G.1 Number Systems

Binary, Octal, and Hexadecimal Number System

Table G.1 lists the integers from 0 to 16, showing their equivalents in the binary ( base 2), octal ( base 8), and hexadecimal ( base 16) number systems. The shaded cells in each column show the digits in each number system.

Table G.1. Number Systems

Decimal (base 10)	Binary (base 2)	Octal (base 8)	Hexadecimal (base 16)
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

In addition to the decimal literals, Java also allows integer literals to be specified in octal and hexadecimal number systems, but not in the binary number system. Octal and hexadecimal numbers are specified with and 0x prefix, respectively. The prefix 0X can also be used for hexadecimal numbers. Note that the leading (zero) digit is not the uppercase letter O . The hexadecimal digits from a to f can also be specified with the corresponding uppercase forms ( A to F ). Negative integers (e.g., -90 ) can be specified by prefixing the minus sign ( - ) to the magnitude, regardless of number system (e.g., -0132 or -0X5A ). The actual memory representation of the integer values is discussed in Section on page 598.

Converting Binary Numbers to Decimals

A binary number can be converted to its equivalent decimal value by computing the positional values of its digits. Each digit in the binary number contributes to the final decimal value by virtue of its position, starting with position 0 (units) for the right-most digit in the number. The positional value of each digit is given by

digit  x   base   ^position

The number 101001 ₂ corresponds to 41 ₁₀ in the decimal number system:

101001 2	= 1 x2 5 + x2 4 + 1 x2 3 + x2 2 + x2 1 + 1 x2
	= 32 + 0 + 8 + 0 + 0 + 1
	= 41 10

Converting Octal and Hexadecimal Numbers to Decimals

Similarly, octal (base 8) and hexadecimal (base 16) numbers can be converted to their decimal equivalents:

0132 = 132 8	= 1 x8 2 + 3 x8 1 + 2 x8	= 64 + 24 + 2	= 90 10	Octal Decimal
0x5a = 5a 16	= 5 x16 2 + a x16	= 80 + 10	= 90 10	Hex Decimal

The same technique can be used to convert a number from any base to its equivalent representation in the decimal number system.