This method, called "Ethiopian multiplication" or "Russian multiplication" or sometimes "Russian peasant multiplication," is a popular topic in Liberal Arts Mathematics and Mathematics History classes. Another popular topic in Liberal Arts Mathematics classes is the binary number system. Interestingly, Russian multiplication is equivalent to the way microprocessors multiply using a binary algorithm, and therefore presages how modern-day computing devices multiply.
Showing students the link between Russian multiplication and microprocessor multiplication gives them a deeper appreciation of electronic computing devices, as well as reviews for them the binary number system and introduces them to a computing algorithm. Moreover, it is an excellent topic with which to begin a discussion of the history of multiplication methods. In the book, Excursions in Number Theory , the authors told the story of an Austrian colonel who wished to purchase seven bulls for 22 Maria Theresa Austrian dollars each in a village in a remote part of Ethiopia in the mid s.
In commenting on this story, a later writer Hayes, explained why he believed this method came to be called "Russian multiplication" in Europe rather than, say, "Ethiopian multiplication," which is gaining favor today :. The multiplicative system just demonstrated is very ancient indeed: it is probably the very earliest mathematical system worthy of the name and was doubtless invented, reinvented and forgotten innumerable times throughout human history.
Since it does not require any form of writing and involves only three operations, pairing off, halving and doubling, which are both easy to carry out and are not troublesome conceptually, the system remained extremely popular with peasants the world over and became known as Russian Multiplication because, until recently, Russia was the European country with by far the largest proportion of innumerate and illiterate peasants.
The binary number system, long a practically useless example of an alternative number system, became vital to the success of present day digital computing devices. Fundamentally, these devices rely on the binary number system where 1 represents "on" and 0 represents "off".
These l's and 0's are called bits. Arithmetic in binary becomes a task of bit manipulation. Following the properties of all number systems, the place values of the binary number system are powers of two, with digits 0 and 1. If you shift a binary number to the left add a 0 on the right , you have multiplied it by 2. If you shift a binary number to the right chop off the far right bit , you have divided it by 2, ignoring the remainder. Odd binary numbers end in 1; even binary numbers end in 0.
Students might want to consider analogies for the decimal number system. The multiplier and multiplicand in a microprocessor are typically stored in binary form.
Say I want to multiply x by Then I go through the same process, except that when you divide the numbers in the left-hand column by 2, just ignore the remainder in case the number is odd. Now to obtain the final answer I need to add x to the number at the bottom of the right-hand column. In this case, as we move down the two columns, the product at the second step is no longer the same as the product at the first, because 17 became 8, which is not actually half of In general, if you're multiplying x by a and go through this process, every time the number in the left-hand column is odd, the product of the two columns one line down will be smaller than the product before.
To compensate, you need to add in the number which is in the right-hand column at this stage when you get to the end. Cross out all the rows where the left-hand column is even, but save the rest.
Now add up all the entries in the right-hand column which are not crossed out and you'll have your answer. This file is also available in postscript, dvi, and idvi formats.
Look at www. Here's my own attempt to explain it: [Please use a non-proportional font so the columns line up. The procedure is described in Jan Gullberg's "Mathematics" and other sources. We will talk about what remains in a minute. Next, cross out the rows in which the number on the left is even. Extend your line to cross out the corresponding number on the right as well. Find the sum of the remaining numbers in the right column. The sum of these numbers is equal to the product you would get from multiplying the original numbers using the standard method.
The Russian peasant multiplication method works because it converts the problem into binary base 2 multiplication, rather than base For example, try multiplying x Make two columns. Using a piece of paper and pen, divide the piece of paper into two columns by drawing a line down the middle of the paper.
Write one of the numbers you want to multiply at the top the each column. Halve the number in the left column repeatedly. Divide the number at the top of the left column by 2 continually until you get to 1. Ignore any remainder each time you halve the number. Write each halved number down in the left column, in order. Double the number in the right column until the columns are the same length.
Multiply the number in the second column by 2 until there are the same amount of numbers here as there are in the first column. In this example: Each column should have 8 numbers in it. This is because it took seven steps of dividing the original number in the right column to reach 1. Cross out rows with an even number in the left column.
Using your pen, strike through those horizontal rows which begin with an even number in the left column. In this example: There are 8 rows.
You will strike through 5 of them. Strike through the rows beginning with , 36, 18, 4, and 2 in the left column, since these are even numbers. Working left to right, this means striking through the first row , 37 , the third row 36, , the fourth row 18, , the sixth row 4, , and the seventh row 2, Please note that you should strike through even numbers, even if they begin with an odd numeral. For example, you should strike through the row beginning with since it is an even number, even though begins with an odd numeral, 1.
Likewise, you should strike though 36, since it is an even number, even though 36 begins with an odd numeral, 3. If you prefer, you can just strike through the numbers in the right side that fall into the rows that begin with an even number on the left side as in the picture above.
In this example, this means striking through the numbers on the right hand side of the first, third, fourth, sixth, and seventh rows: 37, , , , and Find the sum of the remaining numbers in the right column. Add the numbers in the right column that you did not strike through.
The sum of these numbers is equal to the product you would get from multiplying the original numbers using the standard method. In this example: The remaining numbers in the right column are 74, ,
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