It is a little hard to define Jaina mathematics. Jainism is a religion and philosophy which was founded in India around the 6th century BC. To a certain extent it began to replace the Vedic religions which, with their sacrificial procedures, had given rise to the mathematics of building altars. The mathematics of the Vedic religions is described in the article Indian Sulbasutras.Now we could use the term Jaina mathematics to describe mathematics done by those following Jainism and indeed this would then refer to a part of mathematics done on the Indian subcontinent from the founding of Jainism up to modern times. Indeed this is fair and some of the articles in the references refer to fairly modern mathematics. For example in  Jha looks at the contributions of Jainas from the 5th century BC up to the 18th century AD.This article will concentrate on the period after the founding of Jainism up to around the time of Aryabhata in around 500 AD. The reason for taking this time interval is that until recently this was thought to be a time when there was little mathematical activity in India. Aryabhata‘s work was seen as the beginning of a new classical period for Indian mathematics and indeed this is fair. Yet Aryabhata did not work in mathematical isolation and as well as being seen as the person who brought in a new era of mathematical investigation in India, more recent research has shown that there is a case for seeing him also as representing the end-product of a mathematical period of which relatively little is known. This is the period we shall refer to as the period of Jaina mathematics. There were mathematical texts from this period yet they have received little attention from historians until recent times. Texts, such as the Surya Prajnapti which is thought to be around the 4th century BC and the Jambudvipa Prajnapti from around the same period, have recently received attention through the study of later commentaries. The Bhagabati Sutra dates from around 300 BC and contains interesting information on combinations. From about the second century BC is the Sthananga Sutra which is particularly interesting in that it lists the topics which made up the mathematics studied at the time. In fact this list of topics sets the scene for the areas of study for a long time to come in the Indian subcontinent. The topics are listed in  as:-
… the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.The ideas of the mathematical infinite in Jaina mathematics is very interesting indeed and they evolve largely due to the Jaina’s cosmological ideas. In Jaina cosmology time is thought of as eternal and without form. The world is infinite, it was never created and has always existed. Space pervades everything and is without form. All the objects of the universe exist in space which is divided into the space of the universe and the space of the non-universe. There is a central region of the universe in which all living beings, including men, animals, gods and devils, live. Above this central region is the upper world which is itself divided into two parts. Below the central region is the lower world which is divided into seven tiers. This led to the work described in  on a mathematical topic in the Jaina work, Tiloyapannatti by Yativrsabha. A circle is divided by parallel lines into regions of prescribed widths. The lengths of the boundary chords and the areas of the regions are given, based on stated rules.This cosmology has strongly influenced Jaina mathematics in many ways and has been a motivating factor in the development of mathematical ideas of the infinite which were not considered again until the time of Cantor. The Jaina cosmology contained a time period of 2588 years. Note that 2588 is a very large number!
Still the highest enumerable number has not been attained.The whole procedure is repeated, yielding a truly huge number which is called jaghanya- parita- asamkhyata meaning “unenumerable of low enhanced order”. Continuing the process yields the smallest unenumerable number.Jaina mathematics recognised five different types of infinity :-
… infinite in one direction, infinite in two directions, infinite in area, infinite everywhere and perpetually infinite.The Anuyoga Dwara Sutra contains other remarkable numerical speculations by the Jainas. For example several times in the work the number of human beings that ever existed is given as 296.By the second century AD the Jaina had produced a theory of sets. In Satkhandagama various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets. Permutations and combinations are used in the Sthananga Sutra. In the Bhagabati Sutra rules are given for the number of permutations of 1 selected from n, 2 from n, and 3 from n. Similarly rules are given for the number of combinations of 1 from n, 2 from n, and 3 from n. Numbers are calculated in the cases where n = 2, 3 and 4. The author then says that one can compute the numbers in the same way for larger n. He writes:-
In this way, 5, 6, 7, …, 10, etc. or an enumerable, unenumerable or infinite number of may be specified. Taking one at a time, two at a time, … ten at a time, as the number of combinations are formed they must all be worked out.Interestingly here too there is the suggestion that the arithmetic can be extended to various infinite numbers. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion was noted. In a commentary on this third century work in the tenth century, Pascal‘s triangle appears in order to give the coefficients of the binomial expansion.Another concept which the Jainas seem to have gone at least some way towards understanding was that of the logarithm. They had begun to understand the laws of indices. For example the Anuyoga Dwara Sutra states:-
The first square root multiplied by the second square root is the cube of the second square root.The second square root was the fourth root of a number. This therefore is the formula
… the second square root multiplied by the third square root is the cube of the third square root.The third square root was the eighth root of a number. This therefore is the formula
- Hello world!
- Ms.Shilpa Bhasker One of the Directors of Vasista Eduventures as Chief Guest at The Inauguration of Science Forum,@ Maharani's Science College for Women, Mysore
- Vedic Maths Presentation @ Institution of Engineers-India (IEI) Mysuru.
- Vedic Maths Workshop @ Yajaman First Grade College, Periyapatna, Mysore.
- Vedic Maths Workshop @ Madikeri