This article is essentially a duplication. However using your example, I can now make my previous explanation even simpler.
In article <34FCB257.2C7DF882@ufl.edu>, Kevin Fortin <firstname.lastname@example.org> wrote: >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, I >SNIP< >Example: 19 x 54 = X > >(H = "halving column", D = "doubling column"; in column H, any remainder >is discarded after each halving.) > >_H_ _D_ >54 19 (ignored) >27 38 >13 76 > 6 152 (ignored) > 3 304 > 1 608 > >The even numbers in column H are crossed out, along with the >corresponding entries in column D across the way. The remaining numbers >in column D are added up to give the product: >19 x 54 = (38 + 76 + 304 + 608) = 1026
Now, as I explained in my previous article, all you need to do to explain this is notice that when the left-hand column is even, the product of the two columns remains the same when you move down to the next line. (For instance, 54 * 19 = 27 * 38.) However when the left-hand column is odd, this is not quite correct. The discrepancy is in fact the number in the right-hand column. (For instance, 27 * 38 = 13 * 76 + 38.) Now if you take the final entry in the right-hand column and add to it all the discrepancies, i.e. all the other entries in the right-hand column which appear next to odd numbers in the left hand column, you recover the original (unknown) product. (As you indicate, 54 * 19 = 608 + 304 + 76 + 38. The point is that 54 * 19 = 27 * 38 = 13 * 76 + 38 = 6 * 152 + 76 + 38 = 3 * 304 + 76 + 38 = = 1 * 608 + 304 + 76 + 38. )
In my opinion, bringing binary numbers into the explanation just makes it more confusing.
See <Http://www.math.Hawaii.Edu/~lee/elementary/Lazy2.pdf> ("Bride of the Lazy Man").
-- Trying to understand learning by studying schooling is rather like trying to understand sexuality by studying bordellos. -- Mary Catherine Bateson, Peripheral Visions lady@Hawaii.Edu