# Mathematics

By Krishna Maheshwari and Sudheer Birodkar

In ancient India, Mathematics was considered to be the mother of all sciences. It helped in understanding astronomical phenomena and thus aided in the development of calendars and determining the timing of festivals and determining auspicious times for rituals and events. Ritual altars and building construction (to be built according to vastu shaastra) had to be built to stringent specifications for them to have the properties desired. Mathematics and geometry enabled the construction to these specifications.
Unlike today, we have not been able to determine how detailed professions were separated in Ancient India. As a result, most mathematicians are considered to be priest-mathematicians or rishis who also focused on mathematics, etc.

## Contents

## Religion and mathematics

According to the Vedic world view, one can also come to understand the intricacies of the phenomenal world while culturing transcendental knowledge. By the process of knowing the absolute truth, all relative truths also become known. In modern society today, it is often contended that never the twain shall meet: science and religion are at odds. This erroneous conclusion is based on understanding of Abrahamic faiths which are constantly at odds with science. In Sanatana Dharma, science is a small circle within the larger circle of spirituality.
Mathematics served as a bridge between understanding material reality and the spiritual conception. Vedic mathematics differs profoundly from Greek mathematics in that knowledge for its own sake (for its aesthetic satisfaction) did not appeal to the Hindu mind. Vedic mathematicians strongly felt that every discipline must have a purpose, and believed that the ultimate goal of life was to achieve self-realization and self-perfection-as a result, mathematics were often presented in a very different format. Most mathematics was presented using the Sutra method where there would be a list of laws and each law would borrow data/authority from a super-ceding law. These lists were compressed into small poems - with the first and last word and the length of the rule-similar to how hashing based indexing works in Computer Science today. Those practices which furthered this end either directly or indirectly were practiced most rigorously ^{[1]}.
In order to illustrate how secular and spiritual life was intertwined in Vedic India, Bharati Krishna Tirtha Maharaja has demonstrated that mathematical formulas and laws were often taught within the context of mantra. Thus while learning spiritual lessons; one could also learn mathematical rules.
Mathematics represents a high level of abstraction attained by the human mind. In India, mathematics has its roots in Vedic literature which is nearly 4000 years old. Between 1000 B.C. and 1000 A.D. various treatises on mathematics were authored by Indian mathematicians in which were set forth for the first time, the concept of zero, the techniques of algebra and algorithm, square root and cube root.

Even the technique of calculation, called algorithm, which is today widely used in designing software programs (instructions) for computers was also derived from Indian mathematics.

## Numbers in Sanskrit

Bharati Krishna Tirtha Maharaja has pointed out that Vedic mathematicians preferred to use the devanâgarî letters of Sanskrit to represent the various numbers in their numerical notations rather than the numbers themselves, especially where large numbers were concerned. This made it much easier for the students of this math in their recording of the arguments and the appropriate conclusions. Maharaja wrote,

“In order to help the pupil to memorize the material studied and assimilated, they made it a general rule of practice to write even the most technical and abstruse textbooks in sutra or in verse (which is so much easier-even for the children to memorize). And this is why we find not only theological, philosophical, medical, astronomical, and other such treatises, but even huge dictionaries in Sanskrit verse! So from this standpoint, they used verse, sutra and codes for lightening the burden and facilitating the work (by versifying scientific and even mathematical material in a readily assailable form)!”^{[2]}

Sanskrit Consonant | Denotes Number |

ka, ta, pa, ya | |

kha, tha, pha, ra | |

ga, da, ba, la | |

Gha, dha, bha, va | |

jña, na, ma | |

ca, ta, sa | |

cha, tha, and sa | |

ja, da, and ha | |

jha and dha | |

ka |

Vowels make no difference and it is left to the author to select a particular consonant or vowel at each step. This great latitude allows one to bring about additional meanings. For example kapa, tapa, papa, yapa all mean 11. By a particular choice of consonants and vowels one can compose a poetic hymn with double or triple meanings.

