"Vedic Mathematics" is the name given to the ancient system of they made it a general Mar 18, I may try to edit this book or write a new book in future. N.B. for the last option we often found Adobe Acrobat books with file names like " Book Title>aracer.mobi". The ".exe" implying installation of a program on your. the principles of mathematics, but they saw no necessity to realization.” – Vedic Mathematics and the Spiritual Dimension. Dharma Sutras. Law books.

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This book on Vedic Mathematics seeks to present an integrated approach to learning This book makes use of the Sutras and Sub-Sutras stated above for. Vedic Mathematics(ORIGNAL BOOK) - Ebook download as PDF File .pdf), Text File .txt) or read book online. Views of Prof. aracer.mobi about Vedic. Mathematics from Frontline. Neither Vedic Nor Mathematics. Views about the Book in Favour and Against.

Please find below a range of free books on the subject of Vedic Mathematics. A good free introductory ebook in Spanish can be found here. The ". From our point of view, the best situation you can end up with is receiving spam email etc, with the worst scenario's being someone using your credit card or having virus's or spyware installed onto your computer. We only ask that you do not upload them to any other place on the Internet without consulting us first and where the document is hosted on another website, please consult with the original author first. This book is designed for teachers of children in grades 3 to 7. It shows how Vedic Mathematics can be used in a school course but does not cover all school topics see contents. The book can be used for teachers who wish to learn the Vedic system or to teach courses on Vedic mathematics for this level. The Manual contains many topics that are not in the other Manuals that are suitable for this age range and many topics that are also in Manual 2 are covered in greater detail here. Nicholas, K. This is an advanced book of sixteen chapters on one Sutra ranging from elementary multiplication etc.

He takes delight in working out huge problems mentally-sometimes even faster than electronic gadgets like calculators or computers. These methods are also useful for our daily life to calculate anything like numbers, calculations, bills, interest or any kind of transcation. The reader for sure would enter into the world of enchantment for maths with our author Virender Mehta.

Example: 4 52 9 52 3 52 4x5 5x5 9x10 5x5 3x4 5x5 20 25 90 25 12 25 4 3 52 1 1 52 43x44 5x5 11x12 5x5 25 25 Exercise: Find out the square of following numbers 35 55 65 85 95 Written by: Virender Mehta World Record Holder in Memory visit www.

Now write down the result in the answer along with the multiplication of the same second digit of the numbers. Deposit Rs. Description of all memory boosting concepts and methods.

Description of functioning of brain. Personality development from memory power. Importance of self hypnosis, meditation and concentration power. Techniques to remember English vocabulary. Techniques to excel in competitive exams. Techniques to prevent memory loss. Techniques to be successful in interviews. Vedic techniques to enhance memory power. Scientific techniques to remember all subjects in academics.

This Chakra is very powerful to enhance your Memory and Concentration Power. It also symbolizes the arrival of Goddess Laksmi. The current method is too cumbrous and may be tried by the student himself.

The Vedic mental one-line method Sutra is as follows: The current method is too cumbrous. The Vedic mental one-line method by the YavadunamTavxdunzm Sutra is as follows: The Vedic mental one-line method is as follows: Simple Equations: The Vedic method simply says: Every quadratic can thus be broken down into two binomial-factors.

And the same principle can be utilised for cubic, biquadratic, pentic etc.

Summation of Series:. The current methods are horribly cumbrous. The Vedic mental one-line methods are very simple and easy. There are several Vedic proofs thereof every one of them much simpler than Euclid's. I give two of them below: N ote: Apollonius Theorem, Ptolemys Theorem, etc. For finding the equation of the straight line passing through two points whose co-ordinates are given. Say 9, 17 and 7, 2. By the Current Method: But this method is.

But the Vedic mental one-line method by the Sanskrit Sutra Formula , II tffapfcr n Paravartya-Sutra enables us to write down the answer by a mere look at the given co-ordinates. Loney devotes about 15 lines section , Ex.

By the Vedic method, however, we at once apply the Adyamadyena Sutra and by merely looking at the quadratic write down the answer: The Vedic methods are so simple that their very simplicity is astounding, and, as Desmond Doig has aptly, remarked, it is difficult for any one to believe it until one actually sees it.

