānāṅga-sūtra, Decimal system; Approximation of π; Uttarādhyāyana-sūtra, Tiloyapannati, Anuyoga-dvāra-sūtra 8 Āryabhaṭa – Āryabhaṭīyam 476-550 CE; Concise verses; Algorithm for square root, cube root, Place value Kusumapura, system; Sine table; geometry; quadratic equations; Linear near Pataliputra, indeterminate equations; Sums of squares and cubes of numbers; Bihar Planetary astronomy; Plane and spherical trigonometry. 9 Varāha Mihira – Bṛhat Samhitā, 482-565 CE, Summary of five ancient siddhāntas; Sine table, trigonometric Bṛhat-jātaka, Pañca-siddhāntikā Ujjain, Madhya identities; (sin2 + cos2); combinatorics; Magic squares. Pradesh 10 Bhāskara I - Commentary on 600-680 CE; Expanded Āryabhaṭa's work on Integer solution for indeterminate Āryabhaṭīya, Laghu-bhāskarīyam Vallabhi region, equations; Approximate formula for the sine function, Planetary and Mahā-bhāskarīyam Saurashtra Astronomy; 11 Brahmagupta – Brahmasphuṭa- 598-668 CE; Rules of arithmetic operations with zero and negative numbers, siddhānta; Khandakādhyāya Bhillamala in Algebra (Bījagaṇita); linear and quadratic indeterminate equations; Rajasthan Pythagorean triplets, Formula for the diagonals and area of a cyclic quadrilateral; notion of arithmetic mean. 12 Vrihanka -Vṛttajātisamuccaya(in ~600 CE Fibonacci numbers; Moric metres. Prākṛt)  Mathematics:Contributions of Ancient Indians (800 CE to 1500 CE) Sl. Details of the Period, Salient Contributions No. Work/Mathematician Location 13 Śrīdharācārya–Triṣatikā and 870-930 CE; Arithmetic, Algebra, and Commercial Mathematics; Approximation Pāṭīgaṇita Hugli, West of square root of a non-square number; Quadratic equations; Bengal Practical applications of algebra; 14 Mahāvīrācārya–Gaṇita-sāra- 800 -870 CE; A comprehensive, exclusive textbook on mathematics covering saṅgraha Gulbarga arithmetic-geometry- algebra. Continuing the ancient Jaina Karnataka; mathematics tradition; permutations and combinations; arithmetic and geometric series; the sum of squares and cubes of numbers in arithmetic progression; 15 Jayadeva 10th Century CE Cakravāla (cyclic) method for solution of the second-order or earlier indeterminate equation. 16 Śripati – Gaṇita-tilaka, 1019 - 1066 CE; Planetary Astronomy Siddhānta-śekhara, Rohiṇīkhaṇḍa, Dhikoṭidakaraṇa etc. Maharashtra 17 Bhāskarācārya (Bhaskara-II) - 1114 - 1185 CE; Canonical textbooks used all over India, Detailed explanations Līlāvatī on arithmetic and Hailed from including Upapatti (demonstration or proof); addition formula for geometry; Bījagaṇita on Bijjadavīda sine function. Surds; permutations, and combinations; Solution of algebra; Siddhānta-śiromaṇi on indeterminate equations, Ideas of calculus, including mean value astronomy. theorem, planetary astronomy; construction of several instruments; Mathematics: Contributions of Ancient Indians (800 CE to 1500 CE) Sl. No. Details of the Work/Mathematician Period, Location Salient Contributions 18 Nārāyaṇa Paṇḍita – Gaṇita-kaumudī- a 1325 - 1400 CE; Advanced textbooks taking forward the works of treatise on arithmetic and Bījaganita- Bhāskarācārya, further properties of cyclical quadrilaterals, avatāmśa - a treatise on algebra. summation and repeated summations of arithmetic series, theory and construction of Magic squares, further developments in combinatorics. 19 Mādhava of Saṅgamagrāma 1340 - 1425 CE; Founder of Kerala School of Mathematics - a pioneer in the Sangama Grama, development of calculus; Infinite series and approximations in Kerala, for π, Infinite series and approximations for cosine and sine functions 20 Parameśvara, - Dṛggaṇita, 1360 - 1460 CE; Properties of Cyclic quadrilateral; iterative techniques. Siddhāntadīpikā; Commentaries on Alathiyur, (near Āryabhaṭīyam, Mahā-bhāskarīya; Laghu- Tirur), Kerala, bhāskarīya, Līlāvatī, and Sūryasiddhānta 21 Nīlakanṭha Somayājī, Tantra-saṅgraha; 1444 - 1544 CE; Irrationality of π, basic ideas of calculus; revised planetary Āryabhaṭīya-bhāṣya, Siddhānta-darpaṇa, Near Tirur, Kerala theory, which is a close approximation to Kepler’s model; Exact results in spherical astronomy 22 