Russian Peasant MultiplicationDate: 02/26/2001 at 15:40:25 From: Kara Koski Subject: Russian Peasant Multiplication Dear Dr. Math, Why is it that we only add together the doubled numbers of the odd halves on the other side when using the Russian Peasant multiplication algorithm? Thank you for your time, Kara Koski Date: 02/26/2001 at 17:16:14 From: Doctor Paul Subject: Re: Russian Peasant Multiplication This question has already been answered in our list of frequently asked questions at: http://mathforum.org/dr.math/faq/faq.peasant.html Please write back if you still don't understand. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ Date: 02/26/2001 at 19:35:29 From: Kara Koski Subject: Russian Peasant Multiplication Dear Dr. Math, I understand that the odd numbers are chosen because the remainder is one, but I don't understand why. Maybe you'll have to explain in more laymen's terms for me, as I am not getting it at all. Thank you for your time and patience, Kara Koski Date: 02/26/2001 at 20:10:02 From: Doctor Paul Subject: Re: Russian Peasant Multiplication Here's the idea: Let's look at 43 * 27 as an example: 43 27 86 13 -172----6- 344 3 688 1 The peasant's method says that 43 * 27 = 43 + 86 + 344 + 688 Notice that all of these are multiples of 43. We can factor out 43: 43 * 27 = 43 + 86 + 344 + 688 = 43 * (1 + 2 + 8 + 16) = 43 * 27 Basically, the idea is that the peasant's method works because it writes one of the numbers (specifically, the number you choose to halve) as a sum of powers of two. Perhaps another example would be good: 17 * 83 Notice that 17 * 83 = 17 * (64 + 16 + 2 + 1) = 1088 + 272 + 34 + 17 = 1411 Let's use the peasant's method to verify this: 17 83 34 41 ---68---20- --136---10- 272 5 --544----2- 1088 1 Notice that the entries in the "doubling" column whose corresponding entries in the "halving" column are odd are exactly the numbers we used above. If this still isn't clear, please write back. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ |
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