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Mar 18, I may try to edit this book or write a new book in future, reflecting In solar vaia Cambridge International AS and A Level Mathematics Pure. Vedic Mathematics(ORIGNAL BOOK) - Free ebook download as PDF File .pdf), Text File These Slokas have been edited and are being translated into Hindi. Free Vedic Mathematics Books from the Vedic Mathematics Academy. The free books available below are all simple Adobe Acrobat PDF documents. We only ask that you do not upload them HINDI version (click below). GERMAN version .

Similar is the case with regard to the Vedangas i. So, one more than that is 2. Multiplication by two-digit numbers from right to left 8. The second remainder digit O is prefixed to the second quotient digit 5 to form 5 as the third dividend digit. Sankalana-vyavakalanabhyam also a corollary4 8. Only the answer is writ ten automatically down by Vrdhwa Tiryak Sutra forwards or back wards.

And the answer isas we shall demonstrate later onthat, in either method, if and as soon as we reach the difference between the numerator and the denominator i.

Details o f these principles and processes and other allied matters, we shall go into, in due course, at the proper place. In the meantime, the student will find it both interesting and pro fitable to know this rule and turn it into good account from time to time as the occasion may demand or justify.

Second Example: Case 2? By the Current method: The procedures are explained on the next page. Here too, the last digit of the denominator is 9 ; but the penultimate one is 2 ; and one more than that means 3.

So, 3 is our commoni. Our modus-ojperandi-chart herein reads as follows:. And the chart reads as follows: Here too, we find that the two halves are all complements of each other from 9.

So, this fits in too. Our multiplier or divisor as the case may be is now 5 i. So, A. By multi plication leftward from the right by 5, we have. At this point, in all the 3 processep, we find that we have reached 48 the difference between the numerator and the denominator. And yet, the remark able thing is that the current system takes 42 steps of elaborate and cumbrous dividing with a series of multiplications and subtractions and with the risk of the failure of one or more trial digits of the Quotient and so on while a single, straight and continuous processof multiplication or division by a single multiplier or divisor is quite enough in the Vedic method.

The complements from nine are also there. But this is not all. Our readers will doubtless be surprised to learnbut it is an actual factthat there are, in the Vedic system, still simpler and easier methods by which, without doing even the infinitely easy work explained hereinabove, we can put down digit after digit of the answer, right from the very start to the very end,.

We shall hold them over to be dealt with, at their own appropriate place, in due course, in a later chapter. Sutra Pass we now on to a systematic exposition of certain salient, interesting, important and necessary formulae of the utmost value and utility in connection with arithmetical calculations etc.

At this point, it will not be out of place for us to repeat that there is a GENERAL formula which is simple and easy and can be applied to all cases; but there ure also SPECIAL casesor rather, types of caseswhich are simpler still and which are, therefore, here first dealt with.

We may also draw the attention of all students and teachers of mathematics to the well-known and universal fact that, in respect of arithmetical multiplications, the usual present-day procedure everywhere in schools, colleges and universities is for the children in the primary classes to be told to cram upor get by heart the multiplication-tables up to 16 times 16, 20x20 and so on. But, according to the Vedic system, the multiplication tables are not really required above 5 x 5.

And a school-going pupil who knows simple addition and subtraction of single-digit numbers and the multiplication-table up to five times five, can improvise all the necessary multiplication-tables for himself at any time and can himself do all the requisite multiplications involving bigger multiplicands and multipliers, with the aid of the relevant simple Vedic formulae which enable him to get at the required products, very easily and speedilynay, practically, imme diately.

The Sutras are very short; but, once one understands them and the modus operandi inculcated therein for their practical application, the whole thing becomes a sort of childrens play and ceases to be a problem. Let us: The Sutra: We shall give a detailed explanation, presently, of the meaning and applications of this cryptical-sounding formula.

But just now, we state and explain the actual procedure, step by step. Suppose we have to multiply 9 by 7. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. And put i. And -you find that you have got 93 i. OR d Cross-subtract in the converse way i. And you get 6 again as the left-hand side portion of the required answer. This availablity of the same result in several easy ways is a very common feature of the Vedic system and is of great advantage and help to the student as it enables him to test and verify the correctness of his answer, step by step.

The product is 3. And this is the righthand-side portion of the answer. In fact, old historical traditions describe this cross-subtraction process as having been res ponsible for the acceptance of the x mark as the sign of multiplication. This proves the correctness of the formula.

A slight difference, however, is noticeable when the vertical multiplication of the deficit digits for obtaining the right-hand-side portion of the answer yields a product con sisting of more than one digit. For example, if and when we have to multiply 6 by 7, and write it down as usual: This difficulty, however, is easily surmounted with the usual multiplicational rule that the surplus portion on the left should always be carried over to the left.

