Vedic Mathematics Today

Having recently read several articles published in India which are critical of Vedic mathematics it seems appropriate to give a response based on the work carried out within the United Kingdom. I have taught Vedic mathematics for almost twenty years at an independent school in London. Some of the students have Indian ancestry but this is not the reason for our approach to mathematics. All our mathematics teachers have become familiar with the system and use it in their lessons. We regularly hold training workshops for the staff and have written our own text books for resource material. Our associate schools in different parts of the world also teach Vedic mathematics. Both teachers and students have found Vedic mathematics of great benefit because it has so many positive qualities. The most obvious of these are found through experience of working with the sutras. Children and adults alike find the system profound, inspiring and delightful. We have also found that the students gain exceptional examination results.

What is Vedic mathematics really about? What is it about the subject that an English school should use it so thoroughly? Is it really Vedic and what was the purpose of Sri Bharati Krishna Tirthaji’s book on the subject? In this account I attempt to answer some of these questions.

Vedic mathematics is based on sixteen sutras together with a similar number of sub-sutras. Each sutra provides a principle of mental working applicable to many diverse areas of mathematics. There are several reasons for the title Vedic but before considering these it is worth mentioning that the word Veda generally has two meanings. The first is the collection of ancient Indian texts relating to both spiritual and secular knowledge. Due to the once strong oral tradition by which these texts were passed on it is not possible to accurately place a date when they were first composed. This meaning of Veda is the most common. The second way that Veda is used is to describe true knowledge in the present which resides within peoples hearts or minds. This is the meaning which we have accepted. This is just as well since Sri Tirthaji bequeathed to us no accurate or definite reference for the sutras. Locating the sutras within ancient texts has so far proven to be beyond some of the most able scholars in this field. It is quite possible that Sri Tirthaji intuited the sutras from his deep understanding of the subject, the Vedas and the nature of the human mind.

Arising from the first of these two meanings of Veda historians of mathematics have tried to relate the work of Sri Tirthaji to the mathematics of well-known ancient texts. If only this first meaning is accepted then the question as to whether Tirthaji’s system is or is not Vedic becomes an almost insurmountable task. In my experience it is better to approach Vedic mathematics from the second meaning relating to natural laws working within the human psyche. This is a practical approach and certainly most of the work in the UK has followed this line. We have sympathy for the historians but regard any discovery of reference to the sutras as a bonus rather than a necessity.

The introduction to Vedic Mathematics indicates that during the early part of the 20^th century Sri Tirthaji rediscovered or reconstructed Vedic mathematics from stray references within the appendix portions of the Atharvaveda. We do not know whether or not these appendices were published. He evidently spent a large proportion of his life teaching the system but it was only shortly before he passed away that he set down an illustrative volume on the subject. This was published posthumously in 1965 and is the main source of all the serious study on the subject. His book offers a snapshot of the sutraic system. Some of the sutras are applied to relatively elementary topics in arithmetic and algebra giving rise to fast and easy methods of calculation. Some of these methods are developed quite carefully particularly at the beginning of the book. A few sutras are only mentioned once or twice in rather abstruse and specialised topics. The really surprising aspect is contained within one of his introductions where he describes these few sutras as having jurisdiction over the whole of mathematics.

Years ago when I was first involved with various groups studying Vedic mathematics we all thought this statement outrageous and absurd. How could sixteen sutras apply to the whole of mathematics? Our view was strengthened by the text due to the paucity of explanation of some of the rules. For example, there is a sutra, Vyashti Samashti, which is mentioned only once in the text and even then it is given in relation to a very particular type of biquadratic equation. It is the sort of equation that you are highly unlikely to meet when working in mathematics. Nevertheless he uses this equation as the means to provide us with a brief glimpse at how the sutra can be applied. As it turns out this sutra is fundamental to mathematics particularly in statistics and mechanics. It has countless applications because it describes a common mental process. The sutra means Individuality and totality or Being many, being one or Specific and general. One of the simplest  applications occurs when finding the mean or average of a set of numbers. The idea is that an average in some way is a single number representing the whole group. The sutra also carries a much deeper meaning relating to the individual and universal Self. Expressed as As above - so below, it is recognised as one of the fundamental philosophical tenets of Western culture. It finds expression within the art and architecture of ancient Egypt, the philosophy of Plato, both the Judaic and Christian teachings and within the Humanism of the Florentine Renaissance. Many of the sutras are like this. They go beyond the purely mathematical sphere and deal with a more fundamental level of human experience.

