Free Resources
The System of Vedic Mathematics
Home Page
Free Resources
Books
Book Sections
Articles
Miscellaneous
Videos
Sutras
Web Sites
Book Purchasing
Guide
Programs
THE SYSTEM OF VEDIC MATHEMATICS
- A COMPARISON
Kenneth R. Williams
Since the publication of the book Vedic Mathematics1 by Sri Bharati
Krsna Tirthaji in 1965 a great deal has been said about Vedic
Mathematics, especially in the last few years. There has been
controversy about the validity of the title as the Sutras on which the
contents of the book is based have not yet been located in Vedic
scriptures. In this article, though, we will be mainly concerned with Vedic
Mathematics as a system, compared to the system normally taught. So
by Vedic Mathematics we here mean the system outlined by Bharati
Krsna Tirthaji in his book.
The book mainly deals with elementary arithmetic, algebra and some
calculus and geometry. It covers quite a wide range though, and
includes many new methods, some of which have been noted in the
past. But here they are put together as a system. What is this ‘system’
and how does it compare with what we currently have? What
advantages does it offer?
The first thing to note is that the book is an introduction; it is not
supposed to be a full exposition of Vedic Mathematics and was written
in the author’s final years. I was given a copy of the book in 1971 and
was amazed at its style and content. Some of the claims made seemed
preposterous, but, given the integrity of the author and the originality of
the material in the book, it seemed reasonable to study the work
carefully first to see if there could be anything in it. Various extensions of
the techniques became obvious and also applications in other areas of
mathematics; and a clear and beautiful system gradually emerged.
Let us begin by comparing the Vedic and traditional methods for
obtaining the product of two 2-figure numbers.
MULTIPLICATION
Using the Sutra Vertically and Crosswise we can multiply any two
numbers together in one line using this general Vedic method. Suppose
we want to multiply 45 by 63:
45
63
2835
4 1
a) We write the numbers out as shown and begin by multiplying
vertically: 5×3=15, so we put down 5 and carry the 1 as shown.
b) Then we multiply crosswise and add the two results: 4×3 + 5×6 = 42.
To this we add the carried 1, so we put down 3 and carry 4 to the left.
c) Finally we multiply vertically on the left: 4×6 = 24 and adding the
carried 4 we get 28 which we put down.
The simple pattern used makes the method easy to remember and it is
very satisfying to get the answer in one line. It is also easy to see why it
works: the three steps find the number of units, the number of tens and
the number of hundreds in the answer.
Let us compare this with the usual method of ‘long multiplication’.
2
2
8
4
5
6
3
1
3
5
7
0
0
3
5
Here we have two lines of working before we get the answer, we have to
remember to put the zero down on the right on the second line and there
is no memorable pattern as there is with the Vedic method.
LEFT TO RIGHT
The Vedic method though, has many other advantages. For example it
can also be carried out from left to right.
Let us find 45 × 63 from the left:
4
5
2
8
4
6
3
3
5
2
a) Vertically on the left, 4×6 = 24, write down 2 and carry 4 to the right.
b) Crosswise we get 4×3 + 5×6 = 42 (as before), add the carried 4, as
40, to get 82, write down 8 and carry the 2.
c) Finally, vertically on the right 5×3 = 15, add the carried 2, as 20, to
get 35 which we write down.
We always add a zero to the carried figure as shown because the first
product here, for example, is really 40×60 = 2400 and the 400 is 40
tens. So when we are gathering up the tens we add on 40 more. This
does not seem so strange when you realise that a similar thing occurs
when calculating from right to left: in the first calculation above the first
vertical product (on the right) was 15, and although the 1 in 15 stands
for 10 it was counted as one unit in the next column.
