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 + 296 + 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 remainder, 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 of 25 is 11001_2. 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 remainder 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.