Salma Khan
MEC 2012-13
MATHEMATICS PROJECT ON VEDIC MATHEMATICS
SALMA KHAN
0717606
(WORD COUNT 2001)
MATHEMATIC SUBJECT ENHANCEMENT KNOWLEDGE
2012-2013
0
Salma Khan
MEC 2012-13
Early Indian mathematics is richly diverse. Indian mathematicians like
Brahmagupta gave us important concepts such as the number zero and
negative numbers. Furthermore the current system of arithmetic widely
used today comes from the Hindu Arabic number system. Indian
mathematics is, however, often overlooked. Greater focus is placed on
Greek, Egyptian and Islamic mathematics.
The focus of this paper is on Vedic mathematics, because it was a topic i
had never come across before and found interesting. Vedic mathematics
was practised by the Aryans of the Upper Indus Basin, sometime before
1000 B.C. Vedic mathematics was then rediscovered around 1911 and
1918 by Sri Bharah Krishna Tirthaji. He studied the Vedas (all knowledge)
Atharvaveda (Holy Book) for years and was able to derive 16 sutras
(aphorisms or word formulae's) and 13 sub-sutras These sutra's were
seen as ways of making the mind work naturally and helping students to
devise a method of solution.
It is my intention in this essay to investigate mathematically one of the
Sutras from Sri Bharati Krishna Tirthaji's work in detail and to look at how
it works and why it works.
The sutra I have chosen to look into in more detail is sutra two, which is
called the Nikhilam Navatas Caraman Dastah 1 , which translated into
English means ‘all from 9 and the last from 10.’
To be able to fully understand and interpret this sutra, we need to look at
numbers in general. Starting from the number one, each new digit formed
is one more than the previous number and more complex. You could
argue that 9 is the most complex number, as the next number is 10 and
has a zero. Ten symbolises a new unity, which incorporates the previous
9 numbers, hence 9 is the stage before unity is reached.
1
This sutra will be referred to as Nikhilam throughout the essay.
1
Salma Khan
MEC 2012-13
Nikhilam is useful for the multiplication of numbers, which are closer to
bases like 10, 100 and 1000. The rationale for using bases is that it is
easier to visualise and work out mentally. It also helps to avoid adding
and subtracting large numbers and uses tenth powers. The difference
between the number and the base is called the deviation, which can be
positive or negative. Table 1 provides examples of the Nikhilam deviation.
Table 1: Deviation, using the Nikhilam method
Number
Base
Number - Base
Deviation
17
10
17-10
7
6
10
6-10
-4 or ¯4
93
100
93-100
-7
139
100
139-100
39
989
1000
989-1000
-11
1021
1000
1021-1000
21
The ¯4 in the table represents another way that -4 can be written. The
bar on top of the 4 is called the Rekhank.
Before examining how this sutra can be used to work out different
multiplications, here is an simple example. of how this works for base
numbers.
Let us take the number 97 To find the deviation of this number we will
firstly concentrate on the second digit, 7, this is taken from 10 as the
sutra states ‘the last from 10.’ This gives us 3. For the first digit we will
take it away from 9, as the sutra states ‘all from nine,’ which gives us 0.
Hence, we get the answer 03.
Example One
We want to multiply 69 by 96
2
Salma Khan

MEC 2012-13
The base for these numbers will be 100 because it is the nearest
base to these numbers.

We write the numbers, one below the other
69
96

We then take the deviation of both the numbers from the base, (the
base is always subtracted from the number) which are:
69-100 = -31
96-100 = -4
This is written as:

69
-31
96
-4
The answer will have two parts, one on the left hand side (L.H.S)
and the other on the right hand side (R.H.S).
69
-31
96
-4
_______
/

For any of these examples you can start from the L.H.S or the
R.H.S because it works both ways. For the purposes of this example
we will deal with the R.H.S side first.

For the R.H.S we need the product of the two numbers. Whenever
you find the product of the two numbers, the digits that go on the
3
Salma Khan
MEC 2012-13
R.H.S must equal the number of zeros in the base because this
represents the digits in the units section and preserves place value.

For this example the number of zeros in this base are two so the
last two digits of the product will go on the R.H.S, the digit that
remains (called a carrier), will be carried over to the L.H.S and will
added to the answer we get for the R.H.S.

If you find that the product has fewer digits than the number of
zeros in the base, then a zero or zeroes are added to the left side of
the R.H.S. If you add zero to the left nothing changes. Therefore
one could argue that the zero or zeroes are added on to indicate
place value and to ensure each digit is only multiplied once. For
example, if you have a base of 1000 and the product of the R.H.S is
84 a zero is added to the left hand side to get 084.

In this example we arrive at the following, where we have placed
the 1 in a box, as it is the carrier and will be used at the end.
69
-31
96
-4
_______
/24
1

Now we will deal with the L.H.S, which is the sum of one number
with the deviation of the other.