These Vedic techniques can be extended to products of numbers of any
size. The second method, from left to right, is useful for mental
calculation because we write and pronounce numbers from left to right
and so it is easier to get our answers the same way. Another advantage
of calculating from left to right is that we may only want the first one, two
or three figures of an answer, but working from the right we must do the
whole sum and get the most significant figure last. In the Vedic system
all arithmetic operations can be carried out from left to right and this
means we can combine operations: obtain two products and sum them
for example, or obtain two squares, sum them and take the square root.
So we get answers digit by digit from left to right. We can extend this
further to the calculation of sines, cosines, tangents and their inverses
and the solution of polynomial and transcendental equations (which
cannot be calculated from the right as there is no last figure).2
The same vertical and crosswise method can be used for algebraic
multiplications. For example (2x + 5)(3x + 1):
2x + 5
3x + 1
6x2 + 17x + 5
Either direction will do. From the left we have:
vertically: 2x × 3x = 6x2,
crosswise: 2x × 1 + 5 × 3x = 17x,
and vertically: 5 × 1 = 5.
The conventional system does not use the same method for multiplying
algebraic expressions as it uses for arithmetic products. It does
something completely different: as shown below by the curved lines, we
multiply the first by the first, the last by the last, the inner pair and the
outer pair.
This works nicely but is not related to the way the conventional system
multiplies numbers. In the Vedic system however we find that once we
know how to multiply, divide, take a square root etc. of numbers, we can
use the same technique with polynomial expressions. This close link
between algebra and arithmetic in the Vedic system means the practice
of arithmetic builds up skills in algebra too.
This shows that the Vedic system is simpler and more unified than the
modern methods, which seem to be a hotchpotch of techniques that
have come to be the standard ones over the years, but have little real
coherence.
This simplicity also suggests that this method could have been known in
the past (it is in fact given by Leonardo of Pisa in his book "Liber Abaci")
and could well have been used by the enlightened mathematicians of
Vedic times. If a place-value system was then used to represent
numbers the multiplication and division techniques given by Bharati
Krsna Tirthaji could very well be the same as those used in the Vedic
civilisation.
DIVISION
Unlike the conventional multiplication technique the Vedic method can
be simply reversed to give us a one line division method.
Suppose we want to divide 2835 by 63. This means we want to find a
number which, when multiplied by 63 gives 2835, or in other words we
want a and b in the multiplication sum below:
a
b
6
3
2
8
3
5
a) Since we know that the vertical product on the left must account for
the 28 on the left of 2835, or most of it, we see that a must be 4.
This accounts for 2400 of the 2800 and so there is a remainder of 400.
A subscript 4 is therefore placed as shown.
b) Next we look at the crosswise step: this must account for the 43 (43),
or most of it. One of the products gives: 4×3 = 12 and this can be taken
from the 43 to leave 31 for the other product, b×6. Clearly b is 5. The
cross-product is therefore 42 and so there is a remainder of 1, which we
carry to the right.
Now the right vertical product is 5×3=15, and as this is exactly what we
have left there is no remainder and the answer is exactly 45.
In fact Bharati Krsna Tirthaji gives a different arrangement for the
general division method which further simplifies the work.
This can be presented to children in the form of a game or puzzle once
2-figure multiplication is established.
By contrast the conventional long division method, shown below, is
cumbersome and slow and is not obviously related to long multiplication.
We practically need the multiplication table for 63, whereas with the
Vedic method we just divide by 6, the first figure of the divisor. Even with
long divisors we need only divide by the first digit in the Vedic system.
45
63)2 8 3 5
252
315
315
000
This method is easily developed for division of any number by any other
number to any number of significant figures, and even dividing
polynomial expressions. Furthermore the multiplication method
described here simplifies when the numbers being multiplied are the
same, i.e. for squaring numbers. And this squaring method can also be
easily reversed to provide one line square roots: easy to do, easy to
understand.
These illustrations show that not only does Vedic Mathematics have a
system but it is more unified than the conventional system. The same
coherence in the Vedic system can be demonstrated in other areas of
mathematics.
FRACTIONS
For example, addition and subtraction of fractions, which is normally
very difficult for the children to remember and understand can be done
immediately with Vertically and Crosswise: using the simple pattern and
putting the answer straight down. It is also easy to understand. This
gives a unity to the four operations of addition, subtraction, multiplication
and division of fractions.
SPECIAL CASES
The conventional system uses few special cases. For example if you
needed 56×10 you would not use long multiplication, you would just put
the zero on the 56. But the conventional system does not go nearly far
enough: it tends to stick to one method rigidly.
If we needed 88×98, for example, the conventional system uses the
same long multiplication method, and it is not a particularly attractive
sum with those big digits: eights and nine. But these numbers are close
to 100 and you might expect there to be an easy way to get their
product. In the Vedic system there is.
You may be familiar with the Vedic method.
-12
-2
88 × 98 = 86/24
It is not usually set out like this, but seeing the numbers are close to 100
we naturally think of the deficiencies (12 and 2) of the numbers from
100, which here are placed over the numbers (on the flag), the minus
signs indicating that the given numbers are below 100.
The left-hand part of the answer, 86, is found by taking one of the
deficiencies from the other number: 88 – 2 = 86 or 98 – 12 = 86
(whichever you like),
and the 24 is simply the product of the deficiencies: 12 × 2 = 24.
It could hardly be easier.
This technique is capable of an extraordinary range of variations and it
can easily be explained: algebraically, arithmetically or geometrically.
There are many special cases in the Vedic system, but you do not have
to use them as the general method is always available. This adds to the
fun of doing mathematics though: having only one method, which is
always applied, for multiplication, division, solving equations etc. leads
to rigid thinking and is boring for most people. The variety of methods in
the Vedic system on the other hand gives the student a choice about
how to solve a problem and so they start to be creative: they think more
about what they are doing and start inventing their own methods.
Just as in everyday life we are faced with many challenges and each
has to be approached in its own way, similarly every mathematical
problem invites its own unique method of solution.
MENTAL MATHS
Another aspect of the Vedic system is that as the methods are so easy
and can also be carried out from left to right, calculations can be done
mentally. This gives confidence, greater mental agility and improves
memory; students become more flexible and innovative and this has an
effect on their other studies.
Years ago I tutored a fourteen year old boy who had broken his leg and
could not get to school. He was in a large comprehensive school and for
maths he was in the bottom class out of six. He noticed that I tended to
do calculations mentally and began to do the same. When I noticed that
I showed him how to multiply two 2-figures numbers in his head. He was
delighted and could soon multiply the numbers quite quickly. Then I
showed him how to multiply two 3-figure numbers. When it was
explained I gave him two numbers to multiply. After a while he gave the
answer and when I told him it was correct he was really amazed and
delighted. It was a special moment for him: he had done something that
probably no-one in his school could have done, including the teachers.
It is sometimes argued that with the advent of the electronic calculator
the Vedic methods are obsolete. The calculator has its uses but the
teaching of mathematics is about a lot more than just getting an answer.
And although the calculator can save us a lot of time we should still be
able to manage without it if necessary.
EQUATIONS
In the solution of equations we see another example of the contrast
between the Vedic and traditional systems. The children nowadays are
taught to tackle equations by writing down every line of the solution in a
rigid way. They are not challenged to find the answer mentally or to
notice special cases. This must lead to inflexible and dull thinking. But
children are naturally creative and spontaneous and in most cases are
not challenged enough.
To solve an equation like: 7x + 5 = 4x + 20
we insist that they take 5 from both sides: 7x = 4x + 15,
then take 4x from both sides: 3x = 15
and then divide by 3: x = 5
But just looking at the equation we can see that there will be 3x on the
left and 15 on the right so that x must be 5. The Vedic system
encourages us to look at the equation and let the mind just go to the
answer.
The actual working (whether mental or written), and the logic, is the
same in both cases: it is the attitude that is different.
