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What is Russian peasant multiplication? How do I use it? 

The way most people learn to multiply large numbers looks something like this:

86

x 57

------

602

+ 4300

------

4902

If you know your multiplication facts, this "long multiplication" is quick and

relatively simple. However, there are many other ways to multiply. One of 

these methods is often called the Russian peasant algorithm. You don't need

multiplication facts to use the Russian peasant algorithm; you only need to

double numbers, cut them in half, and add them up. Here are the rules:

  Write each number at the head of a column.

  Double the number in the first column, and halve the number in the

second column.

If the number in the second column is odd, divide it by two and drop

the remainder.

  If the number in the second column is even, cross out that entire row.

  Keep doubling, halving, and crossing out until the number in the second

column is 1.

  Add up the remaining numbers in the first column. The total is the

 product of your original numbers.

Let's multiply 57 by 86 as an example:

Write each number at the head of a column. 

57 86

 Double the number in the first column, and halve the number in the

 second column. 

57 86

114 43

 If the number in the second column is even, cross out that entire row. 

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57 86

114 43

 Keep doubling, halving, and crossing out until the number in the second column is 1. 

57 86

114 43

228 21

456 10

912 5

1824 2

3648 1

 Add up the remaining numbers in the first column. 

57 86

114 43

228 21

456 10

912 5

1824 2

+ 3648 1

4902

Real Russian peasants may have tracked their doublings with bowls of pebbles,

instead of columns of numbers. (They probably weren't interested in problems

as large as our example, though; four thousand pebbles would be hard to work 

with!) Russian peasants weren't the only ones to use this method of 

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multiplication. The ancient Egyptians invented a similar process thousands of 

years earlier, and computers are still using related methods today.

From the Math Forum:

Russian Peasant Multiplication 

Russian Peasant Multiplication 

History of Multiplication 

Two suggested books with more information.

Egyptian Method of Multiplication 

Egyptian Division 

From the Web:

Topics in Arithmetic and Elementary Mathematics, E. Lee Lady

"Bride of the Lazy Man" presents Russian peasant multiplication. It is

available in PDF, DVI, and PostScript formats.

Marvelous Old Way of Multiplying, Heidi Burgiel

From the geometry pre-college discussion group

 Back to top 

Why does Russian peasant multiplication work? 

Let's calculate 9 * 8 as an example:

9 8

18 4

36 2

72 1

72 is the only remaining number in the left-hand column, so our answer is 72. Notice that we were multiplying by 2 on one side, and by 1/2 on the other side.

2 * 1/2 = 1, so the overall product did not change:

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9 * 8

= 18 * 4

= 36 * 2

= 72 * 1.

We were grouping numbers in a different way, not changing the answer.

If we multiply 8 * 9, we should get the same answer. Can we explain our 

answer the same way?

8 9

16 4

32 2

+ 64 1

72

When we cut 9 in half, we dropped the remainder because 9 is an odd number.

Because we have "lost" a one, the product of each row should be smaller from

now on. Let's find the difference between the first row and the second row:

8*9 - 16*4= 72 - 64

= 8.

We can rewrite the subtraction as a sum:

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8 * 9

= 16 * 4 + 8.

Because our product has decreased by 8, we have to add 8 back in again at the

end. We can think of the addition as restoring 1 group of 8, for the remainder of 

1 that we dropped earlier. In a different problem, we might have to restore

several different groups of numbers.

From the Web:

Russian Peasant Multiplication: how does it work? E. Lee Lady

From the sci.math newsgroup

Other Algorithms, University of Melbourne, Australia

The lattice and Russian peasant methods for multiplying numbers

 Back to top 

How is Russian peasant multiplication related to binary numbers? 

Binary numbers are numbers written in  base two instead of base ten. This

means that place value depends on powers of two instead of powers of ten:

instead of ones, tens, and hundreds places, base two has a ones place, a twos

 place, a fours place, and so on. For example, fourteen in base two is 1110:

1110 (base 2)

= 1 * 23 + 1 * 22 + 1 * 21 + 0 * 20 

= 8 + 4 + 2 + 0

= 14.

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

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Russian peasant multiplication depends on multiplying and dividing by two. To

see the connection more clearly, let's investigate the problem 12*13.

Halving

You can convert a number to binary form by repeatedly dividing by two andkeeping track of the remainders. Let's try 12:

12/2 = 6 remainder 0

6/2 = 3 remainder 0

3/2 = 1 remainder 1

1/2 = 0 remainder 1.

Reading the remainders from bottom to top, we get 1100, so 12 in base two is

1100.

Why does this conversion method work? Let's try cutting twelve in half again,

the same way. This time, we'll write everything in base two. (Naturally, 2 in

 base two is 10.)

1100/10 = 110 remainder 0

110/10 = 11 remainder 0

11/10 = 1 remainder 1

1/10 = 0 remainder 1.

Dividing by two and then taking the remainder gives us a number's last digit in binary notation.

Here's what we know about 12, so far:

12 = 1100 (base 2)

= 1*23 + 1*22 + 0*2 + 0*1

= 23 + 22 

= 8 + 4.

By halving 12 repeatedly, we have broken it down into powers of two.

