# Vedic Mathematics/Printable version

Vedic Mathematics

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# What is Vedic Mathematics?

 Vedic Mathematics Printable version Next: Sutras

Welcome to the wonderful world of "Vedic" mathematics, a science that its founder claims was lost due to the advent of modern mathematics. Vedic mathematics is said by its founder to be a gift given to this world by the ancient sages of India, though there is no historical evidence whatsoever for this claim. It is a system for limited arithmetic and polynomial calculation which is simpler and more enjoyable than the equivalent algorithms of modern mathematics.

Vedic Mathematics is the name given to a supposedly ancient system of calculation which was "rediscovered" from the Vedas between 1911 and 1918 by Sri Bharati Krishna Tirthaji Maharaj (1884-1960). According to Tirthaji, all of Vedic mathematics is based on sixteen Sutras, or word-formulae. For example, "Vertically and Crosswise" is one of these Sutras. These formulae are intended to describe the way the mind naturally works, and are therefore supposed to be a great help in directing the student to the appropriate method of solution. None of these sutras has ever been found in Vedic literature, nor are its methods consistent with known mathematical knowledge from the Vedic era.

Perhaps the most striking feature of the Tirthaji system is its coherence. The whole system is interrelated and unified: the general multiplication method, for example, is easily reversed to allow one-line divisions, and the simple squaring method can be reversed to give one-line square roots. And, these are all easily understood. This unifying quality is very satisfying, it makes arithmetic easy and enjoyable, and it encourages innovation.

Difficult arithmetic problems and huge sums can often be solved immediately by Tirthaji's methods. These striking and beautiful methods are a part of a system of arithmetic which Tirthaji claims to be far more methodical than the modern system. "Vedic" Mathematics is said to manifest the coherent and unified structure of arithmetic, and its methods are complementary, direct and easy.

The simplicity of the Tirathji system means that calculations can be carried out mentally, though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods; they are not limited to one method. This leads to more creative, interested and intelligent pupils.

Interest in the Tirathji's system is growing in education, where mathematics teachers are looking for something better and finding the Vedic system is the answer. Research is being carried out in many areas including the effects learning the Tirathji system has on children; developing new, powerful but easy applications of these Sutras in arithmetic and algebra.

The effectiveness of the Tirathji system cannot be fully appreciated without practising the system. One can then see why its enthusiasts claim that it is the most refined and efficient calculating system known.

"Vedic Mathematics" refers to a technique of calculation based on a set of 16 Sutras, or aphorisms, as algorithms and their upa-sutras or corollaries derived from these Sutras. Its enthusiasts advance the claim that any mathematical problem can be solved mentally with these sutras.

 Vedic Mathematics Printable version Next: Sutras

# Sutras

 Vedic Mathematics Previous: What is Vedic Mathematics? Printable version Next: Techniques

The Sixteen Sutras and their Corollaries are as follows:

Ekadhikina Purvena - By one more than the previous one (Corollary: Anurupyena)

Nikhilam Navatashcaramam Dashatah - All from 9 and the last from 10 (Corollary: Sisyate Sesasamjnah)

Paraavartya Yojayet - Transpose and adjust (Corollary: Kevalaih Saptakam Gunyat)

Shunyam Saamyasamuccaye - When the sum is the same, that sum is zero (Corollary: Vestanam)

(Anurupye) Shunyamanyat - If one is in ratio, the other is zero (Corollary: Yavadunam Tavadunam)

Puranapuranabyham - By the completion or non-completion (Corollary: Antyayordashake'pi)

Chalana-Kalanabyham - Differences and Similarities (Corollary: Antyayoreva)

Yaavadunam - Whatever the extent of its deficiency (Corollary: Samuccayagunitah)

Vyashtisamanstih - Part and Whole (Corollary: Lopanasthapanabhyam)

Shesanyankena Charamena - The remainders by the last digit (Corollary: Vilokanam)

Sopaantyadvayamantyam - The ultimate and twice the penultimate (Corollary: Gunitasamuccayah Samuccayagunitah)

Ekanyunena Purvena - By one less than the previous one (Corollary: Dhvajanka)

Gunitasamuchyah - The product of the sum is equal to the sum of the product (Corollary: Dwandwa Yoga)

