## Vedic mathematics

*Vedic Mathematics* is the name given to the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to his research all of mathematics is based on sixteen Sutras or word-formulae. For example, ‘Vertically and Crosswise` is one of these Sutras. These formulae describe the way the mind naturally works and are therefore a great help in directing the student to the appropriate method of solution.

Perhaps the most striking feature of the Vedic system is its coherence. Instead of a hotch-potch of unrelated techniques the whole system is beautifully interrelated and unified: the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots.

And these are all easily understood. This unifying quality is very satisfying, it makes mathematics easy and enjoyable and encourages innovation.

In the **Vedic system** ‘difficult’ problems or huge sums can often be solved immediately by the Vedic method. These striking and beautiful methods are just a part of a complete system of mathematics which is far more systematic than the modern ‘system’. Vedic **Mathematics** manifests the coherent and unified structure of mathematics and the methods are complementary, direct and easy.

The simplicity of *Vedic Mathematics* means that calculations can be carried out mentally (though the methods can also be written down). There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘correct’ method. This leads to more creative, interested and intelligent pupils.

Interest in the **Vedic system** is growing in education where mathematics teachers are looking for something better and finding the Vedic system is the answer. Research is being carried out in many areas including the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the *Vedic* Sutras in geometry, calculus, computing etc.

At the Maharishi School in Lancashire we have developed a course on Vedic mathematics for key stage 3 that covers the national curriculum. The results have been impressive: maths lessons are much livelier and more fun, the children enjoy their work more and expectations of what is possible are very much higher. Academic performance has also greatly improved: the first class to complete the course managed to pass their GCSE a year early and all obtained an A grade.

Vedic maths comes from the Vedic tradition of India. The Vedas are the most ancient record of human experience and knowledge, passed down orally for generations and written down about 5,000 years ago. Medicine, architecture, astronomy and many other branches of knowledge, including maths, are dealt with in the texts. Perhaps it is not surprising that the country credited with introducing our current number system and the invention of perhaps the most important mathematical symbol, 0, may have more to offer in the field of maths.

The remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts early last century. The system is based on 16 sutras or aphorisms, such as: "by one more than the one before" and "all from nine and the last from 10". These describe natural processes in the mind and ways of solving a whole range of mathematical problems. For example, if we wished to subtract 564 from 1,000 we simply apply the sutra "all from nine and the last from 10". Each figure in 564 is subtracted from nine and the last figure is subtracted from 10, yielding 436.

This can easily be extended to solve problems such as 3,000 minus 467. We simply reduce the first figure in 3,000 by one and then apply the sutra, to get the answer 2,533. We have had a lot of fun with this type of sum, particularly when dealing with money examples, such as £10 take away £2. 36. Many of the children have described how they have challenged their parents to races at home using many of the Vedic techniques – and won. This particular method can also be expanded into a general method, dealing with any subtraction sum.

We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. By the end of Year 8, I would expect all students to be able to do a "3 by 2" long multiplication in their heads. This gives enormous confidence to the pupils who lose their fear of numbers and go on to tackle harder maths in a more open manner.

This is an advanced book of sixteen chapters on one Sutra ranging from elementary multiplication etc. to the solution of non-linear partial *differential equations*. It deals with (i) calculation of common functions and their series expansions, and (ii) the solution of equations, starting with simultaneous equations and moving on to algebraic, transcendental and differential equations.

This book shows applications of Pythagorean Triples (like 3,4,5). A simple, elegant system for combining these triples gives unexpected and powerful general methods for solving a wide range of mathematical problems, with far less effort than conventional methods use. The easy text fully explains this method which has applications in trigonometry (you do not need any of those complicated formulae), coordinate geometry (2 and 3 dimensions) transformations (2 and 3 dimensions), simple harmonic motion, astronomy etc., etc.

National Curriculum for England and Wales. It consists of three books each of which has a Teacher’s Guide and an Answer Book. Much of the material in Book 1 is suitable for children as young as eight and this is developed from here to topics such as Pythagoras’ Theorem and Quadratic Equations in Book 3. The Teacher’s Guide contains a Summary of the Book, a Unified Field Chart (showing the whole subject of mathematics and how each of the parts are related), hundreds of Mental Tests (these revise previous work, introduce new ideas and are carefully correlated with the rest of the course), Extension Sheets (about 16 per book) for fast pupils or for extra classwork, Revision Tests, Games, Worksheets etc.

This book demonstrates the kind of system that could have existed before literacy was widespread and takes us from first principles to theorems on elementary properties of circles. It presents direct, immediate and *easily understood* proofs. These are based on only one assumption (that magnitudes are unchanged by motion) and three additional provisions (a means of drawing figures, the language used and the ability to recognise valid reasoning). It includes discussion on the relevant philosophy of mathematics and is written both for mathematicians and for a wider audience.

"Entertaining, engaging and eminently ‘doable’, Williams’ pocket volume reveals many fascinating and useful applications of the ancient Eastern system of Vedic Maths. Tackling many number operations encountered between First and Sixth class, Fun with Figures offers several speedy and simple means of solving or double-checking class activities. Focusing throughout on skills associated with mental mathematics, the author wisely places them within practical life-related contexts." "Compact, cheerful and liberally interspersed with amusing anecdotes and aphorisms from the world of maths, Williams’ book will help neutralise the ‘menace’ sometimes associated with maths.

It’s practicality, clear methodology, examples, supplementary exercises and answers may particularly benefit and empower the weaker student." "Certainly a valuable investment for parents and teachers of children aged 7 to 12." Reviewed by Gerard Lennon, Principal, Ardpatrick NS, Co Limerick. The Tutorial below is based on material from this book ‘Fun with Figures’