"It is India that gave us the ingenuous method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity." — French mathematician Pierre Simon Laplace (1749 - 1827) Rig Veda The Rig Veda is the oldest of the Vedas and contains a wealth of knowledge on a variety of subjects. Some of them relate to the broader field of mathematics, of which a subset is described here. Arithmetic: Numbers and decimal system: Hindu mathematicians used a system based on 10. The names for the numbers one to nine found in Rig Veda are eka, dvi, tri, chatur, pancha, shat, sapta, asta, nava. They had a name for each power of 10, and used these names when writing numerals. For example, "1 sata, 3 dasan, 5" to represents the number 135. The names for ten, twenty, ....., ninety occur in Rig-Veda (2.18.5-6). The Rig-Veda (3.9.9) has a number 3339 spelled as three thousand, three hundred and thirty nine. The Rig-Veda (2.14.16) uses the word hundred thousand, the modern lakh. Many lakhs are described as hundreds of thousands in Rig-Veda (1.14.7). Rig Veda has more than a hundred references to numbers. Geometry: Geometry is used throughout the Rig Veda. The word Geometry is a Sanskrit word means measuring the earth. "Jya" in Sanskrit means earth, "miti" means measurement "jyamiti" or geometry means measuring the earth. Some of the hymns, which deal with cosmology, imply that these poets were very familiar with geometry and the planning needed to construct complex objects. Consider for example the following verse in RV (Mandala10, Sūkta130, Verse 3). The words in Italics are the words in the Sanskrit original. It deals with the creation or formation of the universe. Who was the measurer prama? What was the model pratimā? What were the building materials for things offered nidānam ājyam? What is the circumference (of this universe) paridhiĥ? What are the meters or harmonies behind the Universe chhandaĥ? What is the triangle (yoke) praugam [which connects this universe to the source of driving force, the engine]? All the words in Sanskrit, prama, pratimā etc., are geometrical terms which also occur in the later Sulba Sūtras with the indicated meaning. Hence it is safe to assume that these poets were aware of the construction of buildings and other artifacts. Chariots are described in great detail in many different verses in the Rig Veda (1.102.3, 1.53.9, 1.55.7, 1.141.8, 2.12.8, 4.46.2...) and Yajur Veda. Dr. Kulkarni writes: The proficiency in chariot building presupposes a good deal of knowledge of geometry... The fixing of spokes of odd or even numbers require knowledge of dividing the area of the circle into the desired numbers of small parts of equal area, by drawing diameters. This also presupposes the knowledge of dividing a given angle into equal parts. The Rig Veda is full of references to words in rituals whose definitions we find in subsequent Brahmanas and in the Sulba Sutra to be pointing to geometrical figures. For example, three types of fire altars, garhapatya, ahavaniya and dakshina are mentioned in the Rig Veda but defined in the Shatapatha Brâhmana as being square, circular and semi circular, respectively, and also having the same area. Today, what we call Pythagoras Theorem is a mere repetition of what had been said in Baudhaya "SulbaSutras", written five to six hundred years before Pythagoras. Error Correction & Detection Codes: Rig Veda is a book of about 10,000 verses composed several thousands of years ago. Still all the available manuscripts and the authorized audio versions recorded all over India differ from each other in only one syllable. Such a feat is possible because Rig Vedic sages had developed methods of chanting reminiscent of the modern methods of error detecting codes, an advanced mathematical concept and associated technology developed only in the last fifty years of this century. The rishis had focused on developing methods of chanting which can detect any errors in chanting of a mantra, such as omitting a syllable or replacing one syllable by another. For each mantra, there are several different methods of chanting, each method capable of detecting one type of error. As for example: Two wheeled chariot (Àvichakra ratha): This is a code that can handle two verses which end with different words which are phonetically close, i.e. error of type 4. Verse 1.1.1: 1 2 3 4 5 6 7 8 Verse 1.20.1: a b c d e f g h 8: ratnadhātamam g: ratnadhātamaĥ The chanting procedure ensures that word 8 is chained to 7 and the word g is chained to f and no jump can occur from 7 to g or f to 8. Fractions: Rig Veda (10.90) mentions the fractions ¼, ½, ¾ and also the fact ¼ + ¾ = 1. Shatapatha Brāhmaņa mentions these and similar results, In addition it mentions in (4.6.7.3) that 1/3 + 2/3 = 1 Yajur Veda (18.26) mentions the series 1/2 1 1/2, 2, 2 1/2,3, 3 1/2 and 4. Yajur Veda Concept of Infinity: The concept of infinity was also known during Vedic times. They were aware of the basic mathematical properties of infinity and had several words for the concept-chief being