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Old 2010-03-23, 18:49   #1
only_human
 
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"Gang aft agley"
Sep 2002

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Default approximating products

Bits don't just grow on trees you know - when I was little we had to make do. Comes a time we'd know what a bit was without checking...then we sometimes didn't use a bit at all! We just tell ourselves what it should be. To multiply, sometimes we would just stick the numbers together! If bits were the same, sometimes we made them share! Young'uns today! hah! They just throw bits away if'n they don't need them or have a few too many!

Odd numbers always start and end with '1's in binary so we'd jam them together and make them share! And numbers don't get to choose which factor went first! Each number had a turn in first place and a turn in last place and we'd add them together! And the beginning bits and ending bits had to share too!

3 x 5 = 11 x 101 in binary, but the last bit of the first number is the same as the first bit of the second number so we'd just use a single bit and we wouldn't even keep track of where it was! we'd just write it like this:
1101

That's the 11 jammed on the 101 with the connecting parts sharing a bit.
Then we'd write it the other way 101 jammed on 11 as:
1011

But before we'd add them together we knew that the beginning and end of both jammed numbers are always '1' and so we made the beginnings share and made the ends share too! Folks 'd do it different ways but the final result is that you would only add a single beginning bit and a single end bit even though you had were adding two numbers. So
1101 plus
1011 would total as 1111 because the ends only get counted once. If easier you could write it as
1101 plus
0010
which is the same thing with just one beginning and end bit written as a '1' and the other beginning and end got changed to a '0' to prevent accidental adding. 'Cause they had to share you see. Now when you add the columns you can see it totals as 1111. And that is binary for 15 which equals 3 times 5. And 5 x 7 works the same way: 101 times 111 jammed together as 10111. And 7 x 5 with the seven first is 111 times 101 jammed together as 11101. And added together after zeroing the beginning and end bit of the second number:

10111 plus
01100
= 100011 which is 35

7 times 9 is written 111001+000110 = 111111 which is 63
9 times 11 is 1001011+0011000=1100011 which is 99

Although these specific examples have prime numbers in them, having a prime factor is not why these examples have exactly correct results. The procedure is intended to be an approximation of multiplication and always produces a result that is less than or equal to the correct product.

The puzzle is to determine, state or even guess what kind of factors produce correct results.

Extra credit:
Describe, explain or calculate ancillary features like the error amount of inexact results
Suggest how one might have derived this quirky method of approximating a product.

I might not be able to correctly describe the error term or evaluate a supplied answer of that portion of the puzzle but I can explain my understanding of it and describe what portion of a full product calculation is neglected.

Last fiddled with by only_human on 2010-03-23 at 19:20 Reason: minor grammer changes and less ambiguous phrasing
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