83 - Cognitive Patterns in Vedic Mathematics


ISSUE No. 83

A warm welcome to our new subscribers.
Vedic Mathematics is becoming increasingly popular as more and more people are introduced to the beautifully unified and easy Vedic methods. The purpose of this Newsletter is to provide information about developments in education and research and books, articles, courses, talks etc., and also to bring together those working with Vedic Mathematics. If you are working with Vedic Mathematics - teaching it or doing research - please contact us and let us include you and some description of your work in the Newsletter. Perhaps you would like to submit an article for inclusion in a later issue or tell us about a course or talk you will be giving or have given. If you are learning Vedic Maths, let us know how you are getting on and what you think of this system.


This issue's article is by James Glover who has a long experience of teaching Vedic Mathematics in schools together with running public courses on the subject. Until recently he was deputy headmaster at St James Senior Boys' School in London and previously was head of Mathematics there for twenty years. He has written three books on the subject, Vedic Mathematics For Schools 1, 2 and 3, and is currently writing second editions for these. Between 1995 and 2006 James ran several public courses on Vedic Maths at such places as Imperial College and The School Of Economic Science. He has also given numerous lectures and workshops including a lecture tour of five major cities in India.





We are delighted to announce once again the start of a new 9-week, self-paced, certificated teacher training course in Vedic Mathematics. The course is conducted entirely online and will begin on 30th September 2012. Details of the course are available at http://e-gurukul.net/portal/?p=1499 where you can also apply to join. There is no fee for the course.


Lalit Shah, who learnt Vedic Mathematics on our recent teacher training course, was interviewed in the programme "Wide Angle with Ashok Vyas" on 5th August. Congratulations to Lalit for giving such an excellent account of VM. You can see the interview here: http://youtu.be/NPf833cg4M4


As you may recall, the popular Tutorials at www.vedicmaths.org have recently been supplemented with new interactive material.
We have now added the 2nd chapter of 'The Natural Calculator' to the tutorials on the web site and access to this is currently only available to those who have made a donation supporting the development of the website. We have to do this to encourage people to donate, as currently there is no funding to pay for the development of the web site, and we wish to avoid having to place advertising there (we do not wish to submit visitors to propaganda to buy stuff they are not interested in).

We hope to add more chapters from the Natural Calculator book over time. We have a plan to develop a small amount of material that can be promoted to the general public as a mini course. We hope to use this to help popularise Vedic Mathematics. However this will take time as the development of this material is being carried out in my spare time with the support of Kenneth
in testing and reviewing the work, plus I have to have some kind of work-life balance.
We wish to thank those few people who have made a donation supporting the development of the website, as most of the costs and time spent developing the website have been absorbed by ourselves.

Also if you have used the tutorials (especially the updated tutorials), we would be grateful for any feedback you can give us: at .
Please be constructive if you have criticisms of the web site and remember that support for things like languages other than English and mobile web devices are currently ideas for when there is more development time and funding.
Thank You


An article by Dr D.K.R. Babajee titled "Solving Systems of linear equations using the Paravartya rule in Vedic Mathematics" can be viewed at http://www.vedicmaths.org/Journal.asp
This develops Tirthaji's method for solving simultaneous equations, showing how it can be used to solve systems of equations.


The Elementary Teacher's Manual has now been translated into Romanian by Daniela Panait, a teacher in Romania. This is available for free at:


ARTICLE for VM Newsletter 83


By James Glover, July 2012

One of the outstanding features of Vedic Mathematics is that it provides a system for patterns of cognition in both the experience of processing and in the structure of formulae. The aphorisms, or sutras as they are known in Sanskrit, have a wide range of modes of application. They provide general rules for problem solving, specific instructions for calculation and algebraic manipulation, rules of thumb and patterns of working. Additionally, they can have more than one of these modes simultaneously.
Tirtha's book, Vedic Mathematics, is an illustrative volume providing examples of how the sutras work. It never attempts to give a complete and exhaustive description of the extent of any sutra and, considering their multi-faceted characteristics, this is understandable. In several parts of the text he states that topics are to be continued at a later stage and he clearly had the intention of providing further volumes. Unfortunately, he passed away before being able to provide more material.
There are strong indications that the extent of application of the sutras is unknown and yet to be discovered. Not only are there pointers in the text that this is so but penetrative studies have also shown that the sutras have a wider field of context. In his introduction Tirthaji gives the most remarkable comment that the sutras apply to all aspects of mathematics and that there is no part of the subject which does not come within their jurisdiction. Did he have the view that his sutras should somehow be like those of Panini's grammar which cover all aspects of spoken language? We shall never know. The question must be asked as to whether, and in what way, this statement could be true? I believe that pursuing an answer to this reveals a new perspective of what mathematics is, of the nature of the human mind and of the harmony between the human psyche and the world around us.