Jyeṣṭhadeva – Yukti-bhāṣā 1500 - 1575 CE; Hailed as the first textbook of Calculus; detailed Kerala explanations and proofs of the infinite series given by Mādhava  Mathematics: Contributions of Ancient Indians (1600 CE to 1700 CE) Sl. Period, Details of the Work/Mathematician Salient Contributions No. Location 23 Śaṅkaravāriyar- Kriyākramakarī 1500-1569 CE Explanations and Proofs of the results and commentary on Līlāvātī and Kerala procedures given in Līlāvātī commentary on Tantra-saṅgraha 24 Ganeśa Daivajña – Buddhi-vilāsinī 1504 CE; Nandi Explanations and Proofs of the results and procedures (commentary on Līlāvatī); Grama, Nadod, given in Līlāvātī Gujarat 25 Kṛiṣṇa Daivajña – Bījapallva- 1600 CE Delhi Explanations and proofs of results and procedures Commentary on Bījagaṇita of given in Bījagaṇita Bhāskarācārya 26 Munīśvara – Siddhānta- 17th Century Explanations and Proofs of the results and procedures sārvabhauma, commentary on CE; Varanasi given in Līlāvātī; trigonometric identities Līlāvatī; Pātisāra; 27 Kamalākara – Siddhānta-tattva- 1616 - 1700 CE; Addition and subtraction theorems for the sine and viveka Varanasi, Uttara the cosine; Sines and cosines of double, triple, etc., Pradesh angles.  Ancient Indian’s tryst with Mathematics Construction of Śyena-citi (Falcon shaped Altar) - The sacrificial altars used during Vedic times were not a standard shape such as a square or a rectangle. There were more than 70 different shapes of altars used in various sacrifices. These include shapes such as tortoise, falcon, and chariot wheels. B1 B2 B3 B4 B5 The construction of these involved several complex shapes including, the isosceles triangle, rhombus, and circle. Head 1 6 6 1 14 The falcon (see figure) has five components: the head, the body, Body 30 6 10 46 the tail, and the two wings. Wings 30 62 16 108 Five differently shaped bricks have been employed to Tail 8 4 20 32 construct this structure. Total 69 72 52 6 6 200 As evident from the table there are strict constraints in terms of the number of bricks of each type to be used with respect to each part Ancient Indian’s tryst with Mathematics - Construction of Śyena-citi (Falcon shaped Altar)  Number Systems and Unit of Measurement Why do we need? The bedrock of modern scientific discoveries is the use of well-defined number systems, units of measurement, and computational mechanisms Importance of the binary system Vibrant international trade - to transact through the exchange of goods and services, we need standard means of measurement, estimation, and communication This requires a well-defined number system and units of measurement for length, weight, time, etc.  Number System in India: Historical Evidence Laplace remarked, “The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India”. Whilst we use letters for calculation according to their numerical value, the Indians do not use letters at all for arithmetic. (Al-Biruni, 1030 CE) I can only compare their [i.e., Indian] mathematical and astronomical literature, as far as I know it, to a mixture of pear shells and sour dates …. (Al-Biruni, 1030 CE) The street widths in the Indus-Saraswati Civilization were highly standardized. Kalibangan, a city in the Indus-Saraswati Civilization (in Rajasthan, India) had street widths of 1.8 m, 3.6 m, 5.4 m, and 7.2 m. These were built to the standard dimensions of 1, 2, 3, and 4 Dhanuṣ-s  Number System in India Historical Evidence The excavations at Harappa, Mohenjo Daro, Dholavira, and Lothal show that several constructions were done using fired bricks standardised with length x width x depth in the ratio 4:2:1. In the Arthaśāstra, there is a mention of two types of Dhanus as units for measurement of roads and distances : Dhanus = 96 Angulas; Gārhapatya- dhanus = 108 Angulas  Number System in India Historical Evidence A legal document dated 594 CE from the Bharukachcha (or Broach) region in Gujarat contains a number written in the place-value format In an inscription at Gwalior dated ‘Samvat 933’ in the Vikrama calendar (876 CE) the numbers 50 and 270 were recorded with a small