G.4 Representing Integers

Integer data types in Java represent signed integer values, that is, both positive and negative integer values. The values of char type can effectively be regarded as unsigned 16-bit integers.

Values of type byte are represented as shown in Table G.2. A value of type byte requires 8 bits. With 8 bits, we can represent 28 or 256 values. Java uses 2's complement (explained later) to store signed values of integer data types. For the byte data type, this means values are in the range ?128 (?27) to +127 (27?1), inclusive.

Bits in an integral value are usually numbered from right to left, starting with the least significant bit 0 (also called the right-most bit). When applying bitwise operators, the number of the most significant bit (also called the left-most bit) is dependent on the integral type; bit 31 for byte, short, char, and int, and bit 63 for long. The representation of the signed types sets the most significant bit to 1, indicating negative values. Adding 1 to the maximum int value 2147483647 results in the minimum value -2147483648, that is, the values wrap around for integers and no overflow or underflow is indicated.

Table G.2. Representing Signed byte Values Using 2's Complement

Decimal Value

Binary Representation (8 bit)

Octal Value with Prefix 0

Hexadecimal Value with Prefix 0x

127

01111111

0177

0x7f

126

01111110

0176

0x7e

...

...

...

...

41

00101001

0123

0x29

...

...

...

...

2

00000010

02

0x02

1

00000001

01

0x01

0

00000000

00

0x0

-1

11111111

0377

0xff

-2

11111110

0376

0xfe

...

...

...

...

-41

11010111

0327

0xd7

...

...

...

...

?127

10000001

0201

0x81

?128

10000000

0200

0x80

Calculating 2's Complement

Before we look at 2's complement, we need to understand 1's complement. 1's complement of a binary integer is computed by inverting the bits in the number. Thus, 1's complement of the binary number 00101001 is 11010110. 1's complement of a binary number N2 is denoted as ~N2. The following relations hold between a binary integer N2, its 1's complement ~N2, and its 2's complement ?N2:


?N2 = ~N2 + 1
  0 = ?N2 + N2

If N2 is a positive binary integer, then ?N2 denotes its negative binary value, and vice versa. The second relation states that adding a binary integer N2 to its 2's complement ?N2 equals 0.

Given a positive byte value, say 41, the binary representation of -41 can be found as follows:

 

Binary Representation

Decimal Value

Given a value, N2:

00101001

41

Form 1's complement, ~N2:

11010110

 

Add 1:

00000001

 

Result is 2's complement, ?N2:

11010111

?41

Similarly, given a negative number, say -41, we can find the binary representation of 41:

 

Binary Representation

Decimal Value

Given a value, N2:

11010111

?41

Form 1's complement, ~N2:

00101000

 

Add 1:

00000001

 

Result is 2's complement, -N2:

00101001

41

Adding a number N2 to its 2's complement ?N2 gives 0, and the carry bit from the addition of the most significant bits (after any necessary extension of the operands) is ignored:

 

Binary representation

Decimal value

Given a value, N2:

00101001

41

Add 2's complement, ~N2:

11010111

?41

Sum:

00000000

0

Subtraction between two integers is also computed as addition with 2's complement:


N2 ? M2 = N2 + (?M2)

For example, calculating 4110 ? 310 (with the correct result 3810) is computed as follows:

 

Binary Representation

Decimal Value

Given a value, N2:

00101001

41

Add ?M2 (i.e., subtract M2):

11111101

?3

Result:

00100110

38

The previous discussion on byte values applies equally to values of other integer types: short, int, and long. These types have their values represented by 2's complement in 16, 32, and 64 bits, respectively.



     
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