```Date: Mar 3, 1998 8:45 PM
Author: Kevin Fortin
Subject: Re: Russian Peasant Multiplication: Explained!

Thanks to the sci.math readers whose responses helped me to understand"Russian Peasant Multiplication"!Here's my own attempt to explain it:[Please use a non-proportional font so the columns line up.  Also, Iwill use the caret (^) to indicate an exponent, e.g. 2^3 = "two cubed".The procedure is described in Jan Gullberg's "Mathematics" (1997) andother sources.]Example:  19 x 54 = X(H = "halving column", D = "doubling column"; in column H, any remainderis discarded after each halving.)_H_   _D_54     19 (ignored)27     3813     76 6    152 (ignored) 3    304 1    608The even numbers in column H are crossed out, along with thecorresponding entries in column D across the way.  The remaining numbersin column D are added up to give the product:19 x 54 = (38 + 76 + 304 + 608) = 1026After the halving is finished (i.e., you have reached a result of 1),what matters in column H is whether the numbers are even or odd.  Thepattern of even and odd numbers in column H corresponds to a binary (orbase 2) representation of the original number:                Place in                binary                notation54   even   0   (2^0)27   odd    1   (2^1)13   odd    1   (2^2) 6   even   0   (2^3) 3   odd    1   (2^4) 1   odd    1   (2^5)54 base ten, converted to base 2, equals 110110, or (back in base 10):(2^5 + 2^4 + 2^2 + 2^1) = (32 + 16 + 4 + 2) = 54.  [If the first entryin column H is even, there will always be a zero in the unit (2^0) placeof the binary representation; if the first entry is odd, there will be aone in the unit place.]Therefore, 19 x 54 = 19(2^5 + 2^4 + 2^2 + 2^1) = (608 + 304 + 76 + 38) =1026.As 2^0 and 2^3 are not part of the sum shown above that produces 54, thecorresponding products in column D (19 x 2^0 and 19 x 2^3) are alsoignored.  The surviving elements of column D are added up to give theproduct of 54 x 19.Hoping this helps others puzzled (as I was) by this method ofmultiplying,Kevin
```