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 METHODS OF MULTIPLICATION (TWO DIGIT NUMBER)

For example:

        For testing the result of 54 × 32

            54 × 32 → 5432   →    5342

                                               ¯   ¯

                                               ↓    ↓

                                          ↓      15 08 → 1508

                                   34 =>   3 × 4 = 12

                                  52 =>    5 × 2 = 10

                                              22 × 10 = 220

                                        1508 + 220 = 1728

                                    54 × 32 = 1728 

Testing of result:-

                        5 + 4   ×    3 + 2    =     1 + 7 + 2 + 8

                                 ↓          ↓                   ↓

                               9   ×     5        =      18

                                –––––––––             

                                       ↓                         ↓

45             =       18

 ↓                          ↓

                                    4 + 5            =       1 + 8

                                        ↓                          ↓ 

                                        9              =           9

                                   L.H.S.           =        R.H.S.

  L.H.S. ó R.H.S.   Result is true.

  ──────────      

Multiplication:

(1)         This multiplication is for those numbers whose  sum of one’s digit =10 and ten’s digit are same

Example – 35 × 35.  98 × 92 etc.

       Working:                             

                                   Ten’s digit × (ten’s digit + 1)                       one’s digit × one’s digit

            35 × 35                       3   ×   4                                                                    5   ×   5

12                                                                                                                              25

                                                                               1225

            98 × 92                      9   ×   10                                                           8   ×   2

                                                90                                                                16

                                                                              9016          

 NOTE: 

                  Multiplication of three or four digit number can also be done. If sum of one’s digit = 10

and remaining  digits are same. 

This method is based on addition, in this method numbers are added till a single digit number remains and in the end result of multiplication is being tested.

For Example :  346  *  344   =  34  *  35            6  *  4

1190                                       24

                        345  *  344                 =  119024     

(2)      This  is  for those numbers .Which have there ones postion  digits same and  sum of tenth  postion is10.

For Example :  86  *  26 , 52  *  52

                       For Example :  86  *  26  ( tenth * tenth + ones)    (ones *ones)

                                              8 * 2 + 6   = 36

22                             36

            =  2236  

Pascal Triangle

 According to the Pascal (Intelligent, bright, smart, and sharp, mathematician, scientist) triangle there are some interesting facts. From which we can get there is some research left after invention

First we want to know:- From where this triangle came? And which method is behind them?

Let see the triangle:-

1 1

1 2 1

1 3 3 1

1  4  6  4  1

1  5 10 10 5 1

First see how to  write continuously  ? (writing method )

Powerful Digit “9”

If we research on tables, then we got some information.

(1)  For the table of 2 -

By multiplication of numbers by 2 we got the following numbers which repeat themselves at some point ─

    2 4 6 8 10 12 14 16 18 20 22

    2 4 6 8   1   3   5   7   9   2   4

                                                                     Repeat

      Numbers 2 4 6 8 repeated after 9. We write it like this

      2 4 6 8 1 3 5 7 9 2 4 if we add these numbers ─

      2 + 4 + 6 + 8 +1 + 3 + 5 + 7 + 9   = 45   = 4 + 5 = 9

(2)  For the table of 3 -

                        3 6 9 12 15 18 21 24 27 30

                        3 6 9  3   6   9   3   6   9   3

During adding we got (3 6 9) which repeat themselves.

Write it like this -

     3 + 6 + 9 = 18 = 1 + 8 = 9

(3)  For the table of 4 -

                        4 8 12 16 20 24 28 32 36 40 44

                        4 8  3   7   2   6   1   5   9   4   8

                                                                    Repeat

4 + 8 + 3 +7 +2 + 6 + 1 + 5 + 9 = 45 = 4 + 5 = 9

(4)  For the table of 5 -

                        5 10 15 20 25 30 35 40 45 50 55 60

                        5  1   6   2   7   3   8   4   9   5   1   6

                                                                          Repeat

5 + 1 + 6 + 2 + 7 + 3 + 8 + 4 + 9 = 45 = 4 + 5 = 9

(5) For the table of 6 -

                        6 12 18 24 30 36 42 48 54 60

        6   3   9   6    3    9    6    3    9    6

        6  +  3  +  9          = 45   =  4  +  5 =  9

(6)  For  7  table: -

7        14  21  28  35  42  49  56  63  70  77

7     5    3    1    8    6    4    2    9    7    5

7 + 5 + 3 +1 + 8 + 6 + 4 + 2 + 9     = 45    = 4   +  5  =  9

(7)    For  8  table :-

8        16  24  32  40  48  56  64  72  80

8     7    6    5     4   3    2    1    9    8

8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 9    = 45    =  4  +  5  = 9

(8)    For  9  table  :-

9     18   24  36  45  54  63  72  81  90

9      9     9    9    9    9    9    9    9    9 

9        =    9

Note  : 

            By studying this we get knowledge about two things.

(1)              In a  specific table repeating of a number or some  numbers.

(2)              Number 9 in each equation and in the end too.

     (1).     Suppose you want to multiply 25 AND 32 –

                first write 25 × 32 like this –

                after that write it like this - (2352)

    352 = 2 × 3 = 6                                        (ten's digit × hundred's digit)

                            5 × 2 = 10

                                     = 610

                10 (3 × 5 + 2 × 2) = 10 (15 + 4) = 190             (one’s digit × thousand’s digit ×10)

                                             = 610 + 190 = 800

     (2).   

  a.   For multiplication of 25 × 32, we do :-

2 5

3 2

                                    ──

                                    4 0

                                                            (Multiplication of one’s digit)

                                 0 1

                                 6 5

                                                            (Multiplication of ten’s digit)

                              0 1

                            ──────

                                 8 0 0

                            ──────   

    b.   For multiplication of 48 × 35, we do :-

                                    4 8

                                       3 5

                                    ──

                                    0 0

                                 2 4

                                 2 4

 1 2

─────

 1 6 8 0

─────

Methods for testing the result of multiplication of different number of digits :

“PASCAL TRIANGLE”

Pascal was not only the scientist, but also great mathematician . His efforts towards science and math were countless; Pascal triangle is a triangle upon which still some works has to done. I had also done some work on Pascal triangle; there came some points that made his invention           . I studied some more on Pascal triangle then there some points came in front of me that i want to share those points with you.

Firstly we going to learn that, How this triangle is build?

If we add numbers i.e.(1+1=2) (1+2+1=4) (1+3+3+1=8) (1+4+6+4+1=16) we get (2, 4, 8, 16) ,which are even numbers.

Further reading tells, 121 is square of 11, 1331 – 11³, 14641 – (11²)², but for the solution of 11 raise to the power 5, new methods has been developed.

By using this method, we can solve any power of 11.   This triangle can easily be remembered. Just remember 11, in the next number 1-1 is at the both end and just add the middle number. For example 11, 1, 2, 1, 1, (1+2), (2+1), 1 => 1, 3, 3, 1, 1, (1+3), (3+3), (3+1), 1 => 1, 4, 6, 4, 1.

                                                     Figure of Pascal Triangle

11⁵ ->  1     5      10       10     5      1     ->  which is a form pascal’s triangle.

To write  this in 11⁵  from we have to use  this format .

Like this we will find out value of  (11¹⁰) .

First  through  pascal’s triangle we get :

1        10  45  120  210  252  210  120  45  10  1 form . To

find out value we will use this format.