BINARY OR TWO NUMBERS

We are well accustomed to decimal notation in which we use 10 decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and write them in combinations to designate decimal numbers. In binary notation we use two binary digits 0, 1 and write them in combinations to designate binary numbers. For example, the first 17 numbers, from 0 to 16 in the decimal notation, correspond with the following numbers in binary notation:

Decimal	Binary	Decimal	Binary
0	0
1	1	9	1001
2	10	10	1010
3	11	11	1011
4	100	12	1100
5	101	13	1101
6	110	14	1110
7	111	15	1111
8	1000	16	10000

In decimal notation, 101 means one times a hundred, no tens, and one. In binary notation, 101 means one times four, no twos, and one. The successive digits in a decimal number from right to left count 1, 10, 100, 1000, 10000, ...—successive powers of 10 (for this term, see the end of this supplement). The successive digits in a binary number from right to left count 1, 2, 4, 8, 16, ...—powers of 2.

The decimal notation is convenient when equipment for computing has ten positions, like the fingers of a man, or the positions of a counter wheel. The binary notation is convenient when equipment for computing has just two positions, like “yes” or “no,” or current flowing or no current flowing.

Addition, subtraction, multiplication, and division can all be carried out unusually simply in binary notation. The addition table is simple and consists only of four entries.

+	0	1
0	0	1
1	1	10

The multiplication table is also simple and contains only four entries.

×	0	1
0	0	0
1	0	1

Suppose that we add in binary notation 101 and 1001:

Binary Addition	Check
101	5
+ 1001	9
1110	14

We proceed: 1 and 1 is 10; write down 0 and carry 1; 0 and 0 is 0, and 1 to carry is 1; and 1 and 0 is 1; and then we just copy the last 1. To check this we can convert to decimal and see that 101 is 5, 1001 is 9, and 1110 is 14, and we can verify that 5 and 9 is 14.

One of the easiest ways to subtract in binary notation is to add a ones complement (that is, the analogue of the nines complement) and use end-around-carry (for these two terms, see the end of this supplement). A ones complement can be written down at sight by just putting 1 for 0 and 0 for 1. For example, suppose that we subtract 101 from 1110:

Direct Subtraction	Check	Subtraction by Adding Ones Complement
1110	14	1110
- 101	- 5	+ 1010
1001	9	(1)1000
		↓
		⎯→ 1
		1001

Multiplication in the binary notation is simple. It amounts to (1) adding if the multiplier digit is 1 and not adding if the multiplier digit is 0, and (2) moving over or shifting. For example, let us multiply 111 by 101:

Binary Multiplication	Check
111	7
× 101	× 5
111
111
100011	35

The digit 1 in the 6th (or nth) binary place from the right in 100011 stands for 1 times 2 to the 5th (or n-1 th) power, 2 × 2 × 2 × 2 × 2 = 32. The result 100011 is translated into 32 plus 2 plus 1, which equals 35 and verifies.

Division in the binary notation is also simple. It amounts to (1) subtracting (yielding a quotient digit 1) or not subtracting (yielding a quotient digit 0), and (2) shifting. We never need to try multiples of the divisor to find the largest that can be subtracted yet leave a positive remainder. For example, let us divide 1010 (10 in decimal) into 10001110 (142 in decimal):

1110 (14 in decimal)
————
1010)10001110
1010
1111
1010
1011
1010
10 (remainder, 2 in decimal)

In decimal notation, digits to the right of the decimal point count powers of ⅒. In binary notation, digits to the right of the binary point count powers of ½: ½, ¼, ⅛, ¹/₁₆.... For example, 0.1011 equals ½ + ⅛ + ¹/₁₆, or ¹¹/₁₆.

If we were accustomed to using binary numbers, all our arithmetic would be very simple. Furthermore, binary numbers are in many ways much better for calculating machinery than any other numbers. The main problem is converting numbers from decimal notation to binary. One method depends on storing the powers of 2 in decimal notation. The rule is: subtract successively smaller powers of 2; start with the largest that can be subtracted, and count 1 for each power that goes and 0 for each power that does not. For example, 86 in decimal becomes 1010110 in binary:

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]).

Fig. 1. Abacus and notations.