Therefore, in the present case, we keep the 2 of the 12 on the right hand side and cairy the 1 over to the left and change the 3 into 4. We thus obtain 42 as the actual product of 7 and 6.

A similar procedure will naturally be required in respect of other similar multiplications: This rule of multiplication by means of cross-subtraction for the left-hand portion and of vertical multiplication for the right-hand half , being an actual application of the absolute algebraic identity: Thus, as regards numbers of two digits each, we may notice the following specimen examples: The base now required is Note 1: In all these cases, note that both the cross-sub tractions always give the same remainder for the left-hand-side portion of theanswer.

Note 2: Here too, note that the vertical multiplication for the right-hand side portion of the product may, in some cases, yield a more-than-two-digit product; but, with as our base, we can have only two digits on the right-hand side.

We should therefore adopt the same method as before i. Thus 88 12 88 12 91 9 98 2. The rule is that all the other digits of the given original numbers are to be subtracted from 9 but the last i.

Thus, if 63 be the given number, the deficit from the base ' is 37; and so on. This process helps us in the work of ready on-sight subtraction and enables us to pu,t the deficiency down immediately.

A new point has now to be taken into consideration i. What is the remedy herefore? Well, this contingency too has been provided for. And the remedy isas in the case of decimal multiplicationsmerely the filling up of all such vacancies with Zeroes. With these 3 procedures for meeting the 3 possible contingencies in question i.

Y e s ; but, in all these cases, the multiplicand and the multiplier are just a little below a certain power of ten taken as the base. What about numbers which are above it? And the answer is that the same procedure will hold good there too, except that, instead of cross-subtracting, we shall have to cross-add. And all the other rules regarding digit-surplus, digit-deficit etc.

In passing, the algebraical principle involved may be explained as follows: Y e s ; but if one of the numbers is above and the other is below a power of 10 the base taken , what then?

The answer is that the plus and the minus will, on multi plication, behave as they always do and produce a nunus-product and that the right-hand portion obtained by vertical multi.

A vinculum may be used for making this clear. Note that even the subtraction of the vinculumportion may be easily done with the aid of the Sutra under discussion i. Multiples and sub-multiples: Yes ; but, in all these cases, we find both the multiplicand and the multiplier, or at least one of them, very near the base taken in each case ; and this gives us a small multiplier and thus renders the multiplication very easy.

What about the multiplication of two numbers, neither of which is near a con venient base? The needed solution for this purpose is furnished by a small Upasutra or sub-formula which is so-called because of its practically axiomatic character. This sub-sutra consists of only one word Anurupyena which simply means Proportionately. In actual application, it connotes that, inwall cases where there is a rational ratio-wise relationship, the ratio should be taken into account and should lead to a proportionate multiplication or division as the case may be.

A concrete illustration will make the modus operandi clear. Suppose we have to multiply 41 by Both these numbers are so far away from the base that by our adopting that as our actual base, we shall get 59 and 59 as the deficiency from the base. And thus the consequent vertical multiplication of 59 by 59 would prove too cumbrous a process to be per missible under the Vedic system and will be positively inad missible.

We therefore, accept merely as a theoretical base and take sub-multiple 50 which is conveniently near 41 and 41 as our working basis, work the sum up accordingly and then do the proportionate multiplication or division, for getting the correct answer. Our chart will then take this shape:.

OR, secondly, instead of taking as our theoretical base and its half The product of 41 and 41 is thus found to be the same as we got by the first method. OR, thirdly, instead of taking or 10 as our theoretical base and 50 a sub-multiple or multiple thereof as our working base, we may take 10 and 40 as the bases respectively and work at the multiplication as shown on the margin here.

As regards the principle underlying and the reason behind the vertical-multiplication operation on the right-hand-side remaining unaffected and not having to be multiplied or divided proportionately a very simple illustration will suffice to make this clear. We may write down our table of answers as follows: And this is why it is rightly called the remainder fiar stanr: Here 47 being odd, its division by 2 gives us a fractional quotient 23j and that, just as half a rupee or half a pound or half a dollar is taken over to the right-hand-side as 8 annas or 10 shillings or 50 cents , so the half here in the 23J is taken over to the righthand-side as In the above two cases, the J on the left hand side is carried over to the right hand as Most of these examples are quite easy, in fact much easier-by the 3;s fo?

They have been included here, merely for demonstrating that they too can be solved by the Nikhilam Sutra expounded in this chapter. The First Corollary: The first corollary naturally arising out of the Nikhilarh Sutra reads as follows: This evidently deals with the squaring of numbers.

