Russian Peasant Multiplication

     This method involves a process of halving and doubling, which reduces one factor to powers of two and uses the distributive property of multiplication over addition to calculate a product.

Multiplication:     
                              25                    37
                              12                    74
                              6                      148
                              3                      296
                              1                      592
                            -------------------------           Therefore, 37 + 269 + 592 = 925
                                                      925                              ( 25 x 37 = 925 )

Process:
1. Create column beneath each of the factor.
2. Repeatedly halve the number in the left hand column, drop any reminder, 
    until reach to 1.
3. Repeatedly multiply the right column by 2.
4. Erase the rows that have any even number on the left column.
5. Add remaining numbers in the right hand column to get the answer.

Behind the scene:
     The fact that this procedure is dependent on multiplication by two, suggests that this is founded in binary system. Let’s see the binary representation of 25.

Power of 2
24
23
22
21
20
Decimal equivalent
16
8
(4)
(2)
1
Binary representation
1
1
0
0
1


We found binary representation 25 is 110012. Now take another look at our multiplication.

                            Multiplication                       Binary                        Decimal
                            25                    37                    1 x 20                                    1
                            12                    74                    0 x 21                                    0
                            6                      148                  0 x 22                                    0
                            3                      296                  1 x 23                                    8
                            1                      592                  1 x 24                                    16
                          -------------------------                --------                         ----+
                                                    925                  110012                        25

     So, we can see that adding two additional columns clearly show the relationship between the multiplication with the binary system. Because the binary digit represents the reminder in division by power of "two", digit "one" corresponds to odd numbers in the column and "zero" corresponds to even number, thus only the rows with odd numbers will contribute to the multiplication.

     Russian peasant multiplication is actually a quick way to convert two numbers to binary form, multiply them together, and convert back to our number system. The connection is not surprising, because binary numbers use base two, and Russian peasant multiplication depends on multiplying and dividing by two.

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