Going back in time when we first came across Vedic mathematics in London we were impressed by the methods of calculation. They had a different quality to what we were brought up with. Here were methods that were enjoyable, quick, light, easy and refreshing. They related the numbers to their source, unity. They brought the mind out of the wrangles of the past and into the immediacy of the present moment. We found the sutras really brought the subject alive and we still find that all students feel more alive by practising Tirthaji’s methods.

Once our enthusiasm was kindled we studied and practiced the whole of his book. We worked through every sum and read and re-read every word to try and make sense of the system. It became clear that since Sri Tirthaji had only left a specimen we would have to see how the sutras could be applied to other areas of mathematics. What if his statement about the jurisdiction of the sutras was true? Could it be possible that so few rules can take account of all mathematics?

We decided to put his statement to the test and see if there were other applications not mentioned in his book. Initially this was difficult but after a while we discovered a fast method of calculation based on one of the sutras. Over a period of years this work continued and we gradually began to see more and more applications to fast methods of calculation, algebraic manipulations and geometrical theorems. We also discovered the sutras at work within entirely conventional methods, systems and algorithms. For example, although we had found a fast method of subtraction of numbers based on one sutra we also found the correct sutra for the conventional paying and borrowing back method.

This work in turn led to greater understanding of the individual sutras together with a greater confidence in the system. It appeared that the sutras described the mental processes or laws that naturally operate within the mind. The implication is that the human mind has only a limited number of natural processes, channels or principles - each having a myriad of applications.

The next step was to consider topics within mathematics and simply ask, what sutra is working here? For example, what is the sutra working when you bisect an angle or when simplifying an irrational number? The elementary topics are fairly straightforward but what about more sophisticated mathematics? We recently worked through various solutions to the ‘shoe-maker’s knife’ problem in geometry. This is an ancient problem but which has become quite a popular area during the last sixty years or so. The mathematics involved is not easy.

You are given two circles drawn inside a larger circle so that they all touch each other. The area between the circles is the Arbelos or shoe-maker’s knife. The problem is to construct further circles within the Arbelos as shown in the diagram. We worked through some of the conventional solutions to this. There was one particularly elegant solution that seemed the quickest and easiest method. It required transformations of a series of circles.

Whilst looking at this solution it dawned on us that the sutra involved was none other than Transpose and adjust, one of the most common sutras in this Vedic system. This is by way of example and illustrates the way in which a simple principle can have an overriding influence over the specific details of the problem.

These anecdotal instances help describe the nature of what we see as Vedic mathematics. On the one hand you have a series of fast, interesting and delightful algorithms and on the other the sutras supply us with a universal approach by which any mathematical process can be perceived as an integral part of a larger whole.

By looking at Tirthaji’s system from this point of view we have found that the applications of the sutras are not time dependent. In other words they can be applied and are entirely valid not only to known mathematics from Indian antiquity as well as the mathematics that the Shankarcarya himself had to deal with but also modern topics ranging from the theory of matrices to the theories of chaos and catastrophe. The system Tirthaji described and what has been developed from his work is Vedic by way of function independently of historical background. We have found it of immense practical value. It stands as a bona fide system supported by deep logic. This is why it succeeds in the educational ethos of the United Kingdom and elsewhere.

Of late there have been a series of academic papers, books and opinions aired by eminent historians and mathematicians published in the Indian press decrying Vedic mathematics either on the grounds that it is not Vedic or because it is not mathematics. This recent controversy demonstrates the paucity of genuine enquiry into the merits of Vedic mathematics. Of course no one should base educational policy on dogma or upon a political agenda which suffers from the fortunes of popularity. Vedic mathematics is no more political or prejudicial than Newton’s three laws of motion. This can be testified by those outside India who use Vedic mathematics freely and without any cultural association. Nevertheless, it is philosophical since it deals with knowledge and natural law.

Those responsible for developments in education must face practical issues. If, in India, there is a demand for Vedic mathematics in schools then this must be met with the tasks of training teachers, supplying resource materials and altering syllabuses according to a more fundamental and unified system of logic. I hope that that every effort can be made to pursue the study of Vedic mathematics in a practical and open-hearted spirit.