| 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]).