A few elementary examples will suffice to make its meaning and application clear: Suppose we have to find the square of 9. The following will be the successive stages in our mental working: Now, let us take up the case of As 8 is 2 less than 10, we lessen it still further by 2 and get 82 i. We work exactly as before ; but, instead of reducing still further by the deficit, we increase the number still further by the surplus and say: And then, extending the same rule to numbers of two or more digits, we proceed further and say: Thus, if 97 has.

In the present case, if our b be 3, a-f-b will become and ab will become This proves the Corollary. This corollary is specially suited for the squaring of such numbers.

Seemingly more complex and diffi cult cases will be taken up in the next chapter relating to the Vrdhva-Tiryak Siitra ; and still most difficult will be explained in a still later chapter dealing with the squaring, cubing etc. The Second Corollary. The second corollary is applicable only to a special case under the first corollary i. Its wording is exactly the same as that of the Sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents i.

The Sutra now takes a totally different meaning altogether and, in fact, relates to a wholly different set-up and context altogether. So, one more than that is 2. The Algebraical Explanation is quite simple and follows straight-away from the Nikhilam Sutra and still more so from the Vrdhva-Tiryak formula to be explained in the next chapter q. A sub-corollary to this Corollary relating to the squaring of numbers ending in 5 reads: AntyayorDaiake'pi and tells us that the above rule is applicable not only to the squaring of a number ending in 5 but also to the multiplication of two numbers whose last digits together total 10 and whose previous part is exactly the same.

We can proceed further on the same lines and say: At this point, however, it may just be pointed out that the above rule is capable of further application and come in handy, for the multiplication of numbers whose last digits in sets of 2,3 and so on together total , etc. Note the added zero at the end of the left-hand-side of the answer. The Third Corollary: Then comes a Third Corollary to the Nikhilam Sutra, which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc.

The wording of the subsutra corollary Ekanyunena Purvena sounds as. It actually is ; and it relates to and provides fot multiplications wherein the multiplier-digits consist entirely of nines. It comes up under three different headings as follows: The First case: The annexed table of products produced by the single digit multiplier 9 gives us the necessary clue to an under standing of the Sutra: We observe that the left-hand-side is invariably one less than the multiplicand and that the right-side-digit is merely the complement of the left-bandside digit from 9.

And this tells us what to do to get both the portions of the product. As regards multiplicands and multipliers of 2 digits each, we have the following table of products: And this table shows that the rule holds good here too.

And by similar continued observation, we find that it is uniformly applicable to all cases, where the multiplicand aiid the multiplier consist of the same number of digits.

In fact, it is a simple application of the Nikhilam Sutra and is bound to apply. We are thus enabled to apply the rule to all such cases and say, for example: Such multiplications involving multipliers of this special type come up in advanced astronomy e t c ; and this sub formula Ekanyunena Purvena is of immense utility therein. The Second Case: The second case falling under this category is one wherein the multiplicand consists of a smaller number of digits than the multiplier.

This, however, is easy enough to handle ; and all that is necessary is to fill the blank on the left in with the required number of zeroes and proceed exactly as before and then leave the zeroes out.

Thus 7 79 79 99 99 ? To be omitted during a first reading The third case coming under this heading is one where the multiplier contains a smaller number of digits than the multiplicand. Careful observation and study of the relevant table of products gives us the necessary clue and helps us to understand the correct application of the Sutra to this kind of examples.

We note here that, in the first column of products where the multiplicand starts with 1 as its first digit the left-handside part of the product is uniformly 2 less than the multi plicand ; that, in the second column where the multiplicand begins with 2, the left-hand side part of the product is exactly 3 less ; and that, in the third column of miscellaneous firstdigits the difference between the multiplicand and the lefthand portion of the product is invariable one more than the excess portion to the extreme left of the dividend.

The procedure applicable in this case is therefore evidently as follows: This gives us the left-hand-side portion of the product. OR take the Ekanyuna and subtract therefrom the previous i. This will give you the righthand-side of the product. The following examples will make the process clear: The formula itself is very short and terse, consisting of only one compound word and means vertically and cross wise.

The applications of this brief and terse Sutra are manifolci as will be seen again and again, later on. First we take it up in its most elementary application namely, to Multi plication in general. A simple example will suffice to clarify the modus operandi thereof.

Suppose we have to multiply 12 by When one of the results contains more than 1 digit, the right-hand-most digit thereof is to be put down there and the preceding i. The digits carried over may be shown in the working as illustrated below i 15 15 12 2 25 25 40 3 32 32 1 4 35 35 32 5 37 33 32 6 49 49 78 The Algebraical principle involved is as follows: In other words, the first term i.