Questions about scope and jurisdiction
For many years I worked with a group of mathematics teachers researching the Vedic maths system. In addressing the scope of jurisdiction or extent of application of the sutras our studies were guided by two questions. Firstly, when performing mathematical activity what is the sutra operating? Secondly, how do the topics described in the text extend and progress? Of course the first requirement in looking for answers was to become thoroughly familiar with the text and see how Tirthaji uses the sutras as well analysing what they mean. We quickly discovered that practising Vedic mathematics is a delightful activity. It sharpens the mental faculties, brings you into the present moment and has a tremendously satisfying effect. We also discovered that the key to seeing how the sutras work is to look at what the mind is engaged with at any particular moment of mathematical activity. This gives the direction for pursuing answers to the two questions. It appears that they describe or indicate natural mental processes. As an example, consider adding 198 to 247. A simple and natural method is to add 200 and take away 2, giving the answer 445. This process uses the fact that 198 is deficient from 200 by 2. It is an easy enough process although not taught systematically in schools. The sutra indicating this is Yavadunam meaning Deficiency. Taken in its widest context this sutra describes or indicates the mental process involved with any type of problem, mathematical or non-mathematical, which refers to a deficiency.

Patterns of cognition
The relatively small number of sutras leads to the question as to whether the types of mental processes used by the human mind, when performing mathematical activity, is similarly limited. Just as in the musical octave, where a small and limited number of musical and harmonious relationships lead to a vast, perhaps infinite, variety of musical compositions, do the sutras of Vedic maths have this type of potential? If so, then this leads us to look at the possibility of patterns of cognition. Is the mind a haphazard jumble of thoughts sometimes connected through a system of logic and sometimes through intuition or is there a system of processes that operate when performing mathematics?

Giving the same name to different things
Much of mathematics concerns finding patterns. In his book Prelude to Mathematics, W.W.Sawyer states, "...in nature we sometimes find the same pattern again and again in different contexts, as if the supply of suitable patterns were extremely limited". A similar view was expressed by Poincare when he said, "Mathematics is the art of giving the same name to different things". To investigate mathematics in this way is a move towards unity, away from diversity. The sutras of Vedic mathematics powerfully demonstrate the same principle. They are frequently seen to apply in diverse contexts which then become unified. So a single sutra can describe a pattern of mental process, or shape of a formula, which is then repeated in disconnected topics. The topics then become unified, not because the mathematics are logically connected but because they are intuitively connected. The lack of logical connection in terms of hierarchical structure indicates that when dealing with the aphorisms a certain mental looseness or open-mindedness is necessary. This is not a "western" approach, which is largely modeled after Euclid's Elements. In Euclid any proposition is validated by logically stepping from previously determined propositions and eventually back to the postulates, definitions and axioms which he sets out at the beginning. Inasmuch as we understand axiomatic as meaning that a statement is a self-evident principle some of the Vedic mathematical sutras are treated in the same way. The key to understanding this lies in appreciating that the sutras are not abstract statements of logic but are principles experienced at the personal level.

Intuition in mathematics
Personal experience of mathematics shows that some problems are solved intuitively. This can be seen with one sutra which simply states Vilokanainaiva, which is Sanskrit for By simple observation, or By inspection. It deals with the case in which you solve a problem simply by looking. For example, for many people a simple equation like x + 3 = 40, can be solved simply by looking and visualizing the answer. The experience is in seeing the answer straight away and of course it is always then possible to justify the answer through logical steps. But this process is not limited to mathematics. There are many everyday scenarios in which we are presented with a problem and the solution is found just by observation. Intuition plays a key role in this. Solving problems by observation is a common experience but, of course, is an entirely personal affair. What one person can solve merely by looking is not necessarily the same as what another person can do. So it is a common experience but played out in immense variety. Many would argue that this is not mathematics at all but I would say that it is because mathematics includes experience at a personal level.
Mathematics frequently has both a rational and intuitive connectedness. For example, when you see a sequence, such as 5, 8, 14, 23, 35, you may see a pattern but not really be sure if it will continue. A natural response is to go with the flow, so to speak, and then at a later stage see if your hunch is correct and supported by logical argument. Going with the flow is an intuitive process and is as much part of mathematical thinking as the use of logic.