Gunakasamuchyah - The factors of the sum is equal to the sum of the factors (Corollary: Adyam Antyam Madhyam)

# Techniques

 Vedic Mathematics Previous: Sutras Techniques Next: Addition And Subtraction

## Introduction

Whether you believe the stories surrounding the source of the Vedic Mathematics techniques or not, what is important are the techniques themselves. By mastering these techniques you will not only radically improve your numeracy and arithmetic ability, you will also begin to understand that mathematics is a fluid and fascinating subject in which there are usually many different ways to solve any particular problem. You will see that many of the tedious algorithms that you were taught in school, (e.g. long division and multiplication), are but one way to solve a problem. While these 'school' techniques are often general purpose, (e.g. long multiplication allows you to multiply any pair of numbers), in many cases they are very inefficient; their very generality means they have to cover all possibilities and so can't take advantage of the specifics of any particular problem. If instead you use a technique optimised for the particular problem you are working on, you can take advantage of properties that aren't always present in the general case and so solve the problem with a lot less work

It should be noted that, although many of the following techniques may be contained in the 1965 book Vedic Mathematics by Sri Bharati Krsna Tirthaji, they are not unique to that book. Many of the techniques are also part of other arithmetic systems, (e.g. the Trachtenberg system), and most are common knowledge among those who enjoy the challenge of mental arithmetic. In fact the term 'Vedic Mathematics' is now sometimes used to encompass the general idea of solving arithmetic problems with a wide range of different techniques, each optimised for particular circumstances, these techniques not being limited to those in Sri Bharati Krsna Tirthaji's book.

So, do you want to know how to multiply 89 and 97 in your head in seconds? The solution is given below:-

1. First solve 9×7, ie, 63.(Carry over 6){Let 3 here be z}
2. Do cross multiplication & add the carried over no. done in previous step, ie solve (8×7)+(9×9)+6), ie, 143.(Carry over 14){Let 3 here be y}
3. Then solve (8×9)+14, ie, 86.{Let 86 be x}
4. Then join x, y & z to form xyz, ie, here xyz will be 8633.

Further, there is another technique to solve this problem.

 Vedic Mathematics Previous: Sutras Techniques Next: Addition And Subtraction

# Why Does It Work?

## Introduction

It should be understood that there is no magic in the many techniques described in the other chapters, indeed there is no magic in mathematics in general, it can be argued that mathematics is the purest of all sciences as there is no opinion and mathematics needs no experiments or interpretation of results; things are either true, (i.e. they are proven to be true), or they are not. That being the case, there must be sound reasons why all the previously described techniques work.
The reason some of the techniques work is simply that they perform a well understood algorithm (e.g. long multiplication) in a more efficient way, (often due to particular problem properties, e.g. the technique of multiplying any number by 11), even if it is difficult to see this at first. Other techniques work by making use of less widely understood mathematical laws, (e.g. algebra, quadratic equations, modular or 'clock' arithmetic, etc.). In either case, it is not necessary to know why a technique works to be able to use it, (much like you don't need to know how a car works to be able to drive one). It is for this reason, as well as to make the previous chapters more immediately usable, that the description of why each technique works has been omitted.
However, for those that are curious and want to investigate further, this chapter describes why many of the Vedic mathematics techniques work. Remember that some of the descriptions below will require knowledge of areas of mathematics that you may not be familiar with. Hopefully this will give you the impetus to investigate these areas and expand your mathematical knowledge, (this is a very rewarding way to discover new aspects of a subject). However even if this is not the case, you can (and should) still use the techniques and be happy in the knowledge that even if you don't know how the techniques work, they will still improve your numerical and arithmetic skills.
Think of this section as an appendix, useful for further study, but not essential to the understanding of the main theme of the book.

## Multiplication Techniques

### Multiplying by 11

When multiplying by 11 using long multiplication a pattern to the working out can be discovered e.g.

 46     876     4386      432672
11x     11x      11x         11x
--     ---     ----     -------
46     876     4386      432672
460+   8760+   43860+    4326720+
---    ----    -----    --------
506    9636    48246     4759392
---    ----    -----    --------


You can see that in the addition section of each long multiplication above, each column apart from the first and last is the sum of the original digit in the column and the next one (to the right). Once you know this you can just write down the result of multiplying any number by 11.
Working from right to left:

1. Write the rightmost digit down.
2. Add each pair of digits and write the result down right to left (carrying digits where necessary).
3. Finally write down the left most digit.

e.g.

