VEDIC MATHEMATICS









Introduction
The system of Vedic Mathematics offers a new approach to mathematics and mathematics education. This system which is beautifully coherent and direct was rediscovered from ancient Sanskrit texts between 1911 and 1918 by Bharati Krsna Tirthaji (1884-1960). He found that all problems in pure and applied mathematics can be solved easily with the aid of sixteen Sutras, or word-formulae: for example By One More than the One Before, or Vertically and Crosswise. This may sound incredible but the methods described in his book "Vedic Mathematics" are so original and yet so simple, offering a very different approach to mathematics that is both powerful and fun. The word-formulae give a set of natural principles that help to quickly solve all sorts of mathematical problems. The Vedic methods are direct, beautifully interrelated and flexible. The Vedic system is therefore much more unified and flowing than conventional mathematics. Some of the methods are truly amazing in their efficiency and simplicity. The Vedic system is so easy it is really a system of mental mathematics and this, combined with the coherence and flexibility of the system, encourages the development and use of intuition and creativity.

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History of Vedic Mathematics

The ancient system of Vedic Mathematics was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). The word 'Veda' means 'knowledge' and also refers to a set of ancient Indian texts written in Sanskrit. The date these texts were written is unknown but the content of the Vedas was passed on by an oral tradition long before writing was invented. The Vedas are said to cover every aspect and area of knowledge: grammar, architecture, ethics, astronomy etc. are all covered. At the beginning of the twentieth century, when there was a great interest in the Sanskrit texts in Europe, Bharati Krsna tells us some scholars ridiculed certain texts which were headed 'Ganita Sutras'- which means mathematics. They could find no mathematics in the translation and dismissed the texts as rubbish. Bharati Krsna, who was himself a scholar of Sanskrit, Mathematics, History and Philosophy, studied these texts and after lengthy and careful investigation was able to reconstruct the mathematics of the Vedas. 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. The credibility of this extraordinary claim that all of mathematics, pure and applied, is governed by sixteen formulas
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Mental Mathematics

We all make mental calculations from time to time, though we may not always be aware of it. In deciding at exactly what moment and speed to venture across a busy road, for example, our mind judges continuously the positions and speeds of several vehicles and accurately finds the required gap in which to move forward. If our mind can make such complex judgements as this it is certainly able to manipulate a few figures. It is the cumbersome calculating devices we have probably been taught- which require pencil and paper or calculator to work out because of their difficulty- and a lack of encouragement for mental calculation which have prevented us from becoming efficient mental calculators. This need not be so- as Vedic Mathematics shows, mental calculation is easy and to be preferred to pencil and paper or calculator, and has many advantages over these calculating methods. Most people would probably agree that mathematics holds a special position among subjects of study: that it possess qualities of absolute certainty and precision which cannot be attributed to any other subject. On the other hand however mathematics is seen as difficult and remote by most people- the same people who are also aware of its special absolute qualities. This situation has come about because mathematics education has not been effective enough in bringing out the real nature of mathematics. As young students we glimpse the beauty of mathematics but this is usually a passing phenomenon. Though mathematics has applications at many levels it is primarily a mental subject. This being so it is likely that lack of mental calculation is partly responsible for the situation described above, and that a system of mental mathematics could provide students with a lasting link with the realm of mathematics and also engender a deeper understanding of the structure and processes of mathematics, as well as helping to develop other important personal qualities. The following points outline the benefits available from a mental approach to mathematics.
  1. Mental calculation sharpens the mind and increases mental agility and intelligence. This will be evident to anyone who has practised or taught mental calculation or who has seen its effects.
  2. It enhances the precision of thought. Numbers and other mathematical objects are unbiased, giving only one correct answer to which everyone will agree- there is never a contradiction. This absolute precision is unique to mathematics, so dealing intimately with numbers as we do in mental calculation we cultivate fine and careful thinking.
  3. Mental calculation leads naturally to the search for, and discernment of, constancy and law, which are very necessary attributes in a swiftly changing world.
  4. Our mind has the ability to retain several ideas at once so that they can be compared, combined and so on. This facility is enhanced by mental calculation as we practise holding the sum in the mind whilst operating with some of the figures.
  5. Mental calculation improves the memory. Memory depreciates if it is not exercised. Short term, medium term and long term memory are all stimulated by mental calculation.
  6. Because numbers are absolutely dependable and reliable, calculation promotes confidence. In particular, mental calculation creates confidence in oneself and in ones capabilities. To solve a problem, perhaps a difficult one, by mere mental arithmetic without having to rely on some artificial aid is a source of great satisfaction and encouragement.
  7. Mental calculation is a delight to the mind: the intrinsic qualities, relationships and beauty of numbers and the way they create new numbers out of themselves is a source of great enjoyment.
  8. Through mental calculation one becomes familiar with numbers and appreciates their various properties. This leads to a real understanding of number.
  9. In calculating mentally the subtle properties of numbers and their relationships are appreciated much more readily than if the calculation was written down and thereby fixed. Thus mental calculation leads naturally to innovation and to the invention of new methods, thereby developing the student's natural creativity.
  10. Practical uses of mental calculation are many, since we all need to make quick, on the spot, calculations from time to time.
Thus we see that mental calculation has so many advantages and really brings mathematics to life as well as providing motivation and strengthening and enlivening the mind. This is because numbers are mental concepts, they do not exist on paper. Our unhindered mind operates very fast and has a variety of operational properties. With proper training we can use these properties of the mind to our advantage. This is not to say that pencil and paper or calculating instruments are to be totally avoided in mathematics: they certainly have their place, but mental calculation should, it is suggested, be the primary method of calculation.
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Vedic Mathematics Teaching Methodology

