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Old 2009-01-16, 20:40   #5
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"Gang aft agley"
Sep 2002

2×1,877 Posts

Does 99 x 99 = 9801 or any other (b-1) x (b-1) = b2 - 2b + 1 leave you dissatisfied?
Yes. I understand the reason involved.
I am aware that the total discussion is merely exploring a representation system (and about as obvious of one as possible) and not any direct intrinsic property of numbers themselves. I picked the one that allows numbers to be written in the most familiar way of m.s.b. to l.s.b. of traditional reading order. That is why I said that it is not really mathematics. This is also why I picked this particular subforum to place the thread.

... because your representation shows all the intermediate steps before the final summation, whereas the product shows only what's left after the summation.
Yes. Essentially I just tried to be a little more compact than plopping down N grains of rice to represent N. No calculation is done. No adding. No carries. I merely organized it such that a product can be written with one factor along each of its dimensional borders. That allows reading the factors on the edges in a glance. Of course the factors must be known to draw it that way. But it suggests drawing a border around a mess of 1s and 0s and asking oneself if any manipulation of the interior can move the bits such that it can be turned into what I called the canonical form (with factors along the borders and the entire content a valid Boolean "And" table as is intrinsic to the implied binary long multiplication). This is intended to develop a feel about numbers themselves. There are obvious mathematical rules about how to move the 1s around such as that they double or combine when moving to adjacent magnitude positions.

Last fiddled with by only_human on 2009-01-16 at 21:02
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