Convert the decimal number 35 to binary. Solution 0 1 2 4 8 17 35
Binary 1 0 0 0 1 1
3.3
Integer
Representation
Range of Integers
An integer can be positive (正整數) or negative (負整數)
To use computer memory more efficiently, Integers can be represented as unsigned (無號：無正負號) or signed (有號：有正負號)numbers
There are three major methods of signed number representation:
Sign-and-magnitude (符號大小)
One’s complement (1的補數)
Two’s complement (2的補數)
Taxonomy of Integers
-8 = 1001000 11110111 11111000 8 = 00001000
Unsigned Integer
# of Bits ---------
8
16 Range -------------------------------------
0 ~ 255
0 ~ 65,535 Unsigned integer range: 0 … (2N-1)
Storing unsigned integers process:
The number is changed to binary
If the number of bits is less then N, 0s are to the left of the binary number so that there is a total of N bits
Example
Store 7 in an 8-bit memory location. Solution First change the number to binary 111.
Add five 0s to make a total of N(8) bits, 00000111.
The number is stored in the memory location.
Example
Store 258 in an 16-bit memory location. Solution First change the number to binary 100000010.
Add sevent 0s to make a total of N(16) bits, 0000000100000010.
The number is stored in the memory location.
Example of Storing Unsigned Integers in Two Different Computers
Decimal ------------
7
234
258
24,760
1,245,678 8-bit allocation
------------
00000111
11101010
overflow
overflow
overflow 16-bit allocation
----------------------------
0000000000000111
0000000011101010
0000000100000010
0110000010111000
overflow Unsigned numbers are commonly used for
counting and addressing
Example
Interpret 00101011 in decimal if the number was stored as an unsigned integer. Solution Binary 0 0 1 0 1 0 1 1 Weights 128 64 32 16 8 4 2 1
----------------------------------------- 0 + 0 + 32 + 0 + 8 + 0 + 2 + 1 Decimal 43
Sign-and-Magnitude Integers
# of Bits ----------
8
16
32 -127 -0
-32767 -0
-2,147,483,647 -0 +0 +127
+0 +32767
+0 +2,147,483,647 Range
------------------------------------------------------- Range: -(2N-1-1) … +(2N-1-1)
Storing sign-and-magnitude integers process:
The number is changed to binary; the sign is ignored
If the number of bits is less then N-1, 0 are add to the left of binary number so that there is a total of N-1 bits
If the number is positive, 0 is added to the left (to make it N bits). If the number is negative, 1 is added to thel left
Sign-and-Magnitude Integers
In sign-and-magnitude representation, the leftmost bit defines the sign of the number
If it is 0, the number is positive(正數)
If it is 1, the number is negative(負數)
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