The Distributive Property

We are trying to multiply 12 by 13. One way to do this would be to use long

multiplication:

13

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* 12

----

26

+ 130

-----

156

 Notice that we are adding 2*13 and 10*13 to get our final answer. This works

 because of thedistributive property: 

12 * 13

= (2 + 10) * 13

= 2*13 + 10*13.

Of course, we can break 12 down any way we like, and still get the right

answer. Let's use our previous work to split the problem into powers of two:

12 * 13

= (4 + 8) * 13

= (22 + 23) * 13

= 22 * 13 + 23 * 13.

If we can multiply 13 by 2^2 and 2^3, we will be finished.

Doubling

Repeatedly doubling a number multiplies it by powers of two. Let's try

doubling 13:

Number  Multiplications so far  Power of 2 

13 13 20 

26 13*2 21 

52 13*2*2 22 

104 13*2*2*2 23 

Our chart tells us that 22 * 13 + 23 * 13 = 52 + 104 = 156, so 12 * 13 = 156, and

we are done.

Putting It All Together

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We just used repeated halving and doubling to convert 12 to binary form, then

multiply it by 13. Russian peasant multiplication does the same thing, but

 because it leaves out several steps, the process is much faster. Let's combine

our doubling and halving steps to compare the two methods.

Number

doubled Multiplications

so far Power

of 2 Number

halved Division

Problem  Remainder 

13 13 20  12 12/2 = 6 0

26 13*2 21  6 6/2 = 3 0

52 13*2*2 22 3 3/2 = 1 1

104 13*2*2*2 23 1 1/2 = 0 1

The columns used in Russian peasant multiplication are highlighted. Noticethat when the number in the remainder column is 0, the corresponding row for 

Russian peasant multiplication is crossed out.

From Dr. Math:

Russian Peasant Method of Multiplication 

From the Web:

Russian Peasant Multiplication: Explained! Kevin Fortin

From the sci.math newsgroup

http://mathforum.org/dr.math/faq/faq.peasant.html 

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Peasant Multiplication

The multiplication algorithm [Wells, p. 44] discussed below is

commonly known as the Russian Peasant Multiplication. It is evensaid that the algorithm "is still used by peasants in some areas, such as

Russia." However, the source of the Russian Peasant designation is

unexpectedly murky. It probably goes back to a few centuries old

Russian book where the method has been first described in (relatively)

modern times. I may only conjecture that the algorithm has acquired

the Russian part of the designation in the process of translation from

Russian and the Peasant part was appended due to a widely spread

conviction that (at least in older times) it was mostly the peasant

population that exclusively, albeit sparsely, filled the Russian vastness.

The algorithm in fact may have Egyptian roots, as a similar

procedure has been routinely used in the famous Rhind

Papyrus [Midonick, pp. 706-732, Fauvel, pp. 14-16]. It is sometimes

referred to as theEthiopian (Peasant) Multiplication; the linkage could

be explained by the proximity of the two nations and intermixing of 

their cultures. It is curious to note in passing that the great-

grandfather of the illustrious Russian poet Alexander Serge'evich

Pushkin was a blackamoor of Ethiopian origin. However, the spurious

idea that Ibrahim Petrovitch Gannibal, a page to Peter the Great, may

be a historic conduit for the algorithm from North Africa to Russiaclashes with the Peasant part of the designation. A pity.

The applet below allows for experimentation with the algorithm I'll

present shortly. The two blue numbers at the top - the multiplicands -

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can be modified by clicking on their digits. (The digits can be treated

individually or as part of a number depending on the state of the

"Autonomous digits" checkbox.) The number of digits in the

mutiplcands changes from 1 through 4.

Peasant Multiplication

The multiplication algorithm [Wells, p. 44] discussed below is

commonly known as the Russian Peasant Multiplication. It is even

said that the algorithm "is still used by peasants in some areas, such as

Russia." However, the source of the Russian Peasant designation isunexpectedly murky. It probably goes back to a few centuries old

Russian book where the method has been first described in (relatively)

modern times. I may only conjecture that the algorithm has acquired

the Russian part of the designation in the process of translation from

Russian and the Peasant part was appended due to a widely spread

conviction that (at least in older times) it was mostly the peasant

population that exclusively, albeit sparsely, filled the Russian vastness.

The algorithm in fact may have Egyptian roots, as a similar

procedure has been routinely used in the famous Rhind

Papyrus [Midonick, pp. 706-732, Fauvel, pp. 14-16]. It is sometimes

referred to as theEthiopian (Peasant) Multiplication; the linkage could

be explained by the proximity of the two nations and intermixing of 

their cultures. It is curious to note in passing that the great-

grandfather of the illustrious Russian poet Alexander Serge'evich

Pushkin was a blackamoor of Ethiopian origin. However, the spurious

idea that Ibrahim Petrovitch Gannibal, a page to Peter the Great, may

7/27/2019 Kaedah Russian Peasant

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be a historic conduit for the algorithm from North Africa to Russia

clashes with the Peasant part of the designation. A pity.

The applet below allows for experimentation with the algorithm I'llpresent shortly. The two blue numbers at the top - the multiplicands -

can be modified by clicking on their digits. (The digits can be treated

individually or as part of a number depending on the state of the

"Autonomous digits" checkbox.) The number of digits in the

mutiplcands changes from 1 through 4.

http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml