Binary Abacus Information Site

the Moonstick Co.'s first "usable" binary abacus

a preliminary copy of binary abacus instructions including a proposed notation for writing binary numbers

extracting the square root of ½ () on a binary abacus

Larger bits are to the right.  If the 1st, 2nd, 3rd, and 4th rows of the abacus are referred to as registers A, B, C, and D respectively, then notice that during the first part of the calculation it is always true that A-B+C²=½.  The calculation begins with A=½ and all other registers empty and aims to arrive at a state where C= with all other registers empty.  During a later part of the calculation there is insufficient precision (only 32 bits) to maintain exact equality (A-B+C²=½), so register D begins to accumulate an upperbound on the error and thereafter it is true that ½<A-B+C²<½+D.  One can see at the very end that there is insufficient information to determine whether the 30th bit is up or down.  To follow this more closely you can look at it frame by frame.