And, as all arithmetical numbers are merely algebraic expres6. We thus follow a process of ascent and of descent going forward with the digits on the upper row and coming rearward with the digits on the lower row.

If and when this principle of ordinary Algebraic multiplication is properly understood and carefully applied to the Arithmetical multiplication on hand where x stands for 10 , the Urdhva Tiryak Sutra may be deemed to have been successfully mastered in actual practice.

## Vedic Mathematics(ORIGNAL BOOK)

It need hardly be mentioned that we can carry out this tJrdhva-Tiryak process of multiplication from left to right or from right to left as we prefer.

All the diffe-. Owing to their relevancy to this context, a few Algebraic examples of the Vrdhva-Tiryak type are being given. I f and when a power of x is absent, it should be given a zero coefficient; and the work should be proceeded with exactly as before. It may, in general, be stated that multiplications by digits higher than 5 may some times be facilitated by the use of the vinculum.

The following example will illustrate this: But the vinculum process is Miscellaneous Examples: There being so many methods of multiplication one of them the Urdhva-Tiryak one being perfectly general and therefore applicable to all cases and the others the Nikhilarh one, the Yavadunam etc. The digits being small, the general formula is always best. Square Measure, Cubic Measure Etc.

This is not a separate subject, all by itself. But it is often of practical interest and importance, even to lay people and deserves oar attention on that score. We therefore deal with it briefly.

Areas of Rectangles. According to the conventional method, we put both these measurements into uniform shape either as inches or as vulgar fractions of feetpreferably the latter and say: Volumes o f Pandlelepipeds: We can extend the same method to sums relating to 3 dimensions also.

Suppose we have to find the volume of a parallelepiped whose dimensions are 3' 7", 5' 10" and 7' 2". By the customary method, we will say: But, by the Vedic process, we have. Thus, even in these small computations, the customary method seems to have a natural or ingrained bias in favour of needlessly big multiplications, divisions, vulgar fractions etc.

The Vedic Sutras, however, help us to avoid these and make the work a pleasure and not an infliction. The same procedure under the Urdhva-Tiryak Sutra is readily applicable to most questions which come under the headings Simple Practice and Compound Practice , wherein ALIQUOT parts are taken and many steps of working are resorted to under the current system but wherein according to the Vedic method, all of it is mental Arithmetic, For example, suppose the question is: In a certain investment, each rupee invested brings Rupees two and five annas to the investor.

How much will an outlay of Rs. By Means of Aliquot Parts. Total for Rs. Second Current Method. By Simple Proportion Rs. V On Re 1, the yield is Rs. On Rs. By the Vedic one-line method: Total 23 2Tff 9 ii By the current Proportion method. Questions relating to paving, carpeting, ornamenting etc.

For example, suppose the question is: At the rate of 7 annas 9 pies per foot, what will be the ost for 8 yards 1 foot 3 inches? Having dealt with. Multiplication at fairly considerable length, we now go on to Division; and there we start with the Nikhihm method which is a special one. Suppose we have to divide a number of dividends of two digits each successively by the same Divisor 9 we make a chart therefor as follows: Let us first split each dividend into a left-hand part for the Quotient and a right-hand part for the Remainder and divide them by a vertical line.

In all these particular cases, we observe that the first digit of the Dividend becomes the Quotient and the sum of the two digits becomes the Remainder. This means that we can mechanically take the first digit down for the Quotientcolumn and that, by adding the quotient to the second digit, we can get the Remainder.

Next, we take as Dividends, another set of bigger num bers of 3 digits each and make a chart of them as follows: And then, by extending this procedure to still bigger numbers consisting of still more digits , we are able to get the quotient and remainder correctly. And, thereafter, we take a few more cases as follows: As this is not permissible, we re-divide the Remainder by 9, carry the quotient over to the Quotient column and retain the final Remainder in the Remainder cloumn, as follows: We next take up the next lower numbers 8, 7 etc.

Here we observe that, on taking the first digit of the Dividend down mechanically, we do not get the Remainder by adding that digit of the quotient to the second digit of the dividend but have to add to it twice, thrice or 4 times the quotient digit already taken down. In other words, we have to multiply the quotient-digit by 2 in the case of 8, by 3 in the case of 7, by 4 in the case of 6 and so on.

