86 64 64 goes 1 ——— 22 32 does not go 0 16 16 goes 1 ——— 6 8 does not go 0 4 4 goes 1 ——— 2 2 goes 1 2 1 does not go 0 ——— 0 It is a little troublesome to remember long series of 1’s and 0’s; in fact, to write any number in binary notation takes about 3⅓ times as much space as decimal notation. For this reason we can separate binary numbers into triples beginning at the right and label each triple as follows: TRIPLE LABEL 000 0 001 1 010 2 011 3 100 4 101 5 110 6 111 7 For example, 1010110 would become 1 010 110 or 126. This notation is often called _octal notation_, because it is notation in the scale of eight. BIQUINARY OR _TWO-FIVE_ NUMBERS Another kind of notation for numbers is _biquinary notation_, so called because it uses both 2’s and 5’s. Essentially this notation is very like Roman numerals, ancient style. By ancient style we mean, for example, VIIII instead of IX. In the following table we show the first two dozen numbers in decimal, biquinary, and ancient Roman notation: DECIMAL BIQUINARY ROMAN 0 0 1 1 I 2 2 II 3 3 III 4 4 IIII 5 10 V 6 11 VI 7 12 VII 8 13 VIII 9 14 VIIII 10 100 X 11 101 XI 12 102 XII 13 103 XIII 14 104 XIIII 15 110 XV 16 111 XVI 17 112 XVII 18 113 XVIII 19 114 XVIIII 20 200 XX 21 201 XXI 22 202 XXII 23 203 XXIII The biquinary columns alternate in going from 0 to 4 and from 0 to 1. The digits from 0 to 4 are not changed. The digits from 5 to 9 are changed into 10 to 14. We see that the _biquinary digits_ are 0 to 4 in odd columns and 0, 1 in even columns, counting from the right. This is the notation actually expressed by the _abacus_. The beads of the abacus show by their positions groups of 2 and 5 (see Fig. 1). [Illustration: FIG. 1. Abacus and notations.] SOME OPERATIONS OF ALGEBRA One of the operations of algebra that is important for a mechanical brain is _approximation_, the problem of getting close to the right value of a number. Take, for example, finding _square root_ (see the end of this supplement). The ordinary process taught in school is rather troublesome. We can set down another process, however, using a desk calculator to do division, which gives us square root with great speed.