ananta, purnam, aditi, and asamkhyata. Asamkhyata is mentioned in the Yajur Veda, and the Brihadaranyaka Upanishad as describing the number of mysteries of Indra as ananta. These two statements are elaborated in the opening lines of the Isha Upanishad (Shukla Yajur Veda). This sholka is as much metaphysical as it is mathematical. pûrnamadah pûrnamidam pûrnât pûrnamudacyate pûrnâsya pûrnamadaya pûrnamevâvasishyate From infinity is born infinity. When infinity is taken out of infinity, only infinity is left over. The Shukla Yajur Veda (17.2) mentions ayuta (104) niyuta the series of 10 upto 10^12 in steps of powers of 10 namely sahasra (104) niyuta (105), prayuta (106) arbuda (107), hyabuda (108), samudra (109), madhya (1010), anta (104), parardha (1012) Multiplication And Division: Shatapatha Brāhmaņa gives many instances of multiplication . For example (2.3.4.19-20) gives 360 x 2=720, 720 x 80 = 57,600. Again the same book in the section (10.24.2 1-20) gives the result of dividing 720 by all the integers from 2 to 23 which do not give any residue. For instance it considers 720/2, 720/3, 720/4, 720/5, 720/6, 720/8, 720/9, 720/10, 720/12, 720/15, 720/16, 720/18, 720/20, 720/24. Big Numbers: In Yajurveda, numbers starting from four and with a difference of four forming an arithmetic series is discussed. The Yajurveda also mentions the counting of numbers upto 10^18, the highest being named parardha. In the Taittiriya Upanishad, there is a anuvaka (section), that extols the "Beatific Calculus" or a quasi-mathematical relationship between bliss of a young man, who has everything in the world to the bliss of the Brahman, or "realization". Translated roughly as follows, fear is all-pervasive. It continues by assuming that a young, good man who is fit, healthy and strong, and has all the wealth in the world, is one unit of human bliss. The anuvaka provides a precise calculation of a series of multiplications by 100 to give number 10010 units of human bliss that can be had when one attains Brahman. The previous anuvaka exhorts the aspirants to be fearless and strong, as only such a person may realize the absolute within. Atharva Veda Concept of Shunya (Zero) The concept of Shunya, or zero void, was originally conceived as the symbol of Brahman, expressing the sum of all distinct forms. The symbol of zero and the decimal system of notation is described in the Atharva-veda. it describes how the number increases by 10 by writing zero in front of it. While there is no explicit mention of zero, it must have been common knowledge based on how it is used. The concept of zero is referred to as shunya in the early Sanskrit texts and it is also explained in the Pingala’s Chandah Sutra (200 AD). In the Brahma Phuta Siddhanta of Brahmagupta (400-500 AD), the zero is lucidly explained. The earliest recorded date for an inscription of zero (inscribed on a copper plate) was found in Gujarat (585 – 586 AD). Later, zero appeared in Arabic books in 770 AD and from there it was carried to Europe in 800 AD. In 498 AD, Indian mathematician and astronomer Aryabhata stated that "Sthanam sthanam dasa gunam" or place to place in ten times in value, which is the origin of the modern decimal-based place value notation. contd
Approximation of π Aryabhata worked on the approximation for Pi (π), and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes: caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ. "Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached." This implies that the ratio of the circumference to the diameter is {(4 + 100) × 8 + 62000}/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures. Later on Bharati Krishna Tirtha Maharaja, author of Vedic Mathematics, has offered a glimpse into the sophistication of Vedic mathematics. Drawing from the Atharva-veda – “gopi bhagya madhuvrata srngiso dadhi sandhiga khala jivita khatava gala hala rasandara” he has decoded the value of Pi upto 30 decimal accurately that is 0.31415926535897932384626433832792. Trigonometry: In Ganitapada 6, Aryabhata gives the area of a triangle as tribhujasya phalashariram samadalakoti bhujardhasamvargah "for a triangle, the result of a perpendicular with the half-side is the area." Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means "half-chord". For simplicity, people started calling it jya. Later on this concept go through Arabic literature (jaib) to its Latin counterpart, sinus, which means "cove" or "bay". And after that, the sinus became sine in English. Indeterminate equations: A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as diophantine equations. This is an example from Bhāskara's commentary on Aryabhatiya: Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7 That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems is called the kuṭṭaka (कुट्टक) method. Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm. The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulbasutras. Algebra: In Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes: View attachment 228