And this again means that we have to multiply the quotient-digit by the Divisors comple ment from And this suggests that the Nikhilam rule about the sub traction of all from 9 and of the last from 10 is at work ; and, to make sure of it, we try with bigger divisions, as follows: The reason therefor is as follows: A single sample example will suffice to prove this: In this case, the product of 8 and 2 is written down in its proper place, as 16 with no carrying over to the left and so on.

Thus, in our division process by the Nikhilam formula , we perform only small single-digit multiplications; we do no subtraction and no division at all; and yet we readily obtain the required quotient and the required Remainder. In fact, we have accomplished our division-work in full, without actually doing any division at a ll!

Just at present in this chapter , we deal only with big divisors and explain how simple and easy such difficult multiplications can be made with the aid of the Nikhilam SUtra. And herein, we take up a few more illustrative examples relating to the cases already referred to wherein the Remainder exceeds the Divisor and explain the process, by which this difficulty can be easily surmounted by further application of the same Nikhilam method 25 88 1 12 98 Thus, we say:.

This double process can be combined into one as follows: Thus, even the whole lengthy operation of division of by involves no division and no subtraction and consists of a few multiplications of single digits by single digits and a little addition of an equally easy character.

Y es; this is all good enough so far as it gos; but it provides only for a particular type namely, of divisions in volving large-digit numbers. Can it help us in other divisions i. The answer is a candidly emphatic and unequivocal No. An actual sample specimen will prove this:. Suppose we have to divide by This is manifestly not only too long and cumbrous but much more so than the current system which, in this particular case, is indisputably shorter and easier.

In such a case, we can use a multiple of the divisor and finally multiply again by the AnurUpya rule. This we proceed to explain in the next chapter. DIVISION by the Paravartya method We have thus found that, although admirably suited for application in the special or particular cases wherein the divisordigits are big ones, yet the Nikhilam method does not help us in the other cases namely, those wherein the divisor consists of small digits.

The last example with 23 as divisor at the end of the last chapter has made this perfectly clear. Hence the need for a formula which will cover the other cases. And this is found provided for in the Paravartya SiUra, which is a specialcase formula, which reads Paravartya Yqjayet and which means Transpose and apply.

The well-known rule relating to transposition enjoins invariable change of sign with every change of side. Thus-f becomesand conversely ; and x becomes -r and conversely.

In the current system, this law is known but only in its application to the transposition of terms from left to right and conversely and from numerator to denominator and conversely in connection with the solution of equations, the proving of Identities etc. According to the Vedic system, however, it has a number of applications, one of which is discussed in the present chapter.

At this point, we may make a reference to the Remainder Theorem and Horners process and then pass on to the other most interesting applications of the Paravartya Sutra.

The Remainder Theorem: We may begin this part of this exposition with a simple proof o f the Remainder Theorem, as follows: In other words, the given expression E itself with p substituted for x will be the Remainder. Thus, the given expression E i. E with ' p substituted for x. This is the Remainder Theorem. Horners process of Synthetic Division carries this still further and tells us the quotient too. It is, however, only a very small part of the Paravartya formula which goes much farther and is capable of numerous applications in other directions also.

Now, suppose we have to divide 12x28x 32 by x2. We put x 2 the Divsior down on the left as usual ; just below it, we put down the2 with its sign changed ; and we do the multiplication work just exactly as we did in the previous chapter.

A few more algebraic examples may also be taken: At this stage, the student should practise the whole process as a MENTAL exercise in respect of binomial divisors at any rate.

For example, with regard to the division of 12x28x32 by the binomial x2 , one should be able to s a y: Add8 and obtain 16 as the next coefficient of the Quotient. And the student should be able to say. Extending this process to the case of divisors containing three terms, we should follow the same method, but should also take care to reverse the signs of the coefficient in all the other terms except the first:.

But what about the cases wherein, the first coefficient not being unity, fractions will have to be. The answer is that all the work may be done as before, with a simple addition to the effect that every coefficient in the answer must be divided by the first coefficient of the Divisor. The better method therefore would be to divide the Divisor itself at the very outset by its first coefficient, complete the working and divide it all off again, once for all at the end. Note x 2 8 30 42 that the R always 2 4 15 21 - 5 4 remains constant.

Note that R is constant in every case. We shall now take up a number of Arithmetical applications and get a clue as to the utility and jurisdiction of the Nikhilam formula and why and where we have to apply the Pardvartya Sutra. But this is too cumbrous. The Pardvartya formula will be more suitable. This is ever so much simpler. But this is a case where Vilokanenaiva i. Here, as the Remainder portion is a negative quantity, we should follow the device used in subtractions of larger numbers from smaller ones in coinage etc.