## Algebra- The Other Mathematics?

As in the applied sciences like production technology, architecture and shipbuilding, Indians in ancient times also made advances in abstract sciences like Mathematics and Astronomy. It has now been generally accepted that the technique of algebra and the concept of zero originated in India. In India around the 5th century A.D., a system of mathematics that made astronomical calculations easy was developed. In those times its application was limited to astronomy as its pioneers were astronomers. Astronomical calculations are complex and involve many variables that go into the derivation of unknown quantities. Algebra is a short-hand method of calculation and by this feature it scores over conventional arithmetic. In ancient India conventional mathematics termed Ganitam was known before the development of algebra. This is borne out by the name - Bijaganitam, which was given to the algebraic form of computation. Bijaganitam means 'the other mathematics' (Bija means 'another' or 'second' and Ganitam means mathematics). The fact that this name was chosen for this system of computation implies that it was recognized as a parallel system of computation, different from the conventional one which was used since the past and was till then the only one. Some have interpreted the term Bija to mean seed, symbolizing origin or beginning. And the inference that Bijaganitam was the original form of computation derived. Credence is lent to this view by the existence of mathematics in the Vedic literature which was also shorthand method of computation. But whatever the origin of algebra, it is certain that this technique of computation originated in India and was current around 1500 years back. Aryabhatta an Indian mathematician who lived in the 5th century A.D. has referred to Bijaganitam in his treatise on Mathematics, Aryabhattiya. Bhaskaracharya , an Indian mathematician and astronomer, has also authored a treatise on this subject. The treatise dated around the 12th century A.D. is entitled 'Siddhanta-Shiromani' of which one section is entitled Bijaganitam. Thus the technique of algebraic computation was known and was developed in India in earlier times. From the 13th century onwards, India was subject to invasions from the Arabs and other Islamized communities like the Turks and Afghans. Along with these invader came chroniclers and critics like Al-beruni who studied Indian society and polity. The Indian system of mathematics could not have escaped their attention. It was also the age of the Islamic Renaissance and the Arabs generally improved upon the arts and sciences that they imbibed from the land they overran during their great Jihad. The system of mathematics they observed in India was adapted by them and given the name 'Al-Jabr' meaning 'the reunion of broken parts'. 'Al' means 'The' & 'Jabr' mean 'reunion'. This name given by the Arabs indicates that they took it from an external source and amalgamated it with their concepts about mathematics. Between the 10th to 13th centuries, the Christian kingdoms of Europe made numerous attempts to reconquer the birthplace of Jesus Christ from its Mohammedan-Arab rulers. These attempts called the Crusades failed in their military objective, but the contacts they created between oriental and occidental nations resulted in a massive exchange of ideas. The technique of algebra could have passed on to the west at this time. During the Renaissance in Europe, followed by the industrial revolution, the knowledge received from the east was further developed. Algebra as we know it today has lost any characteristics that betray it eastern origin save the fact that the tern 'algebra' is a corruption of the term 'Al jabr' which the Arabs gave to Bijaganitam Incidentally the term Bijaganit is still use in India to refer to this subject. In the year 1816, an Englishman by the name James Taylor translated Bhaskara's Leelavati into English. A second English translation appeared in the following year (1817) by the English astronomer Henry Thomas Colebruke. Thus the works of this Indian mathematician astronomer were made known to the western world nearly 700 years after he had penned them, although his ideas had already reached the west through the Arabs many centuries earlier. In the words of the Australian Indologist A.L. Basham (‘The Wonder That was India’) "... the world owes most to India in the realm of mathematics, which was developed in the Gupta period to a stage more advanced than that reached by any other nation of antiquity. The success of Indian mathematics was mainly due to the fact that Indians had a clear conception of the abstract number as distinct from the numerical quantity of objects or spatial extension." Thus Indians could take their mathematical concepts to an abstract plane and with the aid of a simple numerical notation devise a rudimentary algebra as against the Greeks or the ancient Egyptians who due to their concern with the immediate measurement of physical objects remained confined to Mensuration and Geometry.

