Lecture 2
Lecture 19 Integers, Divisions and Algorithms (Cont.)
Goals and Objectives:
• Understand how signed integers are stored in computer
• Understand 1’s and 2’s complements
• Understand the range of integer
Read 2.4, 2.5, p. 153-179
2. Integer Representations: (32 bits) (Cont.)
2. Signed Integer: use the left most bit as the sign bit
if sign bit=0, then positive #
if sign bit=1, then negative #
A. Positive
(1) Largest
231 - 1
Binary representation: 01111111 11111111 11111111 11111111
(2) Smallest : 1
B. Negative
Decimal change to Binary Algorithm:
(1) Find Binary representation of the positive (or unsigned) value
(2) use the corresponding (1's or 2's complement) technique
(i) 1's complement (change 0 to 1 and 1 to 0)
(ii) 2's complement (the 1's complement representation + 1)
Ex. Using 6 bits to represent integer -12 inside computer
-1210 = ?2
Solution:
(1) Find Binary representation of the positive (or unsigned) value
12 = 0011002
(2)(i) 1's complement
by 1's complement,
Answer: -12 = 110011 in computer using 6 bits
(ii) 2's complement
by 2's complement, -12 = 110011
+ 1
________
110100
Answer: -1210 = 110100 in computer using 6 bits
Example. An 8 bits computer uses 1's complement to represent negative
integers. What are the binary representations of +23 and -23?
What about 2's complement?
Algorithm:
(1) Find the positive signed (or unsigned) representation (Binary)
(Note: the leftmost bit =0 represents + sign)
i.e. +23= 0001 01112
For -23:
(i) 1's complement
apply it to the positive signed integer above using 1's complement
(by interchange 0 with 1 and vice versa)
i.e. -23 = 1110 10002
Note: Computer recognizes 1110 1000 by doing the following:
(a) check left most bit
since the bit is 1, it puts the result sign as -
(b) do 1's complement
the result is 0001 0111, which is 23 in decimal system
Then the answer is -23
(ii) 2's complement
apply it to the positive signed integer above using 2's complement
(by interchange 0 with 1 and vice versa then add 1 to it)
i.e. -23 = 1110 1001
Note: Computer recognizes 1110 1001 by doing the following:
(a) check left most bit
since the bit is 1, it puts the result sign as -
(2) do 2's complement
the result is 0001 0111, which is 23 in decimal system
Then the answer is -23
Disadvantages of using 1's complement for negative integers:
(1) 2 representations for value 0:
Ex. For an 8-bit machine:
+010 = 000000002
-010 = 111111112
However, use 2's complement, 0 has only one representation:
-010 = 000000002 = +010
(2) Do not follow usually arithmetic rules:
Ex. For a 6-bit machine:
-610 = 1110012
-1710 = 1011102
Then
-610 + (-17) 10 = -2310 = 1010002
However,
1110012 + 1011102 [pic] 1010002
On the other hand, use 2's complement, the usual arithmetic
operation such as + follows the usual rule:
-610 = 1110102
-1710 = 1011112
Then
-610 + (-17)10 = -2310 = 1010012
1110102 + 1011112 = 1010012
Summary:
A 8 bits computer have integer range
(1) Unsigned integer
8
0 - 255(= 2 - 1), total 256 representations
The largest integer (255) representation 1111 1111
(2) Signed integer(divide 256 representations into 2 equal parts, one
for + and the other for -)
(i) 1's complement(total 255 representations, missing one representation)
-127 - +127
0111 1111 is the representation of +127
1000 0000 is the representation of -127
(ii) 2's complement(total 256 representations)
-128 - +127
0111 1111 is the representation of +127
1000 0000 is the representation of -128
A 3 bits computer have integer range
(1) Unsigned integer
0 - 7
(2) Signed integer
(i) 1's complement
-3 - +3
(ii) 2's complement
-4 - +3
Ex. Given a 6 bit machine,
(1) What is the largest unsigned integer?
Ans: 26-1 = 63
Ans: 0
(2) What is the smallest unsigned integer?
Ans: 0
(3) What is the largest signed integer using 1's complement?
divide the unsigned integer range (0 - 63) into half of positive and half of negative integers:
we have -31 to -0 and +0 to +31. Note: -0 representation is usually not used.
Ans: 31
(4) What is the smallest signed integer using 1's complement?
Ans: -31
(5) What is the largest signed integer using 2's complement?
divide the unsigned integer range (0 - 63) into half of positive and half of negative integers:
we have -32 to -1 and 0 to +31
Ans: 31
(4) What is the smallest signed integer using 2's complement?
Ans: -32
Assignment #15
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