In other words, take 1 over from the quotient column to the remainder column i. We can thus avoid multiplication by big digits i. Even this is too cumbrous. In both these methods, the working is exactly the same. This work can be curtailedor at least rendered a bit easierby the Anurupyena Sutra. We can take which is one-fourth of or 84 which is one-eighth of it or, better.

The division with as Divisor works out as follows: It will thus be seen that, in all such cases, a fairly easy method is for us to take the nearest multiple or sub-multiple to a power of 10 as our temporary divisor, use the Nikhilarh or the Paravartya process and then multiply or divide the Quotient proportionately.

A few more examples are given below, in illustration hereof: The following examples will explain and illustrate i t: We have already got x by the cross multiplication of the x in the upper row and the 1 in the lower row ; but the coefficient of x in the product is 2. The other x must therefore be the product of the x in the lower row and the absolute term in the upper row. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x.

Hence the second term of the divisor must be 9x. But we have 6x in the dividend. We must therefore get an additional 24x. This can only come in by the multiplication of x by This is the third term of the quotient. We have therefore to get an additional 53 from somewhere. But there is no further term left in the Dividend. This means that the 53 will remain as the Remainder. The procedure is very simple ; and the following examples will throw further light thereon and give the necessary practice to the student: Put zero coefficients for absent powers.

Elementary D iv is io n S e c t io n In these three chapters IV, V and VI relating to Division, we have dealt with a large number and variety of instructive examples and we now feel justified in postulating the following conclusions: At any rate, they do not, in such cases, conform to the Vedic systems Ideal of Short and Sweet ; 3 And, besides, all the three of them are suitable only for some special and particular type or types of cases ; and none of them is suitable for general application to all cases: UrdhvaTiryak9 Sutra , the Algebraic utility there of is plain enough ; but it is difficult in respect of Arithmetical calculations to say when, where and why it should be resorted to as against the other two methods.

All these considerations arising from our detailedin. And the question therefore naturally nay, unavoidably arises as to whether the Vedic Sutras can give us a General to all cases. This astounding method we shall, however, expound in a later chapter under the caption Straight-Division which is one of the Crowning Beauties of the Vedic mathematics Sutras.

Factorisation comes in naturally at this point, as a form of what we have called Reversed multiplication and as a particular application of division. There is a lot of strikingly good material in the Vedic Sutras on this subject too, which is new to the modern mathematical world but which comes in at a very early stage in our Vedic Mathematics.

We do not, however, propose to go into a detailed and exhaustive exposition of the subject but shall content ourselves with a few simple sample examples which will serve to throw light thereon, and especially on the Sutraic technique by which a Sutra consisting of only one or two simple words, makes comprehensive provision for explaining and elucidating a pro cedure hereby a so-called difficult mathematical problem which, in the other system puzzles the students brains ceases to do so any longer, nay, is actually laughed at by them as being worth rejoicing over and not worrying over!

For instance, let us take the question of factorisation of a quadratic expression intd its component binomial factors. When the coefficient of x 2 is 1, it is easy enough, even according to the current system wherein you are asked to think out and find two numbers whose algebraic total is the middle coeffi cient and whose product is the absolute term. And the actual working out thereof is as follows: However, as the mental process actually employed is as explained above, there is no great harm done.

In respect, however, of Quadratic expressions whose first coefficient is not unity e. The Vedic system, however, prevents this kind of harm, with the aid of two small sub-Sutras which say i Anurupyena and ii Adyamadyendntyamantyena and which mean proportionately and the first by the first and the last by the last. The former has been explained already in connection with the use of multiples and sub-multiples, in multiplication and division ; but, alongside of the latter sub-Sutra, it acquires a new and beautiful double application and significance and works out as follows: Now, this ratio i.

And the second factor is obtained by dividing the first coefficient of the Quadratic by the first coefficient of the factor. Thus we say: This sub-Sutra has actually been used already in the chapters on division ; and it will be coming up again and again, later o n i.

But, just now, we make use of it in connection with the factorisation of Quadratics into their Binomial factors. The following additional examples will be found useful: An additional sub-Sutra is of immense ultility in this context, for the purpose of verifying the correctness of our answers in multiplications, divisions and reads: The product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product In symbols, we may put this principle down thus: For example:.

Thus, if and when some factors are known, this rule helps us to fill in the gaps. It will be found useful in the factorisation of cubics, biquadratics etc.

This is obviously a case in which the ratios of the coefficients of the various powers of the various letters are difficult to find o u t ; and the reluctance of students and even of teachers to go into a troublesome thing like this, is quite understandable. The 4 LopanaSthdpana9 sub-Sutra, however, removes the whole difficulty and makes the factorisation of a Quadratic of this type as easy and simple as that of the ordinary quadratic already explained.

