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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. 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 PREFACE LESSON 1 COMPLETING THE WHOLE. cation of the book Vedic Mathematics or 'Sixteen Simple Mathe- humaniay ae such for the aohievemsnt of all-round, complete and perfect succeas.
B E A few more examples may be taken: There is no meaning of it except learning the related information. Amahier Khandesh and similar lectures at several other places like Poona. And herein. As we know the first remainder is 3. By Means of Aliquot Parts.
It is our earnest aim and aspiration. De Morgan etc. XX this series1. There are still other methods and in the latter system whereby even that very small working can be rendered shorter still!
This and the beatific beauty of the whole scheme. To start with. And this process is naturally a little longer. For instance. But it need hardly be pointed out that.. And all that the student of these Sutras has to do is to look for the special characteristics in question. By current method: And then we shall take up the other various parts. By the current method: Only the Vedic mental one-line method is given here.
The current method is notoriously too long. XXIV Division continued: By the Vedic mental one-line-method. The truth-loving student can work it out by the other method and compare the two for himself.
And the same is the case with all such divisions whatever the number of digits may b e: By the mental Vedic one-line method: By the Vrdhwa-Tiryak Sutra 7 Extract the square root of The current method is too cumbrous and may be tried by the student himself. By the Vedic mental one-line method: By the current method. The Vedic mental one-line method Sutra is as follows: But by the Vedic mental one-line method by the Ekadhika-Purva Sutra.
The current method is too cumbrous. The Vedic method simply says: Simple Equations: The same is the case here. The Vedic mental one-line method is as follows: By the Vedic method by the Sunyam-Samuccaya Sutra. Cubing and Cube-Root: The Vedic mental one-line method by the YavadunamTavxdunzm Sutra is as follows: G and H are points on AB. Summation of Series: The current methods are horribly cumbrous.
Formula i. I give two of them below: And the same principle can be utilised for cubic. 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. For finding the equation of the straight line passing through two points whose co-ordinates are given.
Then ABC. Say But this method is Second method using the formula y —y1— is still more cumbrous and confusing. By the Current Method: Solving it for x. By the Vedic method. Loney devotes about 15 lines section The Vedic methods are so simple that their very simplicity is astounding. 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.
Yavadunam Tavadunarp. Ekadhikena Purvena also a corollary 2. Sunyam Samyasamuccaye 6. Paravartya Yojayet 5. Kevalaih Saptakarp. Sankalana-vyavakalanabhyam also a corollary4 8. Anurupyena STOW: Vibkanam Yavadunam Ekanyunena Purvena Gunitasamuccayah Sopantyadvayamantyam Samuccayagunitah Sub-Sutras or Corollaries [Note—This list has been compiled from stray references in the text— e d i t o r.
Gunakasamuccayah SesanyanJcena Caramena Lopanasthdpandbhyam Immemorial tradition has it and historical research confirms the orthodox belief that the Sages. This manifestly out-of-the-common procedure must doubtless have been due to some special kind of historical back-ground.
Seers and Saints of ancient India who are accredited with having observed. Suffice it. This seems to have been the real historical reason why.
They are. And so. From the herein-above described historical back-ground to our Vedic Mathematics. And they help thereby to facilitate mathematical study especially for the small children. M yet. In other words. Brevity may be the soul of w it.
It is in this spirit and from this viewpoint that we now address ourselves to the task before us. In conclusion.
We may also add that. By the Current Method. And the modus operandi is explained in the next few pages. Second method. First method. By the Vedic one-line mental method. And there. First Example: Case 1. The inference is therefore obvious that either multiplication or division must be enjoined. The relevant Sutra reads: Ekadhikena Purvena which. The First method: Its application and modus operandi are as follows: Our modus-operandi-chart is thus as follows: But this has two digits.
We follow this procedure continually xviii until we reach the 18th digit counting leftwards from the right. In passing. And such is actually found to be the case. In the Vedic method just above propounded. The Second method: As already indicated. Its application and modus operandi are as follows i Dividing 1 the first digit of the dividend by 2.
All this lightens. In this context. This 1 1 1 means that the decimal begins to repeat itself from here. We therefore put 1 down as the 5th quotient-digit. But this is exactly what we began with. This is not all. And the answer is—as we shall demonstrate later on—that. Let us put down the first 9 digits of the answer in one horizontal row above and the other 9 digits in another h o r iz o n t a l row just below and observe the fun of it. Note that. And this means that. As a matter of fact.
