alt Hacker NewsI don't know what age range "grade school" is, but I remember being taught that method when I was about 7 or 8, although it didn't really "land" properly until I read the short story "The Feeling of Power" by Isaac Asimov.
What I'm surprised to see left out here (unless I missed it in the page's horrible formatting) is a mention of the way that computers multiply two integers. They use a technique I saw described in a book when I was about 11 as the "Russian Farmer Method" (or something like that, it was in English and I might have misremembered it).
In that you shift the multiplier right and multiplicand left, halving one and doubling the other. If the multiplier is odd, add the multiplicand to the total.
It's really doing the same thing as "long multiplication" like you're taught in primary school but in binary so when you add a 0 to the right for the higher order digits you're doubling, not multiplying by ten. If you write code to do it you'd shift the multiplier first then consider whether or not to add by testing the Carry flag, or "Link bit" if like the author of the book I read you're demonstrating it on a PDP8 ;-)
But let's have a worked example, picking two numbers at random 205 * 707, use the smaller as the multiplier:
205, 707 odd, add 707 to total
102, 1414 even, disregard
51, 2828 odd, add 2828 to the total
25, 5656 odd, add 5656 to the total
12, 11312 even, disregard
6, 22624 even, disregard
3, 45248 odd, add 45248 to the total
1, 90496 odd, add 90496 to the total
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Replies
Russian Peasant Multiplication (https://www.embeddedrelated.com/showarticle/760.php)
> If we're disregarding shifts and adds as completing in negligible time
When considering multiplication algorithms the parameter N is the number of digits of the two numbers. In that model adds do not complete in negligible time, and instead take O(N) time.
Yeah, that shift-and-add algorithm is sometimes used on microcontrollers, either in software if there's no hardware multiplier, or in hardware if you want the bare minimum in acceleration at a tiny cost in area.
Adds are not really considered negligible; the article is just sloppy. (Some shifts might be negligible in some models because a fixed shift requires no logic gates.) The cost of the adds in Karatsuba is significant both theoretically and in practice, and determines the cutoff where Karatsuba is useful. But the exponent in O(n^(log_2 3)) is dominated by the recursive multiplications; the adds only affect the leading constant hidden in the O().