```Date: Mar 3, 1998 2:47 AM
Subject: Re: Russian Peasant Multiplication: how does it work?

In article <34FAEB0F.C90@ufl.edu>, Kevin Fortin  <kfortin@ufl.edu> wrote:>Hello,>>Could someone point me toward a good explanation of how "Russian Peasant>Multiplication" works?  I saw the method in Jan Gullberg's recent book>"Mathematics: From the Birth of Numbers", but I still can't grasp why it>works.  This is really easy to understand on the basis of common sense.Unfortunately, most math books try to give a clever explanation based onthe binary representation of numbers.  This works, but the explanation isharder to follow.  Okay, suppose you want to multiply  x  by  16.  Since  16 = 2^4, what you should do is to double  x  4 times.  Now for a larger number,you might not know quickly what the right exponent is.  Say take 512,which is a power of 2, but you've forgotten which power.  But just writedown 512 and write down  x.  Now divide 512 by 2, getting 256, and at thesame time double  x.  Keep dividing the left-hand column (the one thatstarts with 512) by 2 and doubling the numbers in the right-hand column(which started with  x), until you've reduced the left-hand column to 1.The number at the bottom of the right-hand column is the product you'reseeking.  It's easy to see why this works, because as you move down bothcolumns simultaneously, the product of the numbers in the two columns isalways the same.  Looking at the top entries, this product is clearly 512times x.  Looking at the bottom entry gives you the product. But suppose you want to multiply by a number which is not a power of 2.Say I want to multiply x by 17.  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 thefinal answer I need to add x to the number at the bottom of theright-hand column.  In this case, as we move down the two columns, theproduct at the second step is no longer the same as the product at thefirst, because  17  became  8, which is not actually half of 17.  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 thetwo columns one line down will be smaller than the product before.  Tocompensate, you need to add in the number which is in the right-handcolumn at this stage when you get to the end.  (Cross out all the rowswhere the left-hand column is even, but save the rest.  Now add up allthe entries in the right-hand column which are not crossed out and you'llhave your answer.)If you want a longer explanation, with several examples, look at "Bride of the Lazy Man" at <Http://www.math.Hawaii.Edu/~lee/elementary/Lazy2.pdf>.  (This file isalso available in postscript, dvi, and idvi formats.  Look at www.math.Hawaii.Edu/~lee/elementary/ and click on the format you prefer.)-- Trying to understand learning by studying schooling is rather like trying to understand sexuality by studying bordellos.                               -- Mary Catherine Bateson, Peripheral Visionslady@Hawaii.Edu
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