This will be self-evident from sheer observation. A Further short-cut. By the Current method: Details o f these principles and processes and other allied matters. Second Example: Case 2?
Let us now take another case of a similar type say. In the meantime. The procedures are explained on the next page.
Second Method. First Method i 2 1 B. And we find that. Our modus-ojperandi-chart herein reads as follows: Explanation of the Second Method: The Complements from Nine: Here too.
Explanation of the First Method: And the chart reads as follows: Our multiplier or divisor as the case may be is now 5 i. But this is not all. And yet. This means that half the work of multiplication or division. The complements from nine are also there. Our readers will doubtless be surprised to learn—but it is an actual fact—that there are. We shall hold them over to be dealt with. Sutra Pass we now on to a systematic exposition of certain salient.
We may also draw the attention of all students and teachers of mathematics to the well-known and universal fact that. The Sutras are very short. And a school-going pupil who knows simple addition and subtraction of single-digit numbers and the multiplication-table up to five times five.
At this point. We shall give a detailed explanation. Let us: You get the same answer 6 again. The Sutra: A vertical dividing line may be drawn for the purpose of demarcation of the two parts.
But just now. Suppose we have to multiply 9 by 7. And put 16—10 i. And this is the righthand-side portion of the answer. And you get 6 again as the left-hand side portion of the required answer. And -you find that you have got 9—3 i. The product is 3. This difficulty. We thus obtain 42 as the actual product of 7 and 6. The algebraical A slight difference. The base now required is We should therefore adopt the same method as before i. The rule is that all the other digits of the given original numbers are to be subtracted from 9 but the last i.
Note 2: What is the remedy herefore? A new point has now to be taken into consideration i. And the remedy is—as in the case of decimal multiplications—merely the filling up of all such vacancies with Zeroes.
This process helps us in the work of ready on-sight subtraction and enables us to pu. And all the other rules regarding digit-surplus. What about numbers which are above it? And the answer is that the same procedure will hold good there too..
The answer is that the plus and the minus will. Multiples and sub-multiples: A vinculum may be used for making this clear. What about the multiplication of two numbers. In actual application. Both these numbers are so far away from the base that by our adopting that as our actual base. Suppose we have to multiply 41 by Our chart will then take this shape: A concrete illustration will make the modus operandi clear.
We therefore. The product of 41 and 41 is thus found to be the same as we got by the first method. We may write down our table of answers as follows: They have been included here.
As 8 is 2 less than A few elementary examples will suffice to make its meaning and application clear: The First Corollary: The following will be the successive stages in our mental working: This evidently deals with the squaring of numbers.
We work exactly as before. In the present case, if our b be 3, a-f-b will become and a—b will become This proves the Corollary. This corollary is specially suited for the squaring of such numbers.
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. 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: 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: 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 ? 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.
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. The applications of this brief and terse Sutra are manifolci as will be seen again and again.
A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by First we take it up in its most elementary application namely.. 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: 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 All the diffe-. Owing to their relevancy to this context. 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. It may. The following example will illustrate this: But the vinculum process is 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 feet—preferably 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.
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: M For example. 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: This means that we can mechanically take the first digit down for the Quotientcolumn and that.
In all these particular cases. Multiplication at fairly considerable length. Having dealt with. And then. As this is not permissible. The reason therefor is as follows: A single sample example will suffice to prove this: And herein. This double process can be combined into one as follows: Just at present in this chapter.
The answer is a candidly emphatic and unequivocal No. Can it help us in other divisions i. An actual sample specimen will prove this: This we proceed to explain in the next chapter. Thus-f becomes—and conversely. In the current system. Hence the need for a formula which will cover the other cases. According to the Vedic system.
And this is found provided for in the Paravartya SiUra. The well-known rule relating to transposition enjoins invariable change of sign with every change of side. The Remainder Theorem: We may begin this part of this exposition with a simple proof o f the Remainder Theorem. The last example with 23 as divisor at the end of the last chapter has made this perfectly clear.
A few more algebraic examples may also be taken: In general terms. This is the Remainder Theorem. Add—8 and obtain 16 as the next coefficient of the Quotient. Extending this process to the case of divisors containing three terms. But what about the cases wherein. The better method therefore would be to divide the Divisor itself at the very outset by its first coefficient.
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. The Pardvartya formula will be more suitable. T2 In other words. We can thus avoid multiplication by big digits i. Anurupya and Paravartya will be more suitable. We can take which is one-fourth of or 84 which is one-eighth of it or.
