D.2. Abbreviating Binary Numbers as Octal and Hexadecimal Numbers

The main use for octal and hexadecimal numbers in computing is for abbreviating lengthy binary representations. Figure D.7 highlights the fact that lengthy binary numbers can be expressed concisely in number systems with higher bases than the binary number system.

Figure D.7. Decimal, binary, octal and hexadecimal equivalents.

Decimal number	Binary representation	Octal representation	Hexadecimal representation
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F
16	10000	20	10

A particularly important relationship that both the octal number system and the hexadecimal number system have to the binary system is that the bases of octal and hexadecimal (8 and 16 respectively) are powers of the base of the binary number system (base 2). Consider the following 12-digit binary number and its octal and hexadecimal equivalents. See if you can determine how this relationship makes it convenient to abbreviate binary numbers in octal or hexadecimal. The answer follows the numbers.

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Binary number	Octal equivalent	Hexadecimal equivalent
100011010001	4321	8D1

To see how the binary number converts easily to octal, simply break the 12-digit binary number into groups of three consecutive bits each, starting from the right, and write those groups over the corresponding digits of the octal number as follows:

100 011 010 001
4 3 2 1

Note that the octal digit you have written under each group of thee bits corresponds precisely to the octal equivalent of that 3-digit binary number, as shown in Fig. D.7.

The same kind of relationship can be observed in converting from binary to hexadecimal. Break the 12-digit binary number into groups of four consecutive bits each, starting from the right, and write those groups over the corresponding digits of the hexadecimal number as follows:

1000 1101 0001
8 D 1

Notice that the hexadecimal digit you wrote under each group of four bits corresponds precisely to the hexadecimal equivalent of that 4-digit binary number as shown in Fig. D.7.