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Integer arithmetic. Depends what you mean by “ integer ”. Assume at 3-bit string. Then we define: zero = 000 one = 001 Use zero, one and binary addition: Zero 000 One + 001 001 Zero + one = one. Makes sense!. - PowerPoint PPT Presentation
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Computer Engineering page 1
Integer arithmetic
Depends what you mean by “integer”.
Assume at 3-bit string.– Then we define:
zero = 000
one = 001
Use zero, one and binary addition:
Zero 000
One + 001
001
Zero + one = one. Makes sense!
Computer Engineering page 2
Add one repeatedly, use up all possible patterns:
Zero 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111
Called the;
Unsigned Integer System.
No negative integers!
Computer Engineering page 3
Two additions:
2 010
+ 3 + 011
5
4 100
+ 5 + 101
9
Computer Engineering page 4
Two additions:
2 010
+ 3 + 011
5 101
Yes! 5 = 101
4 100
+ 5 + 101
9 001
But 001 represents one.
is 4 + 5 = 1???
Computer Engineering page 5
Addition of unsigned integers
Error detected by presence of
“carry”
Computer Engineering page 6
How do we subtract unsigned integers?
We need the concept of the;
“ Two’s complement”
Computer Engineering page 7
One’s complement
Take any string;
Invert every bit;
0 1
This is One’s complement. "NOT”.
Computer Engineering page 8
Two’s complement
Given a string;
One's complement;
then add one.
This is called;
two’s complement
Computer Engineering page 9
To subtract unsigned A-B
Perform:
A + 2’s compl (B)
= A + Not (B) + 1
Computer Engineering page 10
Example:
5 101 101
- 3 - 011 + 100
2 + 1
010
3 011 011
- 5 - 101 + 010
- 2 + 1
110
Carry!
=2; Good!
No Carry!
=6; BAD!
Computer Engineering page 11
Subtraction of unsigned integers
Error
detected by
absence of carry!
– Warning: Some machines invert the carry bit on subtraction
– So that "carry" => Error for both add and sub
Computer Engineering page 12
Conclusion
For unsigned arithmetic we are interested in carry
Pay attention!
I never used the word "overflow"that's something completely different.
Also notice:– 3-bit operands gave 3-bit results.
– Don't be tempted to write that 4'th bit down!
Computer Engineering page 13
How about negative numbers?
How should we represent -1?
How would we compute 0 - 1?0 + 2's compl (1)
We choose this as our "-1"
1 = 001
- 1 = 110
+ 1
111
Computer Engineering page 14
Repeatedly add -1:
Zero 000
- 1 111
- 2 110 Less than
- 3 101 zero
- 4 110
- 5 011 No!
High order bit called "sign bit"
Computer Engineering page 15
Signed 3-bit integers
3 011
2 010
1 001
0 000
-1 111
-2 110
-3 101
-4 100Not symmetrical around zero!!!
Computer Engineering page 16
Sign bit
The high order bit in a number
Also called "N"-bit
Value is negative when
this bit is "1"
Computer Engineering page 17
Let's try A + B
1 001 1 001
+2 010 +(-1) 111
3 011 0 *000
Both results is OK
But: Left case: no carry
Right case: carry
Conclusion: For signed addition carry is worthless
Same conclusion for signed subtraction
* carry
Computer Engineering page 18
Some additions A
1 0 0 1 2 0 1 0
+2 0 1 0 + 2 0 1 0
3 4
-1 1 1 1 -1 1 1 1
+(-3) 1 0 1 +(-4) 1 0 0
-4 -5
1 0 0 1 -2 1 1 0
+(-2) 1 0 0 +1 0 0 1
-1 -1
Computer Engineering page 19
Some additions B
1 0 0 1 2 0 1 0
+2 0 1 0 + 2 0 1 0
3 0 1 1 4 1 0 0
-1 1 1 1 -1 1 1 1
+(-3) 1 0 1 +(-4) 1 0 0
-4 C 1 0 0 -5 C 0 1 1
1 0 0 1 -2 1 1 0
+(-2) 1 0 0 +1 0 0 1
-1 1 1 1 -1 1 1 1
Computer Engineering page 20
Some additions C
1 0 0 1 2 0 1 0
+2 0 1 0 + 2 0 1 0