The procedure is as follows: Suppose we have to factorise the following long Quadratic: And that gives us the real factois of the given long expression. The procedure is an argumentative one and is as follows:. By eliminating two letters at a time, we g e t: We could have eliminated x also and retained only y and z and factorised the resultant simple quadratic.

That would not, however, have given us any additional material but would have only confir med and verified the answer we had already obtained. Thus, when 3 letters x, y and z are there, only two eliminations will generally suffice. The following exceptions to this rule should be noted: But x is to be found in all the terms ; and there is no means for deciding the proper combinations.

In this case, therefore, x too may be eliminated ; and y and z retained. By so doing, we have: Here too, we can eliminate two letters at a time and thus keep only one letter and the independent term, each time.

This Lopana-r-Sthdpana method of alternate eli mination and retention will be found highly useful, later on in H. By Simple Argumentation e. We have already seen how, when a polynomial is divided by a Binomial, a Trinomial etc. From this it follows that, if, in this process, the remainder is found to be zero, it means that the given dividend is divisible by the given divisor, i.

And this means that, if, by some such method, we are able to find out a certain factor of a given expression, the remaining factor or the product of all the remaining factors can be obtained by simple division of the expression in question by the factor already found out by some method of division. In this context, the student need hardly be reminded that, in all Algebraic divisions, the Paravartya method is always to be preferred to the Nikhilam method.

Applying this principle to the case of a cubic, we may say that, if, by the Remainder Theorem or otherwise, we know one Binomial factor of a cubic, simple division by that factor will suffice to enable us to find out the Quadratic which is the product of the remaining two binomial factors.

A simpler and easier device for performing this operation will be to write down the first and the last terms by the AdyamaAnd these two can be obtained by the Adyamadyena' method of factorisation. And as the first and last digits thereof are already known to be 1 and 6, their total is 7. This is a very simple and easy but absolutely certain and effective process.

The student will remember that the ordinary rule for divisibility of a dividend by a divisor as has been explained already in the section dealing with the RemainderTheorem is as follows: In other words, x is a factor.

But their total should be 0 the coefficient of x 2. So we must reject the 1, 1, 6 group and accept the 1, 2, 3 group.

Dividing E by that factor, we first use the Adya. And ti is 48 whose factors are, 1, 2, 3, 4, 6,8 12, 16, 24 and Possible factors are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and But the sum of the coefficients in each factor must be a factor of the total S0. Now, the only possible numbers here which when added, total 2 are 3, 4 and 5. Now, test for and verify x 3.

Then, argue as in the first method. And the only combination which gives us the total 2, is 1, 2 and 3. Test and verify for 5. And put down the answer. Now test for and verify Then the quotient is obtainable by the Adyamddyena' and Samuccaycb Sutras.

And that again can be factorised with the aid of the former. The first is by means of factorisation which is not always easy ; and the second is by a process of continuous division like the method used in the G.

The latter is a mechanical process and can therefore be applied in all cases. But it is rather too mechanical and, consequently, long and cumbrous. The Vedic method provides a third method which is applicable to all cases and is, at the same time, free from this disadvantage.

It is, mainly, an application of the Lopana-Sthapana Sutra, the 4 Sankalana-Vyavakalan process and the Adyamadya rule. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multi ples. A concrete example will elucidate the process: Let Ej and E2 be the two expressions. The chart is as follows: The Algebraical principle or Proof hereof is as follows: Let P and Q be the two expressions; H their H.

The H. A few more illustrative examples may be seen below: But how should one know this beforehand and start monkeying or experimenting with it? Thej H. As this has further factors, it the E. Multiply it by x and take it over to the right for subtraction. But the factorisation of the two big biquadratics into two further factorless quadratics each, will entail greater waste of time and energy.

So, the position may be analysed thus: And the beauty of it is that the H. In order to solve such equations, the students do not generally use these basic sub-Sutras as such but almost invari ably go through the whole tedious work of practically proving the formula in question instead of taking it for granted and applying it!

The Vedic-method gives us these sub-formulae in a con densed form like Pardvartya etc. Transpose and adjust The applications, however, are numerous and splendidly use ful.

A few examples of this kind are cited hereunder, as illustrations thereof: Z1 1 the whole process should be a process. Second General Type 2 The above is the commonest kind of transpositions. The second common type is one in which each side the L. The usual method is to work out the two multiplications and do the transpovsitions and say: Third General Type The third type is one which may be put into the general form. The only rule to remember for facilitating this process is that all the terms involving x should be conserved on to the left side and that all the inde pendent terms should be gathered together on the right side and that every transposition for this purpose must invariably produce a change of sign i.

