Vedic Mathematics

kirhmuru

What is VedicMathematics?

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

Perhaps the most striking feature of the Vedic system is itscoherence. Instead of a hotch-potch of unrelated techniques the whole system isbeautifully interrelated and unified: the general multiplication method, forexample, is easily reversed to allow one-line divisions and the simple squaringmethod can be reversed to give one-line square roots. And these are all easilyunderstood. This unifying quality is very satisfying, it makes mathematics easyand enjoyable and encourages innovation.

In the Vedic system 'difficult' problems or huge sums canoften be solved immediately by the Vedic method. These striking and beautifulmethods are just a part of a complete system of mathematics which is far moresystematic than the modern 'system'. Vedic Mathematics manifests the coherentand unified structure of mathematics and the methods are complementary, directand easy.

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

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

But the real beauty and effectiveness of Vedic Mathematicscannot be fully appreciated without actually practising the system. One canthen see that it is perhaps the most refined and efficient mathematical systempossible.

The Vedic Mathematics Sutras

This list of sutras is taken from the book VedicMathematics, which includes a full list of the sixteen Sutras in Sanskrit, butin some cases a translation of the Sanskrit is not given in the text and comesfrom elsewhere.

This formula 'On the Flag' is not in the list given in VedicMathematics, but is referred to in the text.

The Main Sutras

By one more than the one before.

All from 9 and the last from 10.

Vertically and Cross-wise

Transpose and Apply

If the Samuccaya is the Same it is Zero

If One is in Ratio the Other is Zero

By Addition and by Subtraction

By the Completion or Non-Completion

Differential Calculus

By the Deficiency

Specific and General

The Remainders by the Last Digit

The Ultimate and Twice the Penultimate

By One Less than the One Before

The Product of the Sum

All the Multipliers

The Sub Sutras

Proportionately

The Remainder Remains Constant

The First by the First and the Last by the Last

For 7 the Multiplicand is 143

By Osculation

Lessen by the Deficiency

Whatever the Deficiency lessen by that amount and   set up the Square of the Deficiency

Last Totalling 10

Only the Last Terms

The Sum of the Products

By Alternative Elimination and Retention

By Mere Observation

The Product of the Sum is the Sum of the Products

On the Flag

Try a Sutra

Mark Gaskell introduces an alternative
system of calculation based on Vedic philosophy

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

Vedic maths comes from the Vedic tradition of India. TheVedas are the most ancient record of human experience and knowledge, passeddown 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 countrycredited with introducing our current number system and the invention ofperhaps the most important mathematical symbol, 0, may have more to offer inthe field of maths.

The remarkable system of Vedic maths was rediscovered fromancient Sanskrit texts early last century. The system is based on 16 sutras oraphorisms, such as: "by one more than the one before" and "allfrom nine and the last from 10". These describe natural processes in themind 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 "allfrom nine and the last from 10". Each figure in 564 is subtracted fromnine and the last figure is subtracted from 10, yielding 436.

This can easily be extended to solve problems such as 3,000minus 467. We simply reduce the first figure in 3,000 by one and then apply thesutra, 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 toraces at home using many of the Vedic techniques - and won. This particularmethod can also be expanded into a general method, dealing with any subtractionsum.

The sutra "vertically and crosswise" has manyuses. One very useful application is helping children who are having troublewith their tables above 5x5. For example 7x8. 7 is 3 below the base of 10, and8 is 2 below the base of 10.

The whole approach of Vedic maths is suitable for slowlearners, as it is so simple and easy to use.

The sutra "vertically and crosswise" is often usedin long multiplication. Suppose we wish to multiply
32 by 44. We multiply vertically 2x4=8.
Then we multiply crosswise and add the two results: 3x4+4x2=20, so put down 0and carry 2.
Finally we multiply vertically 3x4=12 and add the carried 2 =14. Result: 1,408.

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

All the techniques produce one-line answers and most can bedealt with mentally, so calculators are not used until Year 10. The methods areeither "special", in that they only apply under certain conditions,or general. This encourages flexibility and innovation on the part of thestudents.

Multiplication can also be carried out starting from theleft, which can be better because we write and pronounce numbers from left toright. Here is an example of doing this in a special method for longmultiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by92. 96 is 4 below the base and 92 is 8 below.

We can cross-subtract either way: 96-8=88 or 92-4=88. Thisis the first part of the answer and multiplying the "differences"vertically 4x8=32 gives the second part of the answer.

This works equally well for numbers above the base:105x111=11,655. Here we add the differences. For 205x211=43,255, we double thefirst part of the answer, because 200 is 2x100.

We regularly practise the methods by having a mental test atthe beginning of each lesson. With the introduction of a non-calculator paperat GCSE, Vedic maths offers methods that are simpler, more efficient and morereadily acquired than conventional methods.

There is a unity and coherence in the system which is notfound in conventional maths. It brings out the beauty and patterns in numbersand the world around us. The techniques are so simple they can be used whenconventional methods would be cumbersome.

When the children learn about Pythagoras's theorem in Year 9we do not use a calculator; squaring numbers and finding square roots (toseveral significant figures) is all performed with relative ease and reinforcesthe methods that they would have recently learned.