## Geometry And Algorithm

It would be surprising for us to know that even the rudiments of Geometry, called Rekha-Ganita(Line Computation) in ancient India, were formulated and applied in the drafting of Mandalas for architectural purposes. They were also displayed in the geometric patterns used in many temple motifs. The Sulva Sutras, which literally mean 'Rule of the Chord’, give geometrical methods of constructing altars and temples. The temples layouts were called Mandalas. Some of important works in this field are Apastamba, Baudhayana, Hiranyakesin, Manava, Varaha and Vadhula.

The Arab scholar Mohammed Ibn Jubair al Battani studied Indian use of ratios from Rekha Ganita and introduced them among the Arab scholars like Al Khwarazmi, Washiya and Abe Mashar who incorporated the newly acquired knowledge of algebra and other branches of Indian mathematics into the Arab ideas about the subject. The chief exponent of this Indo-Arab amalgam in mathematics was Al Khwarazmi who evolved a technique of calculation from Indian sources. This technique which was named by westerners after Al Khwarazmi as "Algorismi" gave us the modern term Algorithm, which is used in computer software. Algorithm which is a process of calculation based on decimal notation numbers. This method was deduced by Khwarazmi from the Indian techniques geometric computation which he had studied. Al Khwarazmi's work was translated into Latin under the title "De Numero Indico" which means 'of Indian Numerals' thus betraying its Indian origin. This translation belonging to the 12th century A.D is credited to one Adelard who lived in a town called Bath in Britian. Thus Al Khwarazmi and Adelard could be looked upon as pioneers who transmit Indian numerals to the west. Incidents according to the Oxford Dictionary, word algorithm which we use in the English language is a corruption of the name Khwarazmi which literally means '(a person) from Khawarizm', which was the name of the town where Al Khwarazmi lived. To day unfortunately', the original Indian texts that Al Khwarazmi studied are lost to us, only the translations are available. The Arabs borrowed so much from India in the field of mathematics that even the subject of mathematics in Arabic came to known as Hindsa which means 'from India and a mathematician or engineer in Arabic is called Muhandis which means 'an expert in Mathematics'.