circle appearing at the appropriate positional place for zero  Salient features of the Indian Numeral System The concept of zero and its use beyond being a placeholder A legacy of using large numbers with unique number names for these large numbers Developing a robust place value system for the numerals A decimal system that opened vast possibilities for arithmetic operations Bhakṣāḷi manuscript: https://www.wikiwand.com/en/History_of_the_Hind Unique methods to represent numbers u%E2%80%93Arabic_numeral_system  The Concept of Zero Among the significant contributions of the ancient Indians is the concept of zero Used both as a symbol (or numeral) and a concept meaning the absence of any quantity The number name to indicate zero is Śūnya Allows us to perform calculus, solve complicated equations, and to have invented computer operations using binary digits. Bhāskara II in his Bīja-gaṇita introduced the properties of zero when mathematical operators such as addition and subtraction are operated on it  The Concept of Zero The available evidence shows that the concept of zero was established during the period 500 - 300 BCE and fully developed in India by 600 CE Piṅgala a second century BCE Indian philosopher authored Chandaḥ-śāstra, in which the word śūnya was used, which obtained the mathematical connotation of 0 (रूपे शून्यम्) It later became its proper name as a number Brahmagupta developed a symbol for zero in 628 CE. With this invention, zero could be used as an independent numeral for computational purposes  Spread of Indian Mathematical Concepts The catalogue of the Sui dynasty (610 CE), pointed to several Chinese translations of Indian works on astronomy and mathematics Records from the seventh century (of the Tang dynasty), suggest that Indian astronomers were employed at the Astronomical Board of Chang-Nan to teach the principles of Indian astronomy Indian numerals reached the court of second Abbasid Caliph al-Mansur (753–774 CE) from Sindh Indian decimal place-value system reached at least a century earlier as evident from a passage attributed to Nestorian Bishop Severus Sebokht (662 CE) Zij-al-Sindhind (an earliest astronomical work in Arabic) is the translation of Brāhmasphuṭasiddhānta and Zij-al-Arkand is based on the ideas of Āryabhaṭa and Brahmagupta (Khaṇḍakhādyaka) Sources: a) Joseph (2009), “A passage to Infinity: Medieval Indian Mathematics from Kerala and Its impact”, Sage, Chapter 8. b) Alladin Saleh Mohammed (1988), “Interaction of Arab and Persian Astronomers with India”, The Mythic Society, 1.7.  Spread of Indian Mathematical Concepts In a book written during 820 CE by al-Khwarizmi the operations of addition, subtraction, multiplication, division and the extraction of square roots according to the Indian system was explained In the transmission of Indian numerals to Europe, via the Arab world, Spain played an important role as it was under Muslim rule for many years Documents from Spain and coins from Sicily provide evidence of the spread Fibonacci (1170–1250 CE) learnt the Indian numerals during his travels in North Africa, Egypt, Syria and Sicily and wrote a book bringing these aspects into focus Source: Joseph (2009), “A passage to Infinity: Medieval Indian Mathematics from Kerala and Its impact”, Sage, Chapter 8.  Decimal System The universal way of dealing with numbers today, is through the use of 10 as the base The Indian system of numerals employs the use of a decimal system Decimal number system originated in India much before the 12th - 11th century BCE Dutta and Singh* in their book provide a list of 33 inscriptions and grant plates that contain numerals written in the decimal place- value notation These inscriptions range from 595 CE to 975 CE * Datta, B and Singh, A.N. (1962), “History of Hindu Mathematics: Parts I and II”, Asia Publishing House  Spread of Indian Decimal System 976. The first Arabic numerals in Europe appeared in the Codex Vigilanus in the year 976. 1202. Fibonacci, an Italian mathematician who had studied in Béjaïa (Bougie), Algeria, promoted the Arabic numeral system in Europe with his book Liber Abaci, which was published in 1202. 