It will be our aim in this and the succeeding volumes1 to bring this long-hidden treasure-trove of mathemetical knowledge within easy reach of everyone who wishes to obtain it and benefit by it.

Ekadhikena Purvena also a corollary 2. Paravartya Yojayet 5. Sunyam Samyasamuccaye 6. Sankalana-vyavakalanabhyam also a corollary4 8. Anurupyena STOW: Yavadunam SesanyanJcena Caramena Sopantyadvayamantyam Ekanyunena Purvena Gunitasamuccayah Gunakasamuccayah Gunitauiitccayah Samiumyagunitah zrfesnrfe Vyastisamastih Lopanasthdpandbhyam Vibkanam Samuccayagunitah Sub-Sutras or Corollaries. Suffice it, for our present immediate purpose, to draw the earnest attention of every scientifically-inclined mind and researchward-attuned intellect, to the remarkably extra ordinary and characteristicnay, unique fact that the Vedic system does not academically countenance or actually follow any automatical or mechanical rule even in respect of the correct sequence or order to be observed with regard to the various subjects dealt with in the various branches of Mathe matics pure and applied but leaves it entirely to the con venience and the inclination, the option, the temperamental predilection and even the individual idiosyncracy of the teachers and even the students themselves as to what particular order or sequence they should actually adopt and follow!

This manifestly out-of-the-common procedure must doubtless have been due to some special kind of historical back-ground, background which made such a consequence not only natural but also inevitable under the circumstances in question.

Immemorial tradition has it and historical research confirms the orthodox belief that the Sages, Seers and Saints of ancient India who are accredited with having observed, studied. And, consequently, it naturally follows that, in-as-much as, unlike human beings who have their own personal prejudices, partialities, hatreds and other such subjective factors distorting their visions, warping their judgements and thereby contri buting to their inconsistent or self-contradictory decisions and discriminatory attitudes, conducts etc.

They are, on the contrary, strictly and purely impersonal and objective in their behaviour etc. This seems to have been the real historical reason why, barring a few unavoidable exceptions in the shape of elementary, basic and fundamental first principles of a preliminary or pre requisite character , almost all the subjects dealt with in the various branches of Vedic Mathematics are explicable and expoundable on the basis of those very basic principles or first principles , with the natural consequence that no particular subject or subjects or chapter or chapters need necessarily precede or follow some other particular subject or subjects or chapter or chapters.

Nevertheless, it is also undeniable that, although any particular sequence is quite possible, permissible and feasible. And so, we find that subjects like analytical conics and even calculus differential and integral which is usually the bugbear and terror of even the advanced students of mathematics under the present system all the world over are found to figure and fit in at a very early stage in our Vedic Mathematics because of their being expounded and worked out on basic first principles.

And they help thereby to facilitate mathematical study especially for the small children. And, with our more-than-half-a-centurys actual personal experience of the very young mathematics-students and their difficulties, we have found the Vedic sequence of subjects and chapters the most suitable for our purpose namely, the elimina ting from the childrens minds of all fear and hatred of mathe matics and the implanting therein of a positive feeling of exuberant love and enjoyment thereof!

And we fervently hope and trust that other teachers too will have a similar experience and will find us justified in our ambitious description of this volume as Mathematics without tears. From the herein-above described historical back-ground to our Vedic Mathematics, it is also obvious that, being based on basic and fundamental principles, this system of mathe matical study cannot possibly come into conflict with any other branch, department or instrument of science and scientific education.

And, above all, we have our Scriptures categroically laying down the wholesome dictum: In other words, we are called upon to enter on such a scientific quest as this, by divesting our minds of all pre-conceived notions, keeping our minds ever open and, in all humility as humility alone behoves and befits the real seeker after truth , welcoming the light of knowledge from whatever direction it may be forthcoming.

Nay, our scriptures go so far as to inculcate that even thir expositions should be looked upon by us not as teachings or even as advice, guidance etc. In conclusion, we appeal to our readers as we always, appeal to our hearers to respond hereto from the same stand point and in the same spirit as we have just hereinabove described.

