1. Topic 3a Two’s Complement Representation
Introduction to Computer Systems Engineering
(CPEG 323)
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2. 2’s Complement bn-1 bn-2 ……b1b0
Number and its representation are not the same thing!
Representation is a convention defined for expressing the number.
A number may have different representations, e.g. BCD, hexadecimal, 2’s complement, etc.
2’s complement: a kind of representation defined as follows
Number
Representation
bn-1 bn-2 ……b1b0
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3. Excercises What is 2’s complement representation of +3 ?
Answer: +3 = 0011 => + 3 = 0x(-2^3) = 011
How about 2’s complement representation of -3 ?
Answer: -3 = -(0011) => - 3 => 1x(-2^3) = -5 => 1101
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4. 2’s Complement bn-1 bn-2 ……b1b0 0000 1 0001 2 0010 3 0011 4 0100 5
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 -8
1000 -7
1001 -6
1010 -5
1011 -4
1100 -3
1101 -2
1110 -1
1111
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5. Signed Integers(2’s Complement)
(For 4 bits)
Reference: Katz: Contemporary Logic Design, p243 -1 +0 -2 +1
1111
0000
1110
0001 -3 +2
1101
0010 -4 +3
1100
0011
Why 1111 and 0000 are adjacent?
What is the meaning of bit 3?
What is the sign of 0?
What is the smallest number?
What is the biggest number? -5
1011
0100 +4
What’s name of this? Number wheel??
1010 -6
0101 +5
1001
0110 -7 +6
1000
0111 -8 +7
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6. 2’s Complement (Cont.) bn-1 bn-2 ……b1b0 bn-1bn-2 ……b1b0
Representing a negative number –x
From the representation of x to that of –x
Number
Representation x
bn-1 bn-2 ……b1b0 -x
bn-1bn-2 ……b1b0
Invert each bi, then add 1
Proof of correctness: an exercise!
Shortcut: Invert every bit and add 1 to the least significant bit
Example: - (0100) = = 1100
Example: =  =
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7. Fit a shorter number to longer bits
Why is it necessary?
Compare an integer with a long integer: typecasting
Load a byte to a word
To add an immediate field to a 32-bit number
MIPS ALU only works with values in 32-bit registers
How to deal with smaller sizes?
What about larger sizes?
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8. Fit a shorter number to longer bits
Example (2-bit): 10= 1 *(-21)+0 *20=-210
Representing -210 in 4-bit word: Can you do that? What is the intuition?
copy the sign bit into the other bits (sign extension) > >
Proof? Are 1010 = ?
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9. Addition & Subtraction
Addition: Just like in grade school  0001
Substraction: a-b=a+(-b)
Two's complement operations easy = 01 1 1 1
So = 0001
Have you specially handled the sign bits?
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10. Addition & Subtraction (Cont.)
Wait! There is one extra bit lost! =
10001
Why the 1 can be omitted?
Because 1 1 is
But treated in the addition as
0*23
+ 1*(-23)
1*23
0*23
+ 1*23
1*23 10
1-1=1+1=0 from the viewpoint of 1-bit.
So as long as you do not consider the carry brought by 1+1 (the green 1), you are right!
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11. Addition & Subtraction (Cont.)
Questions:
Do you have an adder and a “subtractor”?
What about unsigned numbers? (Another adder?)
Advantage of using 2’s complement
Sub can share the same logic as add
Sign bit can be treated as a normal number bit in addition.
These are the clever points!
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12. Overflow
Addition:  
overflow!
Overflow!
What is the sign of the input?
What is the sign of the output?
Can overflow occur in adding a positive and a negative number?
Can overflow occur with 0?
Consider the operations A + B, and A – B
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13. Overflow
Addition:  
What is the sign of the input?
What is the sign of the output?
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14. Detecting Overflow Overflow occurs when:
add two positives yields a negative
add two negatives gives a positive
or, subtract a negative from a positive and get a negative
or, subtract a positive from a negative and get a positive
What about addition and subtraction of unsigned numbers?
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15. What should CPU do about overflow?
Ignore it?
Don't always want to detect overflow
Addu, addiu, subu (MIPS: do not generate a overflow)
Generate a trap so that the programmer can try to deal with it?
An exception (interrupt) occurs
Control jumps to predefined address for exception
Interrupted address is saved for possible resumption
Add, sub
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16. Instructions Comparison Example: What are the values of $t0 and $t1?
Unsigned numbers
sltu: set on less than unsigned
sltiu: set on less than immediate unsigned
Signed numbers
slt: set on less than
slti: set on less than immediate
Example: What are the values of $t0 and $t1?
$s0=
$s1=
(1) slt $t0, $s0, $s1 (2) sltu $t1, $s0, $s1
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17. Example (cont’d) What are the values of $s0 and $s1?
Answer:
slt $t0, $s0, $s the result of $t0 is 1 if both are signed
sltu $t1, $s0, $s the result of $t1 is 0 if both are unsigned
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18. Instructions (Cont.) Load/Store Example: lb $s1, 100($s2)
lb: load byte
lbu: load byte unsigned
Example: lb $s1, 100($s2)
What is the values of $s1
When memory[$s2+100] = ?
When memory[$s2+100] = ?
When memory[$s2+100] = ?
Example: lbu $s1, 100($s2)
when memory[$s2+100] = ?
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19. Remarks MIPS 4000 has "Integer Arithmetic Overflow“ exception.
For most of the processors today, integer overflow will not trigger interrupt. Instead, some status bits will be set such that software may detect such situation.
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20. Remarks (cont’d)
Usually, a typical architecture may have status bits like
N - not zero
Z - zero
S - sign
O – overflow
The 4 bits make 16 combinations. They can be used to represent 16 situations (for compare and conditional branch), which includes:
signed: >=, =, <=, !=
unsigned: >=, =, <=, !=
integer overflow, underflow, and etc.
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