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Old 2009-02-21, 01:49   #7
only_human's Avatar
"Gang aft agley"
Sep 2002

1110101010102 Posts

"Step 5 above was of course written for base ten." in the previous message is incorrect. Step 5 is valid for any positive integer base.

The primary reason to in use base 2 for these 2-D representations is so that they could be written with factors on the edges. Also, as explained by Cheesehead and me, each position can correspond to a single digit multiply of multiplier and multiplicand in the fully expanded long multiplication of two factors.

Part of what possibly makes this an intelligible way to represent numbers is that the single binary digit multiplies do not generate carries. (So a 2-D pattern that is the result of a long multiply can be written with a single digit result for each single digit multiply-- deferring all adds to later, or never)

There is a base 3 representation of numbers that also does not generate carries as a result of a single digit multiply: Balanced ternary

Each digit in balanced ternary can be -1, 0, or 1

Rather than representing -1 by using a bar over or under a '1' I am going to use an underscore by itself. I am using these 2-D representations as a way to think about numbers and by using the underscore by itself I gain a small convenience in writing the number and thinking about it. The reason is that if I take a 2-D representation with the most-significant digit in the top-left corner and the least-significant digit in the bottom-right corner, I can visualize holding those two corners and flipping the number over that diagonal axis. Mentally I allow myself to think of the '_' symbols becoming '1's and vice versa. The aggregate value of the pattern changes sign when this is done. If the flipped pattern is added to the original pattern, the result is 0. Flipping the 2-D binary patterns mentioned earlier in the thread in a similar manner does not change the value of the pattern because the flipped 1s are kept as 1s (no symbol change is considered) and in all cases of 2-D representations of a flip over this diagonal axis the source and destination digits have equal magnitude positions.

(more later)
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