The division with as Divisor works out as follows: A few more examples are given below. The other x must therefore be the product of the x in the lower row and the absolute term in the upper row.
The following examples will explain and illustrate i t: This means that the 53 will remain as the Remainder.
But we have 6x in the dividend. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. And i x3 divided by x gives us x2 which is therefore the first term of the quotient.
But there is no further term left in the Dividend. 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. Hence the second term of the divisor must be 9x. We must therefore get an additional 24x. V and VI relating to Division. All these considerations arising from our detailedin comparative-study of a large number of examples add up.
This astounding method we shall. Formula applicable There And the answer is: And the question therefore naturally— nay. There is a lot of strikingly good material in the Vedic Sutras on this subject too.
And the actual working out thereof is as follows: When the coefficient of x 2 is 1. We do not. The Vedic system. And the second factor is obtained by dividing the first coefficient of the Quadratic by the first coefficient of the factor.
In respect. The former has been explained already in connection with the use of multiples and sub-multiples. A Thus we say: X coefficient of that factor. The following additional examples will be found useful: Conjugate Hyperbolas..
This sub-Sutra has actually been used already in the chapters on division. N ote: For example: Bi-quadratics etc. It will be found useful in the factorisation of cubics. Sutra just explained and another sub-Sutra which consists of only one compound word. 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 that gives us the real factois of the given long expression. The procedure is an argumentative one and is as follows: The procedure is as follows: The 4 Lopana—Sthdpana9 sub-Sutra. That would not. By eliminating two letters at a time. 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. 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.
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. 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.
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. 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 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. 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: A few more illustrative examples may be seen below: The H.
Multiply it by x and take it over to the right for subtraction. But how should one know this beforehand and start monkeying or experimenting with it? Thej H. And the beauty of it is that the H. In order to solve such equations. A few examples of this kind are cited hereunder.
The student has to perform hundreds of such transposttion-operations in the course of his work. 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: As examples. Third General Type The third type is one which may be put into the general form. Fourth General Type The fourth type is of the form: And this method can be extended to any number of terms on the same lines as explained above.
As already explained in a previous context. But there are. We begin this section with an exposition of several special types of equations which can be solved practically at sightwith the aid of a beautiful special Sutra which reads: On the contrary. 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. In this sense.
T h ir d M e a n in g a n d A p p l ic a t io n 4 Samuccaya thirdly means the sum of the Denominators of two fiactions having the same numerical numerator. Removing the numerical factor.
And this has to do with Quadratic equations. None need. The two cancelling out. But there are other cases where the coefficients of x2 are not the same on the two sides. Let us take a concrete example and suppose we have to solve the equation? In the first case. But it does not matter. At sight. It is as simple and as easy as the fourth application. In the two instances given above. The Vedic Sutra. And that is all there is to it! A few more instances may be noted: M ed iu m D isguises The above were cases of thin disguises.
We now turn to cases of disguises of medium thickness 1 x —2. All this argumentation cun of course. And that test is quite simple and easy: There must therefore be some valid and convincing test whereby we can satisfy ourselves beforehand on this point and. And this too can be done mentally. By simple division. By either method. And x —6 is the factor under the cube on R. Taking away the numerical factor. The Vedic mathematicians.
Vedic one-line mental answer is: Cancelling common terms out. And this gives us the required clue to the particular characteristic which characterises this type of equations. N1 Dx and D2 Binomials? And this too gives us the same answer as before. The Vedic formula. And here too. On actual cross-multiplication and expansion etc. An instance in point is given below: The student should not. In the fourth case. And we get the same answer as before. But it is not a sufficient condition by itself for the applicability of the present formula.
This really comes in as a corollary-consequence of the A. And this is in confiormity with the lack of the basic condition in question i. This gives us the assurance that. In the example actually now before us. This section may. But what about the x2 coefficients? For them too to vanish. The Algebraical Explanation for this type of equations is: We now proceed to deal with certain types of cases which do not seem to be of this kind but aie really so.
And there is no quadratic equation left for us to solve herein P roof: The x 2 coefficients are: All that we have to do is to re-arrange the terms in such a manner as to unmask the masked terms.
The first type: The first variety is one in which a number of terms on the left hand side is equated to a single term on the right hand side. As we mean to merge the R.. And the process is complete.. S is to be merged. So the resultant new equation after the merger now reads: A few more examples of this sort may be noted: A few illustrations will make this clear: A few more illustrations of this type are given below: YES 2 In these examples.
