Kaedah Russian Peasant
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Transcript of Kaedah Russian Peasant
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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
http://slidepdf.com/reader/full/kaedah-russian-peasant 11/11
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