Similarly, given the equation:
the student of Vedic
Mathematics sees immediately that the equation is linear as crossmultiplication would give x2 on each side and would spot that x = 0 is
therefore the only solution since
. The traditionally taught student
wastes a lot of time in getting x=0 and then realises (perhaps) that it was
actually obvious, but he has never been taught to take a good look at
the problem before starting to solve it.
OTHER EXAMPLES
Similarly trigonometric equations can be solved with much greater ease
using Vedic methods. In coordinate geometry and in applied
mathematics too the Vedic system gives a new unity and new and highly
efficient methods. The book Triples3 shows that the standard formulae
for rotations in a plane, finding the distance of a point from a line, finding
the angle between two lines etc. can be replaced by a single vertical and
crosswise operation for adding or subtracting Pythagorean triples. And
complex numbers can be multiplied just as triples are added and can be
divided as triples are subtracted. The square root of a complex number
can be written straight down.
One textbook shows the standard method of finding
This has 17
lines of working: we call the required result a + bi, square both sides of
the equation thus formed and equate real and imaginary parts to get a²
– b² = 15 and 2ab = 8. Substituting from the second equation into the
first leads to a quartic equation which is then factorised to give two
quadratic equations. This gives the two values of a and the
corresponding values of b can then be found, giving
But using Vedic Mathematics we simply complete the triple: 15, 8, 17
and find 15+17=32, so that the first two elements of the half-angle triple
are 32, 8 or 4, 1 (cancelling down). Since 42 + 12 = 17 (the third element
of the triple) the modulus of the complex number has been ‘squarerooted’, and the argument has been halved, so
again a one line answer.
This is
THE SUTRAS
The Vedic system uses a collection of sixteen Sutras and some subSutras. These are short memorable phrases, sometimes just a single
word. For example, Vertically and Crosswise, Proportionately and On
the Flag. They are clearly considered by Bharati Krsna Tirthaji to be an
important aspect of Vedic Mathematics. It is easy to see that a phrase
like Vertically and Crosswise could be useful for pupils; to remind them
of a specific technique or way of tackling a problem. But they raise many
questions: why are there sixteen Sutras, where do they originate from,
what are sub-Sutras? The question about the authenticity of the Sutras
cannot be answered, at least from a western point of view, by finding
some reference to them in the extant literature. Because they appear in
a book or other text does not make them valid, it just means somebody
once thought they were.
To understand the Sutras, and for them to be useful to us, it is just
necessary to see them as something we can relate to in a natural way.
My view is that the Sutras describe natural functions of mind. So their
use goes beyond the purely mathematical as they help us to know how
to use our mind to solve a particular problem.
CONCLUSION
It is not possible here to make a thorough comparison of the Vedic
system with that currently taught, but it is clear that Vedic Mathematics
has a lot to offer in education and has also opened up new areas of
research. A school course covering the National Curriculum has been
written and trialled in England. It has been available for four years and is
now being printed in India.4
Vedic Mathematics is certainly more integrated, more efficient and more
fun than conventional mathematics. It leads to greater enjoyment of
mathematics, greater flexibility of mind, increased mental agility and
brings out the creative faculty that is in all students. These have been
observed in situations where Vedic Mathematics has been taught.
Further research is needed to scientifically compare the effects of Vedic
Mathematics teaching with conventional mathematics teaching, and to
establish the nature of the Sutras and the full range of their application.
References:
1. Tirthaji, B. K., Vedic Mathematics, Motilal Banarsidass, 1965
2. Nicholas et al, Vertically and Crosswise, Inspiration Books, 1999
3. Williams, K, Triples, Inspiration Books, 1984
4. Williams and Gaskell, The Cosmic Calculator, Motilal Banarsidass,
2002
copyright to the
ACADEMY OF VEDIC MATHEMATICS
Home Page Site Map Contact us