Maths Logics

The Remainders by the Last Digit (Shesanyankena Charamena) is used to express a fraction as a decimal to all its decimal places.It is similar to Ekadhikina Purvena .
Example 1:
a) Express 1/7 as a decimal
1. Add 'zero' to the numerator which makes 1 as 10
2. If the numerator is less than the denominator, add another 'zero' , else proceed as follows ...
3. Divide : (10)/7 = 1 remainder 3
4. Adding zero to the remainder (since it is less than the denominator), we divide as (30)/7= 4 remainder 2
5. We continue taking the remainder the following the steps from the start as follows :
(20) /7 = 2 remainder 6
(60) /7 = 8 remainder 4
(40) /7 = 5 remainder 5
(50) /7 = 7 remainder 1
6. Now note that the remainder is '1' which is same as the numerator '1'.This means we would get the repetition of the answers again and again.So we will stop here.
7. Using the numbers(remainders) 3,2,6,4,5,1 got above , we multiply them with the denominator '7',
7 x 3 = 2 1
7 x 2 = 1 4
7 x 6 = 4 2
7 x 4 = 2 8
7 x 5 = 3 5
7 x 1 = 7
8. Now we take the red shaded numbers from the above steps as sequence ,142857 (which would be our final result)
Therefore, Result (1/7) = 0.142857142857

ONE LESS THAN PREVIOUS-

One less than the previous or One less than the before :As the name indicates this sutra involves subtracting one from the given number to get our final result.This sutra is highly helpful in case of multiplication by 9,99,999....to any other number and in solving fractions of certain numbers like (1/7),(1/13),(1/17)..etc.

PART 1: For solving Multiplications

Example 1:

6 * 9 = ?

Step 1 : Minus one from the number on L.H.S digit above
Step 2 : Minus the answer(result) got from step 1 from R.H.S digit (i.e 5 from number 9)

6-1 = 5
9-5 = 4
Combine them LHS,RHS ,to get 54

Therefore,The Result for 6 * 9 = 54

EXAMPLE 2:
Now lets try the same again with 999

899 * 999 =?

Step 1: 899-1 = 898
Step 2: 999-898 = 101

Combining the above, 898101

Therefore,The Result :899 * 999 = 898101

VERTICALLY AND CROSSWISE(Urdhva-Tiryagbyham)

(b) For numbers above 100

Example 1: 102 * 105

102 - 2
105 - 5 ........> gives 100
---------------
(105+2) / 5 * 2 ...here we have to add the opposite numbers
107/10

Therefore,102 * 105 =10710

Example 2: 111 * 108

111 - 11
108 - 8.......> gives 100
-------------
(111+8) / 11 * 8
119 /88
Therefore , 111 * 108 = 11988

VERTICALLY AND CROSSWISE(Urdhva-Tiryagbyham)

VERTICALLY AND CROSSWISE for multiplying numbers close to 100
Now let us see another simple form of vertically and crosswise method for numbers close to 100.This method will amuse you with its simplicity.

(a) For numbers below 100

Example 1 : 82 * 94
The above numbers are close to 100
82 is 18 below 100 and 94 is 6 below 100.

So,
82 +18
\/
/\
94 + 6
----------------

(82-6) or (94-18) / (6 * 18) ,now by multipying the right side and subracting the left side
76 / 108 ....76+1 / 08 ,by carrying over '1' since it is a two digit number

Therefore ,82 * 94 =7708

Example 2: 88 * 95

88 +12
95 + 5 ------> gives 100
-------------
(88-5)/(12 * 5)
83 / 60

Therefore, 88 * 95 =8360