Learning mathematics should be a delightful experience for all children and they should all succeed in it. The Cosmic Computer Course offers a complete system of mental mathematics which can be taught in a holistic way. The straightforward and beautifully interrelated Vedic methods mean that mathematics can be done mentally and this, together with the many methods of solution which the Vedic system offers, encourages flexibility and innovation. This in turn leads to the development of creativity and intuition. The Vedic system does not insist on a purely analytic approach as many modern teaching methods do. This makes a big difference to the attitude which children have towards mathematics. With its direct, easy and integrated approach this mental system (the methods can also be written down) brings out the naturally coherent and unified structure of mathematics. The Cosmic Computer Course, for example, uses a Unified Field Chart which shows the whole subject at a glance and how the various parts are related and structured.

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Vedic Mathematics Sutras*


Sutras SubSutras or Corollaries
1. Ekadhikena Purvena
(By One More than the One Before) 		Proportionately
2.Nikhilam Navatascharamam Dastah
(All from 9 and the Last from 10) 		The Remainder Remains Constant
3. Urdhwa-tiryagbhyam
(Vertically and Crosswise) 		The First by the First and the Last by the Last
4. Paravartya Yojayet
(Transpose and Apply) 		For 7 the Multiplicand is 143
5. Sunyam Samyasamuchchaye
(If the Samuccaya is the Same it is Zero) 		By Osculation
6. (Anurupye) Sunyamanyat
(If One is in Ratio the Other is Zero) 		Lessen by the Deficiency
7. Sankalana-vyavkalanabhyam
(By Addition and by Subtraction) 		Whatever the Deficiency lessen by that amount and set up the Square of the Deficiency
8. Puranpuranabhyam
(By the Completion or Non-Completion) 		Last Totalling 10
9. Chalana-Kalanabhyam
(Differential Calculus) 		Only the Last Terms
10. Yavdunam
(By the Deficiency) 		The Sum of the Products
11. Vyastisamastih
(Specific and General) 		By Alternate Elimination and Retention
12. Sesanyankena Charmena
(The Remainders by the Last Digit) 		By Mere Observation
13. Sopantyadyaymantyam
(The Ultimate and Twice the Penultimate) 		The Product of the Sum is the Sum of the Products
14. Ekanyunena Purvena
(By One Less than the One Before) 		On the Flag**
15. Gunitasamuchachayah
(The Product of the Sum) 		- - -
16. Gunaksamuchchayah
(All the Multipliers) 		- - -
* This list is taken from Vedic Mathematics, which includes a full list of the sixteen Sutras in Sanskrit, but in some cases a translation of the Sanskrit is not given in the text and comes from elsewhere. ** This formula is not in the list given in Vedic Mathematics, but is referred to in the text.

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An Example
Urdhwa-tiryagbhyam Sutra This sutra says -"Vertically and Crosswise". That's all to multiply two numbers! Till now, you were multiplying like this: Question : Multiply 432 by 617.
Answer :
        432
    X 617
-------------
    3024
    432
2592
==========
266544
==========

More the number of digits in the numbers, more lines and time you consume. No more! Using the Sutra "Vertically and Crosswise", you have
Step 1 : (mentally, don't write on notebook) vertically (last digits) : 2x7=14; write 4 carry 1
Step 2 (mentally) : crosswise (last two digits) : 3x7 +2x1 = 23 +carry 1 = 24; write 4 carry 2
Step 3 : vertically and crosswise (three digits) : 4x7 + 3x1 +2x6 = 43 +carry 2 = 45; write 5 carry 4
Step 4 : (move left; first two digits) : 4x1 +3x6 = 22 +carry 4 = 26; write 6 carry 2
Step 5 : (move left; first digit of each number) : 4x6 = 24 +carry 2 = 26. End.
Write answer : 266544

This is how it appears on notebook :        432
    X 617
-------------
266544
-------------
No matter how big the numbers are, you will need to write only the final answer. All other steps are easily carried out mentally. If the two numbers have different number of digits, write smaller number below the other and pad it on left side with zeros.


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