## The Concept of Zero

The concept of zero also originated in ancient India. This concept may seem to be a very ordinary one and a claim to its discovery may be viewed as queer. But if one gives a hard thought to this concept it would be seen that zero is not just a numeral. Apart from being a numeral, it is also a concept, and a fundamental one at that. It is fundamental because, terms to identify visible or perceptible objects do not require much ingenuity. But a concept and symbol that connotes nullity represents a qualitative advancement of the human capacity of abstraction. In absence of a concept of zero there could have been only positive numerals in computation, the inclusion of zero in mathematics opened up a new dimension of negative numerals and gave a cut off point and a standard in the measurability of qualities whose extremes are as yet unknown to human beings, such as temperature. In ancient India this numeral was used in computation, it was indicated by a dot and was termed Pujyam. Even today we use this term for zero along with the more current term Shunyam meaning a blank. But queerly the term Pujyam also means holy. Param-Pujya is a prefix used in written communication with elders. In this case it means respected or esteemed. The reason why the term Pujya - meaning blank - came to be sanctified can only be guessed. Indian philosophy has glorified concepts like the material world being an illusion Maya, the act of renouncing the material world (Tyaga) and the goal of merging into the void of eternity (Nirvana). Herein could lay the reason how the mathematical concept of zero got a philosophical connotation of reverence. In a queer way the concept of 'Zero' or Shunya is derived from the concept of a void. The concept of void existed in Hindu Philosophy hence the derivation of a symbol for it. The concept of Shunyata, influenced South-east Asian culture through the Buddhist concept of Nirvana 'attaining salvation by merging into the void of eternity' (Ornate Entrance of a Buddhist temple in Laos). It is possible that like the technique of algebra, the concept of zero also reached the west through the Arabs. In ancient India the terms used to describe zero included Pujyam, Shunyam, Bindu the concept of a void or blank was termed as Shukla and Shubra. The Arabs refer to the zero as Siphra or Sifr from which we have the English terms Cipher or Cypher. In English the term Cipher connotes zero or any Arabic numeral. Thus it is evident that the term Cipher is derived from the Arabic Sifr which in turn is quite close to the Sanskrit term Shubra. The ancient India astronomer Brahmagupta is credited with having put forth the concept of zero for the first time: Brahmagupta is said to have been born the year 598 A.D. at Bhillamala (today's Bhinmal in Gujarat, Western India). Much is not known about Brahmagupta's early life. We are told that his name as a mathematician was well established when K Vyaghramukha of the Chapa dyansty made him the court astronomer. Of his two treatises, Brahma-sputa siddhanta and Karanakhandakhadyaka, first is more famous. It was a corrected version of the old Astronomical text, Brahma siddhanta. It was in his Brahma-sputa siddhanta, for the first time ever he had formulated the rules of the operation zero, foreshadowing the decimal system numeration. With the integration of zero into the numerals it became possible to note higher numerals with limited characters. In the earlier Roman and Babylonian systems of numeration, a large number of characters were required to denote higher numerals. Thus enumeration and computation became unwieldy. For instance, as E the Roman system of numeration, the number thirty would have to be written as X: while as per the decimal system it would 30, further the number thirty three would be XXXIII as per the Roman system, would be 33 as per the decimal system. Thus it is clear how the introduction of the decimal system made possible the writing of numerals having a high value with limited characters. This also made computation easier. Apart from developing the decimal system based on the incorporation of zero in enumeration, Brahmagupta also arrived at solutions for indeterminate equations of type ax2+1=y2 and thus can be called the founder of higher branch of mathematics called numerical analysis. Brahmagupta's treatise Brahma-sputa-siddhanta was translated into Arabic under the title Sind Hind. For several centuries this translation was a standard text of reference in the Arab world. It was from this translation of an Indian text on Mathematics that the Arab mathematicians perfected the decimal system and gave the world its current system of enumeration which we call the Arab numerals, which are originally Indian numerals.

## Value of Pi

Here is an actual sûtra of spiritual content, as well as secular mathematical significance:

gopi bhagya madhuvrata sringiso dadhi sandhiga khala jivita khatava gala hala rasandara

While this verse is a type of petition to Krishna, when learning it one can also learn the value of pi/10 (i.e. the ratio of the circumference of a circle to its diameter divided by 10) to 32 decimal places. It has a self-contained master-key for extending the evaluation to any number of decimal places.

The translation is as follows:

O Lord anointed with the yogurt of the milkmaids' worship (Krishna), O savior of the fallen, O master, please protect me

At the same time, by application of the consonant code given above, this verse directly yields the decimal equivalent of pi divided by 10:

pi/10 = 0.31415926535897932384626433832792

Thus one can memorize significant mathematical facts while offering praise to God in devotion.

## Representation of Large Numbers

The place value system is built into the Sanskrit language, whereas in English we only use thousand, million, billion etc. In Sanskrit, there are specific nomenclature for the powers of 10, the most used in modern times are dasa (10), sata (100=10^2), sahasra (1,000 = 10^3), ayuta (10,000 = 10^4), laksha (100,000 = 10^5), niyuta (1 million = 10^6), koti (10 million = 10^7), vyarbuda (100 million = 10^8), paraardha (1 trillian = 10^12) etc.

Results of such a practice were two-folds.

- The removal of special importance of numbers - Instead of naming numbers in groups of three, four or eight orders of units, one could use the necessary name for the power of 10.
- The notion of the term "of the order of" - To express the order of a particular number, one simply needs to use the nearest two powers of 10 to express its enormity (ie: koti koti (10^7 * 10^7 = 10^14)).

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