In this article, we shall discuss the Vedic Math sutra "vertically and crosswise". This is the General Formula which is applicable to all the cases of multiplication. You will found this method very useful in division also that we shall discuss later. In the previous sutras, you subtracted crosswise; now you will multiply crosswise.
A. Multiplication of two digit numbers
For ab x uv
The numbers are written from left to right.
Now multiply:
1) vertically (a x u)
2) crosswise in both directions and add (a x v) + (b x u)
3) vertically (b x v)
The answer has the form: au | av + bu | bv
Following is the example of two-digit multiplication will make this clear:
12 x 23
1 2
2 3
--------------
2 | 3 + 4 | 6
= 2 | 7 | 6
= 276
The previous examples involved no carry figures, so let us consider this case in the next example.
64 x 93
6 4
9 3
-----------------
54 | 18 + 36 | 12
=54 | 54 | 12
=54 | (54 + 1) | 2 (Carry over the 1)
=54 | 55 | 2
=(54 + 5) | 5 | 2 (Carry over the 5)
=59 | 5 | 2
64 x 93 =5952
The Algebraic Explaination is:
Let the two 2 digit numbers be (ax+b) and (ux+v). Note that x = 10. Now consider the product
(ax + b) (ux + v)
= au. x2 + avx + bux + b.v
= au. x2 + (av + bu)x + b.v
The first term i. e. the coefficient of x2 is got by vertical multiplication of a and u
The middle term i. e. the coefficient of x is obtained by the cross-wise multiplication of a and v and of b and u and the addition of the two products
The independent term is arrived at by vertical multiplication of 'b' and 'v'
B. Multiplying three digit numbers by two digit numbers
For abc x uv
The middle parts are obtained by adding the crosswise multiplications for a and b with v and u respectively, then b and c with v and u respectively. The outer parts are vertical multiplication of a with u on the left, c with v on the right.
Multiply:
1) vertically (a x u)
2) crosswise in both directions and add (a x v) + (b x u)
3) crosswise in both directions and add (b x v) + (c x u)
4) vertically (c x v)
The answer has the form: au | av + bu | bv + cu | cv
Let us take an example: 236 x 53
2 3 6
5 3
-------------------------
10 | 6+15 | 9+30 | 18
= 10 | 21 | 39 | 18
Work right to left carry over
= 10 | 21 | (39+1) | 8
= 10 | 21 | 40 | 8
= 10 | (21+4) | 0 | 8
= 10 | 25 | 0 | 8
= (10+2) | 5 | 0 | 8
= 12 | 5 | 0 | 8
236 x 53 = 12508
C. Multiplying four digit by two digit numbers
For abcd x uv
Multiply:
1) vertically (a x u)
2) crosswise in both directions and add (a x v) + (b x u)
3) crosswise in both directions and add (b x v) + (c x u)
4) crosswise in both directions and add (c x v) + (d x u)
5) vertically (d x v)
The answer has the form: au | av + bu | bv + cu | cv + du | dv
Example: 1348 x 74
1 3 4 8
7 4
-------------------------------
7 | 4+21 | 12+28 | 16+56 | 32
= 7 | 25 | 40 | 72 | 32
= 7 | 25 | 40 | (72+3) | 2
= 7 | 25 | 40 | 75 | 2
= 7 | 25 | (40+7) | 5 | 2
= 7 | 25 | 47 | 5 | 2
= 7 | (25+4) | 7 | 5 | 2
= 7 | 29 | 7 | 5 | 2
= (7+2) | 9 | 7 | 5 | 2
= 9 | 9 | 7 | 5 | 2
1348 x 74 = 99752
D. Multiplying five digit by two digit numbers
For abcde x uv
Multiply:
1) vertically (a x u)
2) crosswise in both directions and add (a x v) + (b x u)
3) crosswise in both directions and add (b x v) + (c x u)
4) crosswise in both directions and add (c x v) + (d x u)
5) crosswise in both directions and add (d x v) + (e x u)
6) vertically (e x v)
The answer has the form: au | av + bu | bv + cu | cv + du | dv + eu | ev
Example: 12345 x 58
1 2 3 4 5
5 8
---------------------------------------
5 | 8+10 | 16+15 | 24+20 | 32+25 | 40
= 5 | 18 | 31 | 44 | 57 | 40
= 5 | 18 | 31 | 44 | (57+4) | 0
= 5 | 18 | 31 | 44 | 61 | 0
= 5 | 18 | 31 | (44+6) | 1 | 0
= 5 | 18 | 31 | 50 | 1 | 0
= 5 | 18 | (31+5) | 0 | 1 | 0
= 5 | 18 | 36 | 0 | 1 | 0
= 5 |(18+3) | 6 | 0 | 1 | 0
= 5 | 21 | 6 | 0 | 1 | 0
= (5+2) | 1 | 6 | 0 | 1 | 0
= 7 | 1 | 6 | 0 | 1 | 0
12345 x 58 = 716010
I hope it will help in quick multiplication. If there is any query, please post it here. In next article, we shall continue to discuss this technique for multiplying three digits by three digits numbers.