Mental processes not limited to mathematics
As to whether or not a sutra is applicable only to mathematics or has a wider use is not an issue once we accept the flexible nature of their applications. It merely indicates that the sutras tell us something about how the mind naturally responds in various situations. An example of this can be seen in how an architect deals with designing a loft conversion. An experienced architect may have a mental blueprint or template which forms the basis of the design. This is because very many loft conversions are constructed within houses of similar design. So the architect has the template and, looking at the particular task in hand, makes adjustments to the plan. This process is encapsulated by the Paravartya sutra, Transpose and Apply (or Transpose and Adjust). The full wording in Sanskrit is Paravartya Yojayet, which can be translated literally as turning and uniting. A simple mathematical application is found when solving equations. The process can often involve the transposition of a term from one side of an equation to the other side with the inevitable change of sign, such as plus to minus or divides to times. Another, seemingly unrelated application occurs when finding the vector equation of a line perpendicular to a given line. The gradient vector undergoes a transposition and a change of sign. There are very many similar applications of this sutra in mathematics. But it is not limited to mathematics because it deals with a natural mental process and this and similar processes I call Cognitive Patterns.

Patterns of formulae
So far I have described aspects of patterns of processes, cognitive patterns. But the Vedic system also throws up patterns of form. This appears in the shape and structure of mathematical formulae and formulae are statements of cognition. It is not uncommon to find formulae with the same basic shape but which apply to and deal with entirely different contexts. A simple example is the 'rule of three' in which three variables are related in elementary physics and mathematics. There are a number of formulae which express a simple relationship of proportion. Such examples are speed = distance/time, density = mass/volume and pressure = force/area. The identical shape of each of these formulae render them easy to learn. Even though the specific details of each formula may not be related, the formulae are related and are connected through form or shape.

The Gunita Example
This congruence of form or shape is an intrinsic feature of Vedic mathematics. By naming the form of the formula or process the mathematics of apparently disparate topics becomes connected. This can be seen clearly with the case of the Gunita sutra which means Product/Sum. The full wording of the sutra in Sanskrit is Gunita Samuccaya Samuccaya Gunita and the literal translation is Product Sum, Sum Product. The word Gunita usually means product and a Samuccaya is a sum, heap or collection. This sutra is usually taken as The product of the sum equals the sum of the product but it is an overarching descriptor for a number of processes or patterns of working as well as for various mathematical formulae.
A simple numerical application applies to the product of two numbers wherein the product of the digital roots is equal to the digital roots of the product.
The digital root of a number is the summation of its digits continued until there is one digit remaining. For example the digital root of 35 is 3 + 5 = 8. The digital root of 769 is 4, 7 + 6 + 9 = 22 and 2 + 2 = 4. The digital root is also the remainder when the number is divided by 9. 769 ÷ 9 is 85 remainder 4.
With the product of two numbers, such as , the digital roots of the two numbers are 8 and 3. The product of 8 and 3, 24, has digital root 6. The digital root of 735 is also 6.
This formula is also applicable to all cases of addition, subtraction and division. A more useful application, perhaps, is for checking factors and products within algebraic identities. Here the wording the formula is slightly different and this is the example Tirthaji gives in his book.
The product of the sum of the coefficients in the factors equals the sum of the coefficients in the product.
(2x+3)(5x-7) = 10x2 + x - 21

The product of the sum of the coefficients in the factors is 5 x -2 = -10.. The sum of the coefficients in the product is 10 + 1 - 21 = -10.
Below is a list of some of the applications of the Gunita sutra all with a common form:

1. In complex numbers, Argz x arg w = arg(z x w) and mod(zw) = mod(z) x mod(w)
2. In matrices det(AxB) = detA x detB
3. In logarithms, log(AB) = logA + logB
4. De Moivre's theorem, (cosx + isinx)n = cosnx + isinnx
5. In set theory,  and vice versa
6. In calculus,
7. Centre of mass,

Leading to unity
Looking at the mental processes taking place whilst performing mathematics and also at similarities within the shapes or structure of formulae requires taking a step back away from the involvement with results and functionality. I believe that careful questioning about the nature of the Vedic maths sutras leads us in this direction. It leads to an appreciation of an underlying unity which connects the conscious processes of mathematics, the structure of mathematics and the world around us to which it is applied and through which we appreciate its reality.

End of article.

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Editor: Kenneth Williams

Visit the Vedic Mathematics web site at: http://www.vedicmaths.org

6th September 2012


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