1. COMS 361 Computer Organization
Title: Number Representations
Date: 10/28/2004
Lecture Number: 16

2. Announcements
Homework 7
Due Tuesday, 11/02/2004

3. Review Addressing modes Representing numbers in binary
Unsigned (magnitude)
Sign-magnitude
Addition
Subtraction

4. Outline Representing numbers in binary Problems One’s complement
Unsigned (magnitude)
Sign-magnitude
One’s complement
Two’s complement

5. Problems Unsigned (magnitude) Sign-magnitude
No representation for negative numbers
Sign-magnitude
Two representations for zero, +0 and -0
Arithmetic operations do not always produce the correct answer in the sign-magnitude representation

6. One’s 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
Negative numbers in n-bit one’s complement has the same binary representation as the unsigned binary number given by:

7. One’s compliment Unsigned decimal One’s complement Signed decimal 000
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. One’s 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. One’s complement

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

11. Two’s Complement
Positive integers have the most significant bit (leftmost) equal to 0
The magnitude of positive numbers is the same as signed magnitude and one’s complement representations
Negative integers have the most significant bit (leftmost) equal to 1
Negative numbers in n-bit one’s complement has the same binary representation as the unsigned binary number given by:

12. Two’s Complement Example: let n = 8, and N = 5
There is a relationship between one’s and two’s complement numbers
ones’ complement
two’s complement
Two’s complement representation can be computed from a numbers one’s complement representation simple by adding one

13. Two’s Complement
It is simple to determine the representation of a negative number in one’s complement given the unsigned (magnitude) representation
Use the formula
Invert the bits
It is easy to convert a one’s complement representation to a two’s complement representation by simply adding 1 to the one’s complement representation

14. Two’s Complement

15. Two's Complement Subtraction
Two's complement operations easy
Subtraction using addition of negative numbers
X - Y = X + (-Y)
Build an adder for addition, use it for subtraction!
X = 6, Y = 5
X-Y = 6+(-5)
0110
+1011
10001

16. Arithmetic Two's complement mathematical operations
Compute the correct answer
In the two’s complement representation
Two’s complement is used in nearly all computers today
X = 4, Y = 3
X - Y = = 4 + (-3)
0100
+1101
0001
X = 5, Y = 6
X - Y = = 5 + (-6)
0101
+1010
1111

17. Two’s complement Sign Bit Sign Extension
All negative numbers have a 1 in the most significant bit
Hardware need test only this bit to determine if the number is positive or negative
Sign Extension
Duplicate the sign in all bits to the left in a larger representation of a number
>
>

18. Two’s complement
Load byte (lb) assumes an signed byte and sign extends the 24 left most bits
Load byte unsigned (lbu) do not sign extend

19. Two’s complement Addresses Signed integers represent values
Start at 0 and continue to the largest address
Represented with an unsigned integer
Signed integers represent values
Important when comparing two numbers
A 1 in the sign bit may mean the number is negative and is less than all numbers with a 0 in the sign bit
A 1 in the sign bit may mean the number is positive and larger than all numbers with a 0 in this position

20. Two’s complement Comparing numbers Signed comparison
Set on less than (slt)
Set on less than immediate (slti)
Unsigned comparison
Set on less than unsigned (sltu)
Set on less than unsigned immediate (sltui)

21. Two’s complement Overflow A value to large for its representation
Represent the number 300 using 8 bits
The sign bit is 0 when the number is negative or a 1 when the number is positive

22. Two's ComplementOverflow
Overflow cannot occur
Adding two operands of opposite sign
Both operands must be represented
Sum can be no larger than one of the operands
Subtracting two operands of the same sign
Subtraction done by negating and add
Adding numbers of opposite sign

23. Two's Complement Overflow
Overflow can occur
Adding two positive numbers with a negative result
Adding two negative numbers with a positive result
Needed one extra bit to contain the sign
Real sign bit is set to a value not to a sign
Only the sign bit is wrong
Carry out into the sign bit 1
4-bit two’s complement numbers

24. Two's Complement Overflow
Overflow can occur
Subtracting a negative number from a positive number with a negative result
Subtracting a positive number from a negative number with a positive result

25. Unsigned Overflow Ignore overflow in unsigned numbers
MIPS has different arithmetic instructions for two’s complement and unsigned numbers
Signed (Overflow)
add, addi, sub
(Unsigned) No overflow
addu, addiu, subu
C/C++ ignores overflow
Compilers always generate unsigned instructions

26. Overflow detection MIPS using an exception (interrupt)
Unscheduled function call
Next instruction executed is the first in the exception routine
Usually a predefined address
Upon completion of the exception, normal program execution continues