before studying the principle and
its application discussed here.
Carrying out tedious and cumbersome
arithmetical operations, easily, speedily
and in some cases executing them mentally
comes under the realm of High Speed maths.
The methods of arithmetical operations
applied are totally unconventional.
People who are deeply rooted in the
conventional methods, may find it difficult,
at first reading, to understand the methods.
But, these methods, based on the Sutras
(aphorisms or Formulas) of Ancient Indian
Vedic Mathematics are simple and easy to understand,
remember and apply, even by little children.
Here we deal with the principles appled to
Multiplication. For the application to other
topics of math, go to
Vedic Maths eBook. for more details, see
near the bottom of the page.
The Urdhva Tiryak Sutra (meaning : Vertically and cross-wise)
This is a general Formula applicable
to all cases of multiplication.
Using this principle, we can find
the product of two numbers easily.
Multiply vertically and crosswise
to get the digits of the product.
Examples will clarify the method.
Before seeing the examples, let us see
the formula for finding the product of
two digit numbers.
Let us say the two digits of the first number
be 'a' (tens' digit) and 'b'(units' digit).
And those of the second number be
'p' (tens' digit) and 'q'(units' digit).
Write the digits of the two numbers
one below the other as follows.
a b
p q
The product of these numbers has three parts
which are given below seperated by '/'.
a b
p q
------
ap/(aq+pb)/bq
------
'ap' is the Hundreds' part which is
the vertical product of the first column.
(aq+pb) is the Tens' part which is the
sum of the cross-wise products 'aq' and 'pb'.
'bq' is the units' part which is
the vertical product of the second column.
Let us see the method by examples.
Example 1 of Vedic Mathematics
To find 21 x 13
21
13
-----
2x1/2x3+1x1/1x3
= 2/7/3
21 x 13 = 273
The product has three parts and each
part is seperated by slash (/).
First part of the product
= product of first vertical digits
= 2x1 = 2
Second part of the product
= sum of the cross-wise products
= 2x3+1x1 = 6+1 = 7
Third part of the product
= product of second vertical digits
= 1x3 = 3
Thus 21 x 13 = 273.
Example 2 of Vedic Mathematics
To find 41 x 51
41
51
----
4x5/4x1+5x1/1x1
=20/9/1
41x51 = 2091
The product has three parts and each
part is seperated by slash (/).
First part of the product
= product of first vertical digits
= 4x5 = 20
Second part of the product
= sum of the cross-wise products
= 4x1+5x1 = 4+5 = 9
Third part of the product
= product of second vertical digits
= 1x1 = 1
The product has three parts and each
part is seperated by slash (/).
First part of the product
= product of first vertical digits
= 2x4 = 8
Second part of the product
= sum of the cross-wise products
= 2x2+4x3 = 4+12 = 16
This is a two digit number.
Units digit (6) is retained and tens' digit (1)
is carried over to the immediate left place.
See that in 16, 1 is written in small letters
indicating its carry over to the immediate
left place and 6 is written normally
indicating it is retained in its place.
Third part of the product
= product of second vertical digits
= 3x2 = 6
Thus 23 x 42 = 966.
To explain the procedure clearly,
so many steps are shown.
In practice we can do the calculations mentally
and write the answer in fewer steps as follows.
23
42
----
8/16/6
=9/6/6 (worked out from right)
23 x 42 = 966.
The product has three parts and each
part is seperated by slash (/).
First part of the product
(before carry over from second part)
= product of first vertical digits
= 3x2 = 6
Second part of the product
(before carry over from third part)
= sum of the cross-wise products
= 3x9+2x7 = 27+14
Third part of the product
= product of second vertical digits
= 7x9 = 63
This is a two digit number.
Units digit (3) is retained and tens' digit (6)
is carried over to the immediate left place.
See that in 63, 6 is written in small letters
indicating its carry over to the immediate
left place and 3 is written normally
indicating it is retained in its place.
Now,
Second part of the product
(after carry over from third part)
= 27+14+6 = 47
This is a two digit number.
Units digit (7) is retained and tens' digit (4)
is carried over to the immediate left place.
See that in 47, 4 is written in small letters
indicating its carry over to the immediate
left place and 7 is written normally
indicating it is retained in its place.
Now,
First part of the product
(after carry over from second part)
= 6+4 = 10
Thus 37 x 29 = 1073.
To explain the procedure clearly,
so many steps are shown.
In practice we can do the calculations mentally
and write the answer in fewer steps as follows.
37
29
----
6/41/63
=10/7/3 (worked out from right)
37 x 29 = 1073.
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