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Originally Posted by only_human
This isn't quite mathematics

False modesty. Of course it's mathematics!
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 write a binary number
 pick another binary number and write it vertically starting with every "1" in the first number
 every space not explicitly written in this rectangle contains a 0

... and if you were to do standard binary long multiplication, you'd find that you would write the same sequences of 0s and 1s one after the other in each intermediate row (before the final sum), except that the rows would occur in the opposite order, and each row would be shifted left one place from the row above it.
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The pattern written in this representation will have a total value equal to the product of the row value times the column value.

Multiplication!
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when added using the positional values for each "1" as listed below.
The positional values in this represent:
Code:
2^{8} 2^{7} 2^{6} 2^{5} 2^{4}
2^{7} 2^{6} 2^{5} 2^{4} 2^{3}
2^{6} 2^{5} 2^{4} 2^{3} 2^{2}
2^{5} 2^{4} 2^{3} 2^{2} 2^{1}
2^{4} 2^{3} 2^{2} 2^{1} 2^{0}
The most significant bit is in the top left corner. The least significant bit is in the bottom right corner.

... and if you flip that matrix vertically, then shift each row one place to the left from the row above it, you find that the powers of 2 would match up exactly with the value of each bit in the longmultiplication intermediate rows, like this:
Code:
2^{4} 2^{3} 2^{2} 2^{1} 2^{0}
2^{5} 2^{4} 2^{3} 2^{2} 2^{1}
2^{6} 2^{5} 2^{4} 2^{3} 2^{2}
2^{7} 2^{6} 2^{5} 2^{4} 2^{3}
2^{8} 2^{7} 2^{6} 2^{5} 2^{4}
So, the 23
^{2} looks like:
Code:
10111
x 10111

10111
10111
10111
00000
10111

Now, notice that in your powersof2 matrix, each power is repeated along an upperright to lowerleft diagonal, the same number of times that that power of 2 can occur in the intermediate lines of the long division. That is, the 2
^{0} occurs just once, and in the long multiplication it occurs only as the rightmost bit of the top intermediate row. The 2
^{1} occurs twice on the next higher diagonal, and in the long multiplication it occurs as both the nexttolast bit of the top intermediate row and the last bit of the second intermediate row, which are in the same vertical column there. And so on.
Each diagonal in your powersof2 matrix corresponds to a vertical column across the intermediate lines of a standard long multiplication. The number of powersof2 added up along a diagonal of your matrix is the same as the number of powersof2 added up in the corresponding column of the intermediate lines of long multiplication.
But your bitproducts matrix is more compact than long multiplication's intermediate lines with their shifted rows and their unused spaces at upper left and lower right, as can be seen above.