This can be done a number of ways, as shown below.
69
-31
96
-4
(The diagonals represent
the fact that we will be
cross-adding the two
sums)
4
Salma Khan
MEC 2012-13
a) 69 + -4 = 65 (Cross adding the first diagonal)
b) 96 + -31 = 65 (Cross adding the second diagonal)

This can also be done in the opposite way.
c) (69+96) - 100 = 65 (subtract the base from the sum of the
numbers)

The remainder of 1 from the R.H.S is added to the L.H.S, which
gives us 65+1=66

We can now put the L.H.S and R.H.S side together to give us the
answer:
69
-31
96
-4
66/24
Hence, 69 x 96 = 6624
Example One
Suppose we want to multiply 8 by 7.
Then:
(1) Select the base (power of 10) which is the nearest to the numbers to
be multiplied (in our example 10 itself).
(2) Put the numbers to be multiplied, above and below on the left- hand
side of a table as:
8
7
Subtract each from the base (8-10= -2 and 7-10=-3) and write down
the remainders on the right-hand side of the table as:
5
Salma Khan
MEC 2012-13
8–2
7–3
The result of the multiplication will be written under the line “_____”. A
vertical (/) dividing line is to separate the left digit from the right digit of
the product.
(3) The right-hand digit of the product can be obtained by vertical
multiplication (-2 x -3 =6).
8–2
7–3
/6
(4) The left-hand side digit can be obtained by adding the numbers
diagonally (8 + -3= 5
or
7+ -2= 5).
8–2
7–3
5 /6
Thus, the result is 56.
Another example: Multiply 98 and 97.
To multiply 98 and 97 we follow the same steps.
(1)
The base has to be ‘100’ (power of 10 which is nearest to the
numbers).
(2)
The Sutra (all from 9 and the last from 10) is used in order to
workout the deviation (100-98= -02 and 100-97= -03) and thus
determines the numbers in the right column.
6
Salma Khan
MEC 2012-13
It is important to note that, in this particular example, the righthand side digit must be a two-digit number because our base
(100) has two zeros and for that reason we will write the deviation
as ‘-02’ and ‘-03’ and hence preserving the place value. Hence, by
multiplying vertically, we get
-02 x -03 = 06
and adding diagonally, we get
98 + -03 = 95
or
97 + -02 = 95.
Thus:
98 – 02
97 - 03
95 / 06
Thus, the result is 9506.
Yet another example, if the numbers 99999 and 99994 must be
multiplied, then the base is 10000 (power of 10) and following the same
steps:
99999 – 00001
99994 - 00006
99993 / 00006
And the result is 9999300006.
When the vertical multiplication gives a product consisting of more than
one digit, then the surplus portion of the left must be “carried” over to
the left of the dividing line and added to the right-hand result.
For multiplying 7 times 6 then
7–3
6-4
3 /12
7
Salma Khan
MEC 2012-13
In the above example ‘1’, will be carried over to the left of the dividing
line. The result will become 3 + 1 /2 so we arrive at the result 42.
The Mathematics Behind Nikhilam Multiplication Using Bases
There are two ways I will further highlight the mathematics of
Nikhilam. Firstly I will compare it to the modern day equivalent we
use today, the grid method. Secondly I will attempt to explain it
through algebra and the importance of place value.
We will be multiplying 69 by 96
Grid Method
X
60
Nikhilam Method
7
9
-3
6
-4
multiplying
vertically
____________
90
5400
810
4/2
1
6
360
54
The grid method splits the numbers into tens and units, from which
we use the partitioning rule and then add the answers we get in
each part of the grid to arrive at 6624.
By using the Nikhilam method we always vertically multiply the
R.H.S and cross-add the diagonals to get our answer to the L.H.S. If
you look at how the diagonals are positioned, this is where the
multiplication sign originated.
The 1 in the box represents the 1
that has been carried across to the L.H.S.
8
Salma Khan
MEC 2012-13
As you can see the main differences are that the grid method uses a
partitioning and addition rule, which preserves place value. The
Nikhilam uses a vertical multiplication and cross-add diagonally
method and by ensuring the base number zeros is same as the
number of digits obtained by multiplying vertically, the place value
is preserved. The two methods vary, but both lead us to the same
answer. It is common wisdom that the grid method when used
correctly provides the correct answer. So this points to the accuracy
of the Nikhilam method.
Now I will use algebra and a example to show how the Nikhilam
method works.
Example we will use is 98 x 97
Let a and b be the deviations, which are less than the base (100).
Hence, we get the following expression
(100-a) (100-b)
(100-02) (100-03)
We will multiply it out to get
=> 10000 - 100b - 100a + ab
=>10000-200-300+6
This can be factorised to
=> 10000 - 100 (b+a) + ab
=> 10000-100 (2+3) +6
The ab represents the product we
(The 6 represents the
get on the R.H.S of our Nikhilam
unit place value, 06)
multiplication and the unit place.
9
Salma Khan
MEC 2012-13
We still have
10000 - 100 (b+a)
10000-100 (2+3)
This can be factorised further to
=> (100 (100 - (b+a))
=> (100 (100- (2+3))
The 100 points to the
so the hundredth place
fact that multiplying by
value
100 will make the answer
=100 (95)
100 times bigger, as well as
=9500
the place value being changed
by a factor of 100
Hence giving us the answer:
9500
+ 06
9506
Thus, we have shown algebraically why the Nikhilam method of
multiplication works and where the numbers come from.
Vedic mathematics is seen by some as a useful and accurate tool and the
way forward for mathematics. On the Vedic Math-E-Magic (2008) website
it states:
"Vedic Mathematics is far more systematic, simplified and unified
than the conventional system. It is a mental tool for calculation that
encourages the development and use of intuition and innovation,
while giving the student a lot of flexibility, fun and satisfaction"
10
Salma Khan
MEC 2012-13
The website goes on further to say that it is a powerful checking tool for
students and saves time in examinations.
Joseph Howse in his book Maths or Magic believes that Vedic mathematics
is a easier and quicker way to perform sums then the traditional way used
in schools. He believes that some of our greatest findings in mathematics
has come from the teachings of Vedic mathematics, notably the
multiplication sign, that I used in the Nikhilam multiplication as well as
"The greatest Vedic invention, the nought, without which most
calculations would be impossible". (Howse, 1976, Preface)
However, there are some people that would criticise the whole concept of
Vedic mathematics and question its validity as Vedic. On a website, Open
features the following was noted by Bal (2010)
......"but
the
evidence
does
nothing
to
support
it.
The
16 sutras expounded by Tirthaji do not appear in any known edition
of the Atharvaveda Veda"
He goes on to say that although this type of multiplication helps with
making addition and multiplication faster. Students should really grasp
the
concept
of
multiplication
and
how
it
works,
as
well
as
understand "how it is in essence an act of repeated addition" Bal (2010).
Kandasamy and Smarandache (2006) provide this commentary on Vedic
mathematics
"This obsession with the 'Vedic' results from the fundamentalist
Hindu organizations need to claim their identity as Aryan (and
hence Caucasian origin) and hence superior to the rest of the native
inhabitants of India" (Kandasamy and Smarandache, 2006, p. 6)
They also go on to refute the claim that Vedic mathematics is derived
from the Atharvaveda and state it is the more the imagination of one
man's take on mathematics.
11
Salma Khan
MEC 2012-13
Nonetheless, one could argue that, Vedic mathematics (as I think it is
mathematics not magic!) has millions of followers around the world who
believe in its fast, reliable and easy to visualise techniques.
However, as useful as it may be, will it become ubiquitous? Or will it
remain a largely Indian phenomenon?
Despite the speed with which calculations can be done the mathematics
behind it is arguably more complex and hard to follow. The grid method
that is widely used today, by comparison, goes through each step and is
self-explanatory. Also the Nikhilam method only deals with numbers near
bases of 10, 100 and 1000, whereas normal multiplication can be used
with any number.
The future take up of Vedic mathematics remains to be seen. It does,
however, represent an alternative way to visualise and engage in
mathematics – one that is being used by many in one of the most
populous countries and emerging economies in the world. Who knows,
this ancient technique may indeed engage the masses and the next
generation of mathematicians.
12
Salma Khan
MEC 2012-13
References