1482. The system did not come into wide use in Europe, however, until the invention of printing. (See, for example, the 1482 Ptolemaeus map of the world printed by Lienhart Holle in Ulm, and other examples in the Gutenberg Museum in Mainz, Germany.) 1512. The numbers appear in their modern form on the titlepage of the “Conpusicion de la arte de la arismetica y juntamente de geometría" written by Juan de Ortega. 1549. These are correct format and sequence of the "modern numbers" in titlepage of the Libro Intitulado Arithmetica Practica by Juan de Yciar, the Basque calligrapher and mathematician, Zaragoza 1549. https://www.wikiwand.com/en/History_of_the_Hindu%E2%80%93Arabic_numeral_system  Bhāskarācārya’s Līlāvatī - Decimal System and Place value एक-दश-शत-सहस्र-अयु त-लक्ष-प्रयु त-कोटयः क्रमशः। अर्ुुदमब्जं खर्ु-धिखर्ु-महापद्म-शङ्कर्स्तस्मात्॥ जलधिश्चान्त्यं मध्यं पिािुधमधत दशगुिोत्तिं संज्ञा:। संख्यायाः स्थािािां व्यर्हािाथं कृताः पूर्ःव ॥ eka-daśa-śata-sahasra-ayuta-lakṣa-prayuta-koṭayaḥ kramaśaḥ । arbudhaṃ-abjaṃ-kharva-nikharva-mahāpadma-śaṃkavastasmāt ॥ jaladhiśca-antyaṃ madhyaṃ parārdhamiti daśaguṇottaraṃ saṃjñā: । saṃkhyāyāḥ sthānānāṃ vyavahārārthaṃ kṛtāḥ pūrvaiḥ || एकम् (eka) – 1 (100) प्रयुतम् (Prayuta) – 106 (Million) महापद्मः (Mahāpadma) – 1012 (Trillion) दश (daśa) – 101 कोट ः (Koṭi) – 107 शङ्ुः (Śaṅka) – 1013 शतम् (śata) – 102 अर्ुदु म् (Arbuda) – 108 जलध ः (Jaladhi) – 1014 सहस्रम् (sahasra) – 103 अब्जम् (Abja)‍‍– 109 (Billion) अन्त्यम् (Antya) - 1015 (Zillion) अयुतम् (Ayuta) – 104 खर्ुः (Kharva) – 1010 मध्यम् (Madhya)‍‍-‍1016 लक्षम् (Lakṣa) – 105 धिखर्ुः (Nikharva) – 1011 परा ुम् (Parārdha) - 1017  Bhūta-saṃkhyā system A structured approach to express numbers 0 to 9 using words representing certain entities Two parts to the compound word; bhūta and saṃkhyā bhūta denotes some entity and saṃkhyā a number The entities are things, beings, or concepts that are commonly known If we use the word “eye” we can spontaneously associate this with the number “two” To broaden the scope of using this system, synonyms are also used For example, “chandra (moon)” is used for the number “one”. The other synonyms of the moon such as “śaśi”, “vidhu”, “soma” and “indu” are also used Is an open-ended list, and it is up to the user to choose an appropriate word User to pick an appropriate word pertaining to the occasion Permits the user to beautifully merge aesthetics and literature into mathematical ideas that he/she wants to communicate through some verses Bhūta-saṃkhyā system Entities used for denoting numbers Word name for the number itself – śūnya, ekam, dvi, trini … nava Physical entities earth, moon, stars, mountain, fire, sky, direction Examples from the animal kingdom elephant, horse, snake Parts of the body eyes, limbs, seven dhātus, etc. Names of Gods Śiva, Viṣṇu, Indra, Manu, Agni, etc. Other concepts such as seasons, months, days, five bhūtas  Bhūta-saṃkhyā system – An example Mādhavācārya’s approximation to π विबुध-नेत्र-गजावि-हुताशन-वत्रगुण-िेद-भ-िारण-बाििः । निवनखिवविते िृवतविस्तरे पररवधिानविदं जगदुबुवधाः ॥ vibudha-netra-gaja-ahi-hutāśana-triguṇa-veda-bha-vāraṇa-bāhavaḥ | nava-nikharva-mite vṛti-vistare paridhimānam-idaṃ jagadurbudhāḥ || In the first line of the verse, the value 2,827,433,388,233 is provided using the bhūta-saṁkhya system. vibudha - Devas (33); netra – Eyes (2); gaja – Elephant (8); ahi - Snake (8); hutāśana – Agni (3); tri – (3), guṇa – (3); veda – (4) bha – stars (27); vāraṇa – Elephant (8); bāhu – Hands (2); In the second line of the verse, there is again a mention of a number; nava – (9); nikharva – 1011. This number is 9x1011. The balance part of the śloka mentions that this is the ratio of the circumference (paridhimānam) to the diameter (vṛtivistare) of the circle. Taking this ratio will yield us the value of π. 2827433388233 The value of π = = 3.14159265359222 9𝑋1011 Kaṭapayādi System A system to convert the numerals to words by associating a number to one or more alphabets Once this association is established, the numbers can be replaced