If you like the article, you may contribute by:
A. Multiplication of two digit numbers
For ab x uv
The numbers are written from left to right.
Now multiply:
1) vertically (a x u)
2) crosswise in both directions and add (a x v) + (b x u)
3) vertically (b x v)
The answer has the form: au | av + bu | bv
Following is the example of two-digit multiplication will make this clear:
12 x 23
1 2
2 3
--------------
2 | 3 + 4 | 6
= 2 | 7 | 6
= 276
The previous examples involved no carry figures, so let us consider this case in the next example.
64 x 93
6 4
9 3
-----------------
54 | 18 + 36 | 12
=54 | 54 | 12
=54 | (54 + 1) | 2 (Carry over the 1)
=54 | 55 | 2
=(54 + 5) | 5 | 2 (Carry over the 5)
=59 | 5 | 2
64 x 93 =5952
The Algebraic Explaination is:
Let the two 2 digit numbers be (ax+b) and (ux+v). Note that x = 10. Now consider the product
(ax + b) (ux + v)
= au. x2 + avx + bux + b.v
= au. x2 + (av + bu)x + b.v
The first term i. e. the coefficient of x2 is got by vertical multiplication of a and u
The middle term i. e. the coefficient of x is obtained by the cross-wise multiplication of a and v and of b and u and the addition of the two products
The independent term is arrived at by vertical multiplication of 'b' and 'v'
B. Multiplying three digit numbers by two digit numbers
For abc x uv
The middle parts are obtained by adding the crosswise multiplications for a and b with v and u respectively, then b and c with v and u respectively. The outer parts are vertical multiplication of a with u on the left, c with v on the right.
Multiply:
1) vertically (a x u)
2) crosswise in both directions and add (a x v) + (b x u)
3) crosswise in both directions and add (b x v) + (c x u)
4) vertically (c x v)
The answer has the form: au | av + bu | bv + cu | cv
Let us take an example: 236 x 53
2 3 6
5 3
-------------------------
10 | 6+15 | 9+30 | 18
= 10 | 21 | 39 | 18
Work right to left carry over
= 10 | 21 | (39+1) | 8
= 10 | 21 | 40 | 8
= 10 | (21+4) | 0 | 8
= 10 | 25 | 0 | 8
= (10+2) | 5 | 0 | 8
= 12 | 5 | 0 | 8
236 x 53 = 12508
C. Multiplying four digit by two digit numbers
For abcd x uv
1) vertically (a x u)
2) crosswise in both directions and add (a x v) + (b x u)
3) crosswise in both directions and add (b x v) + (c x u)
4) crosswise in both directions and add (c x v) + (d x u)
5) vertically (d x v)
The answer has the form: au | av + bu | bv + cu | cv + du | dv
Example: 1348 x 74
1 3 4 8
7 4
-------------------------------
7 | 4+21 | 12+28 | 16+56 | 32
= 7 | 25 | 40 | 72 | 32
= 7 | 25 | 40 | (72+3) | 2
= 7 | 25 | 40 | 75 | 2
= 7 | 25 | (40+7) | 5 | 2
= 7 | 25 | 47 | 5 | 2
= 7 | (25+4) | 7 | 5 | 2
= 7 | 29 | 7 | 5 | 2
= (7+2) | 9 | 7 | 5 | 2
= 9 | 9 | 7 | 5 | 2
1348 x 74 = 99752
D. Multiplying five digit by two digit numbers
For abcde x uv
1) vertically (a x u)
2) crosswise in both directions and add (a x v) + (b x u)
3) crosswise in both directions and add (b x v) + (c x u)
4) crosswise in both directions and add (c x v) + (d x u)
5) crosswise in both directions and add (d x v) + (e x u)
6) vertically (e x v)
The answer has the form: au | av + bu | bv + cu | cv + du | dv + eu | ev
Example: 12345 x 58
1 2 3 4 5
5 8
---------------------------------------
5 | 8+10 | 16+15 | 24+20 | 32+25 | 40
= 5 | 18 | 31 | 44 | 57 | 40
= 5 | 18 | 31 | 44 | (57+4) | 0
= 5 | 18 | 31 | 44 | 61 | 0
= 5 | 18 | 31 | (44+6) | 1 | 0
= 5 | 18 | 31 | 50 | 1 | 0
= 5 | 18 | (31+5) | 0 | 1 | 0
= 5 | 18 | 36 | 0 | 1 | 0
= 5 |(18+3) | 6 | 0 | 1 | 0
= 5 | 21 | 6 | 0 | 1 | 0
= (5+2) | 1 | 6 | 0 | 1 | 0
= 7 | 1 | 6 | 0 | 1 | 0
12345 x 58 = 716010
I hope it will help in quick multiplication. If there is any query, please post it here. In next article, we shall continue to discuss this technique for multiplying three digits by three digits numbers.
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