1. COMS 161 Introduction to Computing
Title: Numeric Processing
Date: October 27, 2004
Lecture Number: 25

2. Announcements
Homework 7
Due Wednesday, 11/03/2004

3. Review Integers Numeric Processing Magnitude representation
Sign-magnitude representation

4. Outline Integers Numeric Processing Real numbers
Ones’s complement representation
Two’s complement representation
Real numbers

5. Sign-Magnitude
Mathematical operations sometimes give an incorrect result
4 – 3 = = 1
+(-3) 1 1 1 1 1 -7

6. Ones compliment
Positive integers have the most significant bit (leftmost) equal to 0
The magnitude of positive numbers is the same as the signed magnitude representation
Negative integers have the most significant bit equal to 1
The magnitude of a negative number is the same as the magnitude representation of a different number

7. Ones compliment Unsigned decimal Ones complement Signed decimal 000 1
000 1
001 2
010 3
011 4
100 -3 5
101 -2 6
110 -1 7
111 -0
-3 = 23 – 1 – 3 = 8 – 4 = 4
-2 = 23 – 1 – 2 = 8 – 3 = 5
-1 = 23 – 1 – 1 = 8 – 2 = 6
-0 = 23 – 1 – 0 = 8 – 1 = 7

8. Ones compliment Example: let n = 8, and N = 5
There is something special here and in the previous table
Negatives are made from the positive by inverting each bit!
Simpler hardware for arithmetic operations
Still two representations of zero

9. Ones complement

10. Ones complement
Mathematical operations sometimes give an incorrect result using this representation
Two representations of zero
4 – 3 = = 1
4 – 2 = = 2
+(-3)
+(-2) 1 1 2 1

11. Twos Complement
Positive integers have the most significant bit (leftmost) equal to 0
The magnitude of positive numbers is the same as signed magnitude and ones complement representations
Negative integers have the most significant bit (leftmost) equal to 1
Negative numbers can be computed as:

12. Two’s Complement Example: let n = 8, and N = 5
There is something special here
ones’ complement
two’s complement
It is simple to determine the representation of a negative number in ones complement given the positive
It is easy to convert a ones complement representation to a twos complement representation by simply adding 1

13. Twos Complement