E BINARY NUMBERS

This appendix briefly introduces binary numbers and the simple logic and arithmetic operations that can be performed with them. These form the basis of all digital systems and coding operations discussed in earlier chapters.

E.1 INTRODUCTION TO BINARY NUMBERS

A decimal number can be expanded into its constituent components, with each digit being shown with its associated power of ten (the subscript in the Equations (E.1) and (E.2) indicates the base of the number system—10 for decimal, 2 for binary):

4327_{10} = (4 \times 1 0^{3}) + (3 \times 1 0^{2}) + (2 \times 1 0^{1}) + (7 \times 1 0^{0})

(E.1)

The location of a digit in a decimal number therefore serves as a place holder. The further towards the left of a number the digit lies, the larger the power of ten by which it is multiplied, and thus the greater its contribution to the value of the number (or, we could say, the greater its significance in the total value).

A similar expansion can be written for a binary number (that is, when it is represented in base-2), for example:

1101 0_{2} = (1 \times 2^{4}) + (1 \times 2^{3}) + (0 \times 2^{2}) + (1 \times 2^{1}) + (0 \times 2^{0})

(E.2)

The binary number 11010 is therefore equivalent to the decimal number 26. Table E.1 lists all five-bit binary numbers and their decimal equivalents.

Table E.1. 5-bit binary numbers and their decimal equivalents.

Binary	Decimal	Binary	Decimal
00000	0	10000	16
00001	1	10001	17
00010	2	10010	18
00011	3	10011	19
00100	4	10100	20
00101	5	10101	21
00110	6	10110	22
00111	7	10111	23
01000	8	11000	24
01001	9	11001	25
01010	10	11010	26
01011	11	11011	27
01100	12	11100	28
01101	13	11101	29
01110	14	11110	30
01111	15	11111	31

An n-bit binary number can represent values in the range between 0 and 2ⁿ–1:

A four-bit binary number can be used to represent values between 0 and 15 (2⁴–1).
An eight-bit binary number can represent values between 0 and 255 (2⁸–1).
and so on.