• Multiply 712x11
${\displaystyle {\begin{matrix}&7&&1&&2\\&\swarrow \searrow &+&\swarrow \searrow &+&\swarrow \searrow \\7&&8&&3&&2\end{matrix}}}$


712x11=7832

The reason for working from right to left instead of the more usual left to right is so any carries can be added in as you go along. e.g.

• Multiply 8738x11
${\displaystyle {\begin{matrix}&8&&7&&3&&8\\&\swarrow \searrow &+&\swarrow \searrow &+&\swarrow \searrow &+&\swarrow \searrow \\9&\leftarrow _{1}&6&\leftarrow _{1}&1&\leftarrow _{1}&1&&8\end{matrix}}}$


8738x11=96118

### Multiplying numbers close to a power of 10

In the techniques section it is shown that the Vertically and Crosswise sutra can be used to easily multiply numbers that are close to 100. It is then shown that the same technique can be used to multiply any numbers near a power of 10, and that in fact the general technique will work for any numbers near any base, the key factor being that the technique is useful if the initial subtractions result in numbers that are easier to multiply. To understand why this technique works, you need a basic understanding of algebra, and quadratic equations.

Consider two numbers A and B that are to be multiplied together and a third number X that is close to both numbers (we will call X the 'base' ). We assume that the numbers A and B are difficult to multiply and so we are looking for an easier alternative that only involves addition, subtraction and the multiplication of easier (e.g. smaller or simpler) numbers. The key is to realise that since X is close to both numbers we can generate smaller numbers (that are hopefully easier to work with) related to A and B by subtracting each from X (we will call these smaller numbers a and b). i.e.

 {\displaystyle {\begin{aligned}&a=X-A\\&b=X-B\\Thus:\\&A=X-a\quad (1)\\&B=X-b\quad (2)\end{aligned}}}


We can multiply A and B by substituting for them using equations (1) and (2) above, i.e.

 {\displaystyle {\begin{aligned}AB&=(X-a)(X-b)\\&=X^{2}-aX-bX+ab\\&=X(X-a-b)+ab\end{aligned}}}


Now we have something we can work with! You can see from the equation above that we can replace the multiplication of A and B with some subtractions of small numbers (X-a-b) a multiplication of the result of this subtraction by the 'base' number X and then the addition of a small multiplication ab. (Remember a and b are small because X is close to A and B and a=X-A, b=X-B). The only multiplication that might be difficult is the multiplication of X by the result of the initial subtraction (X-a-b), however if we choose X carefully (e.g. by making X a power of 10) we can make sure that this multiplication is simple too. With this knowledge we can now make sense of the Vertically and Crosswise multiplication technique. i.e.

${\displaystyle {\begin{matrix}A&\longrightarrow &(X-A)\\\\B&\longrightarrow &(X-B)\\\hline \quad \end{matrix}}\quad \Rightarrow \quad {\begin{matrix}A&\longrightarrow &a\\\\B&\longrightarrow &b\\\hline \quad \end{matrix}}\quad \Rightarrow \quad {\begin{matrix}A&&a\\&&\downarrow \\B&&b\\\hline &&ab\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}\quad \quad A&&a\\&\nwarrow &\\\quad \quad B&&b\\\hline (A-b)&&ab\end{matrix}}\quad \Rightarrow \quad {\begin{matrix}\quad \quad A&&a\\&&\\\quad \quad B&&b\\\hline (X-a-b)&&ab\end{matrix}}}$


Perhaps the cleverest bit is that if the base number X is an appropriate power of 10, both the multiplication of (X-a-b) by X and the subsequent addition of ab is handled automatically by the positional shift of the digits caused by appending the ab digits to the end of the (X-a-b) digits. The only potential problem left is if the product ab is equal to or larger than the base X. In this case the positional shift of the (X-a-b) digits will be one too many, so instead the leading digit of the product ab must be 'carried' and then added to the (X-a-b) value.