For instance: In such cases. T h e Sutra applies and can be applied immediately without bothering about the L. S of the same equation. A few more illustrations will be found helpful: This will be explained later. This equation can be solved in several ways all of them very simple and easy: N is also 24 The Sutra applies. In the final derived equation. The Sutra applies. The student may. By Pardvartya devisioD twice over.
The Vedic method by the Pardvartya Rule enables us to give the answer immediately by mere mental Arithmetic. And this. But even here. And this gives us our Numerator. And the Sutra says that. The Algebraical Proof is this: This gives us two simple equations in y.
An example will make the meaning and the application clear: If one is in ratio. And a repetition of the same. T h i s rule is also capable of infinite extension and may be extended to any number of unknown quantities. And the whole work can be done mentally. And we can say. As this is simple and easy to remember and to apply.
A few of them are shown below. There are other types of miscellaneous linear equations which can be treated by the Vedic Sutras. The Sutra. Let d be the common difference. C and D are in AP. Another Algebraical proof. But the Sunyam Samuccaye Sutra does not apply beacnse the number of factors in the original shape is 2 on the L. B E A few more examples may be taken: We therefore deal with this special type here.
The case is exactly like the one above. And this proves the proposition in question. Practice Lesson 3. Practice Lesson 4. Practice Lesson 5. Practice Lesson 6. Practice Lesson 7. Practice Lesson 8. Practice Lesson 9. Practice Lesson Published by: Road Ind. Area, Ghaziabad U. Anil Kumar Teotia Sr. Publication Team Navin Kumar, Ms. Radha, Jai Baghwan Pages Addition - Completing the whole 2. Addition from left to right 3. Addition of list of numbers - Shudh method 4.
Subtraction - Base method 5. Subtraction - Completing the whole 6. Subtraction from left to right. Base Method 2. Sub Base Method 3.
Vinculum 4. Multiplication of complimentary numbers 5. Multiplication by numbers consisting of all 9s 6. Multiplication by 11 7. Multiplication by two-digit numbers from right to left 8. Multiplication by three and four-digit numbers from right to left.
Squaring 1. Squaring numbers ending in 5 2. Squaring Decimals and Fraction 3. Squaring Numbers Near 50 4. Squaring numbers near a Base and Sub Base 5. General method of Squaring - from left to right 6. Number splitting to simplify Squaring Calculation 7. Algebraic Squaring Square Roots 1. Reverse squaring to find Square Root of Numbers ending in 25 2.
Square root of perfect squares 3. General method of Square Roots. John L. Lehet www. Why Vedic Mathematics? Gunita Samuccayah: Three Proofs of Fermat's Last Theorem. Details Author: Kapoor Pages Author John M Muehlman, Multiplication Skill 77 Hypothesis Two: Checking Skill 80 Hypothesis Three: Multiplication and Checking Affect 81 Hypothesis Four: Automaton; 1 edition December 5, Language: English ISBN Amazon UK Kindle Link.
Amazon US Kindle Link. Dani about Vedic Mathematics from Frontline 33 2. Repositories of Ancient Indian Lore 58 2. The Vedas are considered divine in origin and are assumed to be revelations from God. In traditional Hinduism, the Vedas were to be learnt only by the upper caste Hindus.
The lower castes Sudras and so-called untouchables who were outside the Hindu social order were forbidden from even hearing to its recitation. In recent years, there have been claims that the Vedas contain the cure to AIDS and the production of electricity.
Here the authors probe into Vedic Mathematics that gained renown during the revivalist Hindutva rule in India and was introduced into school syllabus in several states ; and explore if it is really Vedic in origin or Mathematics in content. To gain a better understanding of its imposition, we interviewed students, teachers, parents, educationists and activists.
The issue of imposition of Vedic Mathematics into the school curriculum involves religious politics, caste supremacy, apart from elementary arithmetic so we use fuzzy and neutrosophic techniques to gain acute insight into how students have been affected because of this politically motivated syllabus revision. We believe this paper was a reaction by some in the academic community to certain elements of the Hindu community trying to use Vedic Mathematics to promote Hinduism by teaching Vedic Mathematics in schools.
It is our understanding that much of this initial teaching of Vedic Mathematics may have been implemented poorly by people who did not have a good grounding in their subject matter as they may have been more interested in promoting Hinduism than Vedic Mathematics, hence the academic reaction.