We may also add that, inasmuch as we have since long promised to make these volumes2 self-contained , we shall make our explanations and expositions as full and clear as possible.

First Example: Case 1. By the Current Method. By the Vedic one-line mental method. First method. And the modus operandi is explained in the next few pages. The relevant Sutra reads: Ekadhikena Purvena which, rendered into English, simply says: By one more than the previous one. Its application and modus operandi are as follows: For, in the case of addition and subtraction, to and from respectively would have been the appropriate preposition to use. But by is the preposition actually found used in the Sutra.

The inference is therefore obvious that either multiplication or division must be enjoined. And, as both the meanings are perfectly correct and equally tenable according to grammar and literary usage and as there is no reasonin or from the text for one of the meanings being accepted and the other one rejected, it further follows that both the processes are actually meant.

And, as a matter of fact, each of them actually serves the purpose of the Sutra and fits right into it as we shall presently show, in the immediately following explanation of the modus operandi which enables us to arrive at the right answer by either operation. The First method:. The first method is by means of multiplication by 2 which is the Ekadhika Purva i. For, the relevant rule hereon which we shall explain and Expound at a later stage stipulates that the product of the last digit of the denominator and the last digit of the decimal equivalent of the fraction in question must invariably end in 9.

Therefore, as the last digit of the denominator in this case is 9, it automatically follows that the last digit of the decimal equivalent is bound to be 1 so that the product of the multi plicand and the multiplier concerned may end in 9. We, therefore, start with 1 as the last i. Our modus-operandi-chart is thus as follows: But this has two digits. We therefore put the 6 down imme diately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step as we always do in all multiplication e.

But, as this is a single-digit number with nothing to carry over to the left , we put it down as our next multiplicand. We therefore put up the usual recurring marks dots on the first and the last digits of the answer for betokening that the whole of it is a Recurring Decimal and stop the mul tiplication there.

Our chart now reads as follows.

In passing, we may also just mention that the current process not only takes 18 steps of working for getting the 18 digits of the answer not to talk of the time, the energy, the paper, the ink etc. In the Vedic method just above propounded, however, there are no subtrac tions at all and no need for such trials, experiments etc. All this lightens, facilitates and expedites the work and turns the study of mathematics from a burden and a bore into a thing of beauty and a joy for ever so far, at any rate, as the children are concerned.

In this context, it must also be transparently clear that the long, tedious, cumbrous and clumsy methods of the current system tend to afford greater and greater scope for the children s making of mistakes in the course of all the long multiplications, subtractions etc. The Second method: As already indicated, the second method is of division instead of multiplication by.

And, as division is the exact opposite of multiplication, it stands to reason that the operation of division should proceed, not from right to left as in the case of multi plication as expounded hereinbefore but in the exactly opposite direction i. And such is actually found to be the case.

Its application and modus operandi are as follows i Dividing 1 the first digit of the dividend by 2, we see the quotient is zero and the remainder is 1. We, therefore, set 0 down as the first digit of the quotient and prefix the Remainder 1 to that very digit of the Quotient as a sort of reverse-procedure to the.

We, therefore, put 2 down as the third digit of the quotient and prefix the remainder 1 to that quotient-digit 2 and thus have 12 as our next Dividend. So, we set 6 down as the fourth digit of the quotient; and as there is no remainder to be prefixed thereto, we take the 6 itself as our next digit for division. We therefore put 1 down as the 5th quotient-digit, prefix the 1 thereto and have 11 as our next Dividend.

But this is exactly what we began with. This 1 1 1 means that the decimal begins to repeat itself from here. So, we stop the mentaldivision process and put down the usual recurring symbols dots. Note that, in the first method i. A Further short-cut This is not all. As a matter of fact, even this much or rather, this little work of mental multiplication or division is not really necessary.

This will be self-evident from sheer observation. Let us put down the first 9 digits of And this means that, when just half the work has been completed by either of the Vedic one-line methods , the other half need not be obtained by the same process but is mechanically available to us by subtracting from 9 each of the digits already obtained! 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.

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.