CSC
133  Discrete Mathematical Structures 
Each of these data types for representing integers is signed: left most bit 0 means positive, left most bit 1 means negative. The following table describes the integer data types:
Primitive data type 
Memory allocation 
Min / Max value 
byte 
1 byte 
128 / 127 
short 
2 bytes 
32,768 / 32,767 
int 
4 bytes 
2,147,483,648 / 2,147,483,647 
long 
8 bytes 
9,223,372,036,854,775,808 / 9,223,372,036,854,775,807 
Two's complement is the standard convention used to represent signed binary integer data types in many programming languages including Java, C, and most others. Python, however uses arbitrary length integer type so word size is only specified at the lower levels after interpretation takes place. Using two's complement avoids having +0 and 0 values, which could complicate hardware requirements. To represent a number in two's complement first the word size must be determined so the position of the left most bit is known, next note that all positive numbers have a 0 as the left most bit. Inverting all bits in the positive number and adding 1 derives a negative number. To read a negative number reapply the same algorithm to get the positive value and insert the negative sign. A shortcut for applying two's complement: copy all the bits from the least significant (rightmost) bit up to and including the first 1, then invert the rest. See Complements in Radix r for information on how to apply this technique for radix other than 2. Consider the table below which shows the binary representation of Java byte data type (spaces added to binary numbers to improve readability):
Byte Data Type  

Decimal 
8 bit Binary 
invert 
add 1 for 
Decimal 
0 
0000 0000 
 
 
 
+1 
0000 0001 
1111 1110 
1111 1111 
1 
+2 
0000 0010 
1111 1101 
1111 1110 
2 
+3 
0000 0011 
1111 1100 
1111 1101 
3 
+4 
0000 0100 
1111 1011 
1111 1100 
4 
+5 
0000 0101 
1111 1010 
1111 1011 
5 
¯ 



¯ 
124 
0111 1100 
1000 0011 
1000 0100 
124 
125 
0111 1101 
1000 0010 
1000 0011 
125 
126 
0111 1110 
1000 0001 
1000 0010 
126 
127 
0111 1111 
1000 0000 
1000 0001 
127 
 
 
 
1000 0000 
128 
Complements in Radix r  Base Conversion  
Primitive Data Types:  Reals  Number Systems  Practice 