Fourth General Type The fourth type is of the form: After all the L. In fact, the application of this process may, in due course, by means of practice, be extended so as to cover cases involving a larger number of terms. For instance,. And this method can be extended to any number of terms on the same lines as explained above.

Special Types of Equations The above types may be described as General types. But there are, as in the case of multiplications, divisions etc,, particular types which possess certain specific characteristics of a SPECIAL character which can be more easily tackled than the ordinary ones with the aid of certain very short SPECIAL processes practically what one may describe as mental one-line methods. As already explained in a previous context, all that the student has to do is to look for certain characteristics, spot them out, identify the particular type and apply the formula which is applicable thereto.

The mere fact that x occurs as a common factor in all the-terms on both sides [or on the L. This is practically axiomatic. And this is applicable not only to x or other such unknown quantity but to every such case. On the contrary, we can straightaway say: The word 'Samuccaya' has, as its second meaning, the product of the independent terms. Fourthly, Samuccaya means combination or total. In this sense, it is used in several different contexts ; and they are explained below: In fact, as soon as this characteristic is noted and the type recognised, the student can at once mentally say: And this has to do with Quadratic equations.

None need, however, go into a panic over this. It is as simple and as easy as the fourth application ; and even little children can understand and readily apply this Sutra in this context, as explained below.

In the two instances given above, it will be observed that the cross-multiplications of the coefficients of x gives us the same coefficient for x2. In the first case, we had 4x2 on both sides ; and in the second example, it was 6x2 on both sides.

The two cancelling out, we had simple equations to deal with. But there are other cases where the coefficients of x2 are not the same on the two sides ; and this means that we have a T quadratic equation befoje us.

But it does not matter. For, the same Sutra applies although in a different direction here too and gives us also the second root of the quadratic equation. The only differenc e is that inasmuch as Algebraic Samuccaya9 includes sub traction too, we therefore now take into account, not only the sum of Nt and N2 and the sum of Dx and I 2 but also the differences between the numerator and the denominator on each side ; and, if they be equal, we at once equate that difference to Zero.

## Vedic Mathematics

Let us take a concrete example and suppose we have to solve the equation? And so, according to the present application of the same Sutra, we at once say: With the same sense total5 of the word Samuccaya but in a different application, we have the same Sutra coming straight to our rescue, in the solution of what the various texfc-books everywhere describe as Harder Equations , and deal with in a very late chapter thereof under that caption.

In fact, the label Harder has stuck to this type of equations to such an extent that they devote a separate section thereto and the Matriculation examiners everywhere would almost seem to have made it an invariable rule of practice to include one question of this type in their examination-papers! Now, suppose the equation before us is: And, after 10 or more steps of working, they tell you that 8 is the answer.

The Vedic Sutra, however, tells us that, if other elements being equal , the sum-total of the denominators on the L.

And that is all there is to it! A few more instances may be noted: The above were plain, simple cases which could be readily recognised as belonging to the type under consideration.

There, however, are several cases which really belong to this type but come under various kinds of disguises thin, thick or ultra-thick! But, however thick the disguise may be, there are simple devices by which we oan penetrate and see through the disguises and apply the Sunya Samuccaye' formula Thin D isguises 1 1 x8 1 X5 1 x 12 l x9. Here, we should transpose the minuses, 4 terms are plus ones: If the last two examples with so many literal coefficients involved were to be done according to the current system, the labour entailed over the L.

But, by this Vedic method, the equation is solved at sight! M ed iu m D isguises. The above were cases of thin disguises, where mere transposition was sufficient for enabling us to penetrate them. By dividing the Numerators out by the Denominators, we h a v e: Now, this process of division can be mentally performed very easily, thus: All this argumentation cun of course, be done mentally.

So, we say: Either by simple division or by simple factorisation both of them, mental , we note: But we note that the coefficient of x in the four de nominators is not the same. So, by suitable multi plication of the numerator and the denominator in each term, we get 6 the L. Thus, we h ave: But we cannot gamble on the possible chance of its being of this type and go through all the laborious work of L.

There must therefore be some valid and convincing test whereby we can satisfy ourselves beforehand on this point and, if convinced, then and then only should we go through all the toil involved. And that test is quite simple and easy: And this too can be done mentally. By simple division, we put this into proper shape, as follows: For easy learning of different types of information related to few events, religion, famous personalities, etc author has used some ridiculous associations. There is no meaning of it except learning the related information.

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Description of functioning of brain. Personality development from memory power. Importance of self hypnosis, meditation and concentration power. Techniques to remember English vocabulary.