with a corresponding alphabet Using the alphabet in the place of numbers, one can construct words Deciphering one alphabet at a time will reveal the number expressed in the word The advantage lies in representing long (or large) numbers using a nice word, which can be easily remembered  Letter Numerals of the Kaṭapayādi System Governing Rules The vowels when standing alone indicate the number zero 1 2 3 4 5 6 7 8 9 0 In all other cases when they are क ख ग घ ङ च छ ज झ ञ conjoined with consonants, they merely (ka) (kha) (ga) (gha) (ṅa) (ca) (cha) (ja) (jha) (ña facilitate pronunciation of the ) consonants ठ ड ढ ण त थ द ि Each consonant is uniquely associated (ṭa) (ṭha) (ḍa) (ḍha) (ṇa) (ta) (tha) (da) (dha) (na) with a number from 0 to 9 प फ र् भ म More than one consonant may be (pa) (pha) (ba) (bha) (ma) associated with each of the numerals य र ल र् श ष स ह When more than one consonant is used in (ya) (ra) (la) (va) (śa) (ṣa) (sa) (ha) conjunction, only the terminal consonant preceding a vowel is to be considered A standalone consonant will have to be ignored Letter Numerals of the Kaṭapayādi System Governing Rules 1 2 3 4 5 6 7 8 9 0 क ख ग घ ङ च छ ज झ ञ (ka) (kha) (ga) (gha) (ṅa) (ca) (cha) (ja) (jha) (ña) ठ ड ढ ण त थ द ि (ṭa) (ṭha) (ḍa) (ḍha) (ṇa) (ta) (tha) (da) (dha) (na) प फ र् भ म (pa) (pha) (ba) (bha) (ma) य र (ra) ल र् श ष स ह (ya) (la) (va) (śa) (ṣa) (sa) (ha)  Kaṭapayādi System Examples भिवत (bhavati) Splitting them into separate letters bha-va-ti. After ignoring all the vowels and reading from the table the corresponding numbers, we get 4 – 4 – 6. Therefore, the number is 644. आयुरारोग्यसौख्यि् (āyurārogyasaukhyam - ā-yu-rā-ro-gya-sau-khya-m) In this case, there is a standalone vowel (ā), which will indicate 0. There is a standalone consonant at the end (m), which will have to be ignored. The digits are 0 – 1 – 2 – 2 – 1 – 7 – 1. Therefore, the number is 1,712,210 गोपीभाग्य िधुव्रात शृङ्गीशोदवधसवधधग । खलजीवितखाताि गलिालारसधधर ॥ Gopībhāgya madhuvrāta sṛṅgīśodadhisandhiga | Khalajīvitakhātāva galahālārasandhara || If the letters are replaced by the corresponding numbers, i.e. 'go' by 3, 'pi' by 1, 'bha' by 4, 'ya by 1' and so on, the following result is obtained: 31415926535897932384626433832792  Piṅgala and the Binary system The basic building block of poetry is a syllable A syllable is a vowel or a vowel with one or more consonants preceding it There are two types of syllables defined by Piṅgala in the Chandaḥ-śāstra Laghu (Short Syllable) - Any syllable with a short vowel Guru (Long Syllable) can be of four varieties: Any syllable with a long vowel - यदा Any short syllable followed by conjunction of consonants - ग्रस्त Any short syllable followed by “ṃ” known as anusvāra or visarga denoted by “:” - अंशः The last syllable in a meter (optionally) किमकुर्वत सञ्जय । Replacing Laghu with “1” and Guru with “0” we can obtain a binary representation of the idea  Identification of the laghu and guru as per rules of Chandaḥ śāstra An example यदा यदा टह मुस्य ग्लाधिभुर्धत भारत । अभ्यु्थािम मुस्य तदा्मािं सृजाम्यहम् ॥ yadā yadā hi dharmasya glānirbhavati bhārata | abhyutthānamadharmasya tadātmānaṃ sṛjāmyaham || ya dā ya dā hi dha rma sya glā ni rbh va ti bhā ra ta a L G L G L G G G G G L L L G L G a bhy tth na ma dha rma sya ta dā tmā naṃ sṛ jā my ham u ā a G G G L L G G L L G G G L G L G Let us denote laghu by the number “1” and guru by the number “0”. This will convert the above table into a binary word of length 16  Mnemonic for the 8 “ganas” of Piṅgala Piṅgala first defined groups of three using the laghu and the guru as the basic building block for the rules for various meters. There are eight unique binary words one can obtain; Each one is called a “gana” Sl. No. Gana Name Binary Word* 1 “Ya” Gana 100 2 “Ma” Gana 000 3 “Ta” Gana 001 4 “Ra” Gana 010 5 “Ja” Gana 101 6 “Bha” Gana 011 7 “Na” Gana 111 8 “Sa” Gana 110 Binary Cycle of length 3 य-मा-ता-रा-ज-भा-न-स-ल-गम ् Ya-mā-tā-rā-ja-bhā-na-sa-la-gam Very recently, this method of string generation was discovered using combinatorial mathematics known as the De Bruijn sequence