Bal. S (2010). Open Features- The Fraud of Vedic Maths. [Online].
[Accessed 7 June 2013]. Available at:
http://www.openthemagazine.com/article/art-culture/the-fraud-of-vedic-maths

Dani, S (2011). Understanding ancient Indian mathematics.
[Online]. [Accessed 7 June 2013]. Available at:
http://www.thehindu.com/sci-tech/science/understanding-ancientindian-mathematics/article2747006.ece

Das, S and Sadasivan, M (2013). What are Vedas? [Online].
[Accessed 7 June 2013]. Available at:
http://hinduism.about.com/cs/vedasvedanta/a/aa120103a.htm

Howse, J (1976). Maths or Magic. First edition. London: Watkins
Publishing
13
Salma Khan

MEC 2012-13
Kandasamy, V and Smarandache, F (2006). Vedic Mathematics ‘Vedic’ or ‘Mathematics’: A Fuzzy & Neutrosophic Analysis. First
edition. Los Angeles: Automaton.

Vedic Mathematics-Methods (No date) [Online]. [Accessed 7 June
2013]. Available at:
http://www.vedamu.org/Veda/1795$Vedic_Mathematics_Methods.p
df

Vedic Math-E-Magic (2008). What is Vedic Mathematics. [Online].
[Accessed 20 June 2013]. Available at:
http://www.vedicmathsonline.com/index.html

Williams, K (2003). The System of Vedic Mathematics-A
Comparison. [Online]. [Accessed 7 June 2013]. Available at:
http://vedicmaths.org/Free%20Resources/Articles/SystemVM/syste
m of Vedic mathematics.asp
14