Multiplication by Nikhilam Sutra
Suppose you need 8 x 7
8 is 2 below 10 (the base)
and 7 is 3 below 10.
Think of it like this:
8 - 2
X 7 - 3
--
5 6
The answer is 56. This is how you get it. Multiply 2 x 3 = 6 the last figure of the
answer.8 - 3 = 5 the first figure of the answer.
But you already know 7x8=56? Okay smarty man genius!!! Give me 96*98!!
Same method, this time take the base as 100.
96 - 4
98 - 2
---
94 08
4 x 2= 08 -The last part;
96 - 2 = 94 -Which is the first part.
More Examples:
97 -3 102 2 888 -112
X 98 -2 X 104 4 X997 -003
95,06 106,08 885,336
When the number is greater thclassan the base, add instead of subtracting as in the second
example. (102 + 104)
For cases when the numbers are closer to the middle of the base, use 50 as the base and
divide the result by 2.
48 -2 (base/2 = 50)
X46 -4
---
44, 08 => 22,08
In the example above, you can achieve the same result by doing 46 - 2 =44.
The same 44 = 48 + 46 -50 (the base)! This is the most striking feature of the Vedic
system, how the whole system is beautifully interrelated and unified.
More multiplication Sutras:
Ekadhikena Purvena
An interesting application of this sutra is in computing squares of numbers ending in
five. Consider:
35x35 = (3x(3+1)) 25 = 12,25
The latter portion is always 25 and the previous portion is multiplied by one more than
itself (3 by 4) resulting in the answer 1225.
Similarly 852 = 7225 because 8 x 9 = 72
Urdhva-tiryagbhyam (Vertically and cross-wise)
This sutra applies to all cases of multiplication of one large number by another large
number.
2 3
X 2 1
-------
4 8 3
There are 3 steps to this:
A) Multiply vertically on the left:
2 x 2 = 4. This gives the first figure of the answer.
b) Multiply crosswise and add: 2 x 3 + 1 x 2 = 8
This gives the middle figure.
c) Multiply vertically on the right: 1 x 3 = 3
This gives the last figure of the answer. That's all there is to it !
21 x 26 = 546
2 1
2 6
---
4 4 6
1
---
5 4 6
P.S. When get a 2-figure number,
( 2 x 6 + 1 x 2= 14) the number is carried over to the left
Vedic Mathemagic:
1203579 (This is possible with numbers like 999... just put x-1 followed by the difference
between 1203579 and 9999999) -----x-----
1203578 8796421
How long would this take even on a calculator??
Conversion of a vulgar fraction (1/19) to a decimal would take say 18, 28 to 42 steps of
cumbersome work! Here's the short cut!
Start with 1 and keep doubling it till 18'th digit when it starts repeating.
052631578 947368421
Heres an extra shortcut to this.
052631578
947368421 ------x-----
999999999
The first 9 digits complement the other 9! so you can do one half of the computation
reducing your work by 50% !
Say it with me:
Vedic Maths RULEZ!!@!
Multiplying a number by 11.
To multiply any 2-figure number by 11 we just put the total of the two figures between the
2 figures.
26 x 11 = 286
Notice that the outer figures in 286 are the 2& 6.
And the middle figure is just 2 and 6 added up.
So 72 x 11 = 792
The simplicity of Vedic Mathematics 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. You'll never rely on
calculators to do arithmetic once you have the techniques down, and it'll stay with you
for life.. no batteries needed. Cubing made easy 13 3
1 3 9 27
6 18 ---------x----------
2 1 9 7
1..3...9....27
Some interesting facts about
recurring decimals
A denominator containing only 2 or 5 as factors
gives us a non-recurring decimal function. For example,
1/2 = 0.5
1/4 = 1/(2*2) = 0.25
1/8 = 1/ (2*2*2) = 0.125
1/5 = 0.2
1/25 = 1/(5*5) = 0.04 and so on
Denominators containing only 3, 7, 11 or higher prime numbers as factor
and not even a single 2 or 5 given recurring decimals. For example
1/3 = 0.3
1/7 = 0.142857
1/9 = 0.1
1/11 = 0.09
1/13 = 0.076923
1/17 = 0.05882352/94117647 and so on
A denominator with factors partly of the first type and party of the
second type, give us a mixed, i,e. party recurring and party non-recurring decimal. For
example,
1/6 = 1/(2*3) = 0.16
1/15 = 1/ (3*5) = 0.06
1/18 = 1/ (2*9) = 0.05
1/22 = 1/ (2*11) = 0.045 and so on.
Denominator containing all digits 9.
It is very easy to find the decimal equivalent of a
faction with denominator containing all nines. It is recurring decimal with digits in the
numerator. For example,
1/9 = 0.1
2/9 = 0.2
5/9 = 0.5
23/99 = 0.23
50/99 = 0.50
112/999 = 0.112 and so on.
Vedic mathematics sutras for recurring decimals:
The following sutras along with the Ekanyunen sutra
are used remember the digits of the recurring decimals, in cases of denominators 7, 13 and
17.
Kevalaih Saptakam Gunyaat, or, in the case of seven the
multiplicand should be 143.
Kulau Kshudasasaih, or in the case of the 13 the
multiplicand should be 077.
Kamse Kshaamadaaha-khalairmalaih, or, in
the case of 17 the multiplicand should be 05882353. ( By the way, the literal meaning of
this result is "In king Kamsa’s reign famine, and unhygienic conditions
prevailed." - -not immediately obvious what it had to do with Mathematics. These
multiple meanings of these sutras were one of the reasons why some of the early
translations of Vedas missed discourses on vedaangas.)
These are used to correctly identify first half of a recurring decimal
number, and then applying Ekanyuna to arrive at the complete answer mechanically.
Consider for example the following visual computations.
1/7 = 143 * 999/999999 = 132857/999999 = 0.42857
1/13 = 077*999/999999 = 076923/999999 = 0.076923
1/17 = 05882353 * 99999999/9999999999999999 = 0.0588235294117647
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