Vertically and cross wise Sutra : It is also called as Urdhva – tiryagbhyam which is the general formula applicable to all cases of multiplication and also in the division of a large number by another large number.
(a) Multiplication of two 2 small digit numbers.
Lets take two numbers,say 15 x 23
Here start the multiplication from Right to Left
Step :1
1 5

2 3

5 x 3 =15 ......keep the last digit '5' and carry over '1'
Step 2:
1 5
\/
/\
2 3

(1 x 3 ) + (2x5) = 3+10 =13
With carry over '1' from step 1 ,we get 13+1 =14.....keep the last digit '4' and carry over '1'
Step 3:
1 5

2 3

1 x 2=2
With carry over '1' from step 2, we get 2+1=3
So our result from step 1 to step 3 = 345
Therefore, 15 x 23 =345
(b) Multiplication of two 2 large digit numbers
With the number 72 x 88 ,We do all the above three steps from (a)
7 2
 \/ 
 /\ 
8 8

7x8 / ( (7x8)+(2x8)) / 2x8
56 / ( 56 +16) /16
56 / 72 / 16
= 6336 , by keeping the last digits and carrying over the rest
Therefore, 72 x 88 = 6336
Algebric Proof :
(ax + b) (cx + d) = ac.x^{2} + adx + bcx + b.d
= ac.x^{2} + (ad + bc)x + b.d
where in our example 15 x 23 , a=1 ,b=5 ,c=2,d=3
and so ac.x^{2} + (ad + bc)x + b.d = 2 x^{2} +13 x +15 where x=10 meaning the place value
= 345
by keeping the last digits and carrying over the rest.
by keeping the last digits and carrying over the rest.
Now consider the multiplication of two 3 digit numbers.
Let the two numbers be (ax^{2} + bx + c) and (dx^{2} + ex + f). Note that x=10Now the product is
ax^{2} + bx + c multiply
dx^{2} + ex + f
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
ad.x^{4}+bd.x^{3}+cd.x^{2}+ae.x^{3}+be.x^{2}+ce.x+af.x^{2}+bf.x+cf
= ad.x^{4} + (bd + ae). x^{3} + (cd + be + af).x^{2} + (ce + bf)x + cf
From the above expression, we see that we multiply from left to right as per the expression
Example :
Find the result of 234 x 615 ?
ad.x^{4} + (bd + ae). x^{3} + (cd + be + af).x^{2} + (ce + bf)x + cf where a= 2,b=3,c=4 and d=6 ,e=1,f=5
=2x6 +( 3x6 +2x1 ) +(4x6+3x1 +2x5) +(4x1+3x5)+ 4x5...lets keep X's outside
=12 +(18+2) +(24+3+10)+(4+15)+20
=12x^{4}+20x^{3}+ 37x^{2} +19x+20 ..inserting X's now
And so keeping the last digits and carrying over the rest, we get 143910
Therefore, 234 x 615 =143910
Nice ,as the proverb "Practicing keeps a Man perfect " ,practicing the above sutras makes your multiplication easier .Keep practicing..that's the Veda of Vedic Maths
Interesting!GOOD WORK UMA
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