3 0 1 1 4 1 0 0
3 -4
OK BAD
-1 1 1 1 -1 1 1 1
+(-3) 1 0 1 +(-4) 1 0 0
-4 C 1 0 0 -5 C 0 1 1
-4 3
OK BAD
1 0 0 1 -2 1 1 0
+(-2) 1 0 0 +1 0 0 1
-1 1 1 1 -1 1 1 1
-1 -1
OK OK
Computer Engineering page 21
Some additions D
1 0 0 1 2 0 1 0
+2 0 1 0 + 2 0 1 0
3 0 1 1 4 1 0 0
3 -4
OK BAD
-1 1 1 1 -1 1 1 1
+(-3) 1 0 1 +(-4) 1 0 0
-4 C 1 0 0 -5 C 0 1 1
-4 3
OK BAD
1 0 0 1 -2 1 1 0
+(-2) 1 0 0 +1 0 0 1
-1 1 1 1 -1 1 1 1
-1 -1
OK OK
Computer Engineering page 22
Error during signed addition:
R = A + B
A, B same sign
and
R opposite sign
called overflow
Notice: Mathematically, signed addition is the same as unsigned addition
The same is true for signed subtraction and unsigned subtraction
A - B –> A + (-B) –> A + 2's compl (B)
Computer Engineering page 23
Some subtractions A
3 0 1 1 3 0 1 1
- 1 + 1 1 1 -(-1) + 0 0 1
2 4
-1 1 1 1 1 0 0 1
- (-1) + 0 0 1 - (-1) + 0 0 1
0 2
- 3 1 0 1 - 4 1 0 0
- 1 + 1 1 1 - 1 + 1 1 1
-4 - 5
Computer Engineering page 24
Some subtractions B
3 0 1 1 3 0 1 1
- 1 + 1 1 1 -(-1) + 0 0 1
2 C 0 1 0 4 1 0 0
-1 1 1 1 1 0 0 1
- (-1) + 0 0 1 - (-1) + 0 0 1
0 C 0 0 0 2 0 1 0
- 3 1 0 1 - 4 1 0 0
- 1 + 1 1 1 - 1 + 1 1 1
-4 C 1 0 0 - 5 C 0 1 1
Computer Engineering page 25
Some subtractions C
3 0 1 1 3 0 1 1
- 1 + 1 1 1 -(-1) + 0 0 1
2 C 0 1 0 4 1 0 0
2 -4
OK BAD
-1 1 1 1 1 0 0 1
- (-1) + 0 0 1 - (-1) + 0 0 1
0 C 0 0 0 2 0 1 0
0 2
OK OK
- 3 1 0 1 - 4 1 0 0
- 1 + 1 1 1 - 1 + 1 1 1
-4 C 1 0 0 - 5 C 0 1 1
-4 3
OK BAD
Computer Engineering page 26
Some subtractions D
3 0 1 1 3 0 1 1
- 1 + 1 1 1 -(-1) + 0 0 1
2 C 0 1 0 4 1 0 0
2 -4
OK BAD
-1 1 1 1 1 0 0 1
- (-1) + 0 0 1 - (-1) + 0 0 1
0 C 0 0 0 2 0 1 0
0 2
OK OK
- 3 1 0 1 - 4 1 0 0
- 1 + 1 1 1 - 1 + 1 1 1
-4 C 1 0 0 - 5 C 0 1 1
-4 3
OK BAD
Computer Engineering page 27
Error during signed subtraction:
R = A - B
A, B different sign
and
B, R same sign
called overflow
Computer Engineering page 28
Arithmetic- logic unit (ALU)
C = carry
V = overflow
N = sign bit of R
Z = 1 if R = 0
32
32
32
A
B
C
Operation
Condition codesC, V, N, Z
Computer Engineering page 29
Compare two unsigned numbers?
is A < B ?
Easy! Compute A - B and examine carry
But – to compare two signed numbers?
is A < B ?
Most common mistake:– Compute R = A - B, then look at sign of
R.
– If R < 0 then A < B (N-bit)
Not good enough!
Computer Engineering page 30
To compare two signed numbers:
What about– A = - 4
– B = 3
– - 4 100
– - 3 + 101
– c 001
“If R neg then A < B”–
We conclude A ≥ B, that is - 4 ≥ 3
Wrong!
Computer Engineering page 31
Some examples A
3 0 1 1 3 0 1 1
- 1 + 1 1 1 -(-1) + 0 0 1
-1 1 1 1 1 0 0 1
- (-1) + 0 0 1 - (-1) + 0 0 1
- 3 1 0 1 - 4 1 0 0
- 1 + 1 1 1 - 1 + 1 1 1
Computer Engineering page 32
Some examples B
3 0 1 1 3 0 1 1
- 1 + 1 1 1 -(-1) + 0 0 1
2 C 0 1 0 4 1 0 0
3 < +1? No! 3 < -1? No!
-1 1 1 1 1 0 0 1
- (-1) + 0 0 1 - (-1) + 0 0 1
0 C 0 0 0 2 0 1 0
-1 < -1? No! 1 < -1? No!
- 3 1 0 1 - 4 1 0 0
- 1 + 1 1 1 - 1 + 1 1 1
-4 C 1 0 0 - 5 C 0 1 1
-3 < +1? Yes! -4 < 1? Yes!
Computer Engineering page 33
Some examples C
3 0 1 1 3 0 1 1
- 1 + 1 1 1 -(-1) + 0 0 1
2 C 0 1 0 4 1 0 0
3 < +1? No! 3 < -1? No!
N = 0, V = 0 N = 1, V = 1
-1 1 1 1 1 0 0 1
- (-1) + 0 0 1 - (-1) + 0 0 1
0 C 0 0 0 2 0 1 0
-1 < -1? No! 1 < -1? No!
N = 0, V = 0 N = 0, V = 0
- 3 1 0 1 - 4 1 0 0
- 1 + 1 1 1 - 1 + 1 1 1
-4 C 1 0 0 - 5 C 0 1 1
-3 < +1? Yes! -4 < 1? Yes!
N = 1, V = 0 N = 0, V = 1
Computer Engineering page 34
To compare signed numbers:
Compute R = A - B
A < B true if N and V are different
A<B = exor(N,V) after computation