1. Addition and Subtraction

2. Learning Objectives
Understand 2’s complement system is the most efficient to perform addition and subtraction
Learn to draw logic circuits
Way to detect overflow

3. Basic Idea Hardware can only deal with binary digits, 0 and 1.
Must represent all numbers, integers or floating point, positive or negative, by binary digits, called bits.
Devise electronic circuits to perform arithmetic operations (add, subtract, multiply and divide) on binary numbers.

4. Positive Integers = 32 + 8 +1 = 41
Decimal system: made of 10 digits, {0,1,2, , 9}
41 = 4× ×100
255 = 2× × ×100
Binary system: made of two digits, {0,1}
= 0× ×26 + 1×25 + 0× ×23 + 0× ×21 + 1×20
= = 41
= 255, largest number with binary digits, 28-1

5. Writing Numbers 2 5 5 0 0 1 0 1 0 0 1 Most significant digit
Most significant digit
Least significant digit
Least significant bit
(LSB)
Most significant bit
(MSB)

6. Base or Radix For decimal system, 10 is called the base or radix.
Decimal 41 is also written as 4110 or 41ten
Base (radix) for binary system is 2.
Thus, 41ten = or two
Also, 111ten = two
and two = 7ten

7. Signed Integers
Use left-most bit (called most significant bit or MSB) for sign:
0 for positive
1 for negative
Example: +18ten = two
- 18ten = two

8. Why Not Use Sign Bit
Sign and magnitude bits should be differently treated in arithmetic operations.
Addition and subtraction require different logic circuits.
Overflow is difficult to detect.
“Zero” has two representations:
+0ten = two
- 0ten = two
Signed-integers are not used in modern computers.

9. Integers With Sign – Other Ways
Use fixed-length representation, but no sign bit
1’s complement: To form a negative number, complement each bit in the given number.
2’s complement: To form a negative number, start with the given number, subtract one, and then complement each bit, or
first complement each bit, and then add 1.
2’s complement is the most preferred representation.

10. 1’s-Complement
To change the sign of a binary integer simply complement (invert) each bit.
Example: 3 = 0011, -3 = 1100
n-bit representation: Negation is equivalent to subtraction from 2n – 1
Infinite
universe
-3 0 3
0 3 6
Modulo-16
universe

11. 2’s-Complement Add 1 to 1’s-complement representation.
Some properties:
Only one representation for 0
Exactly as many positive numbers as negative numbers
Slight asymmetry – there is one negative number with no positive counterpart

12. Three Representations
Sign-magnitude
000 = +0
001 = +1
010 = +2
011 = +3
100 = - 0
101 = - 1
110 = - 2
111 = - 3
1’s complement
000 = +0
001 = +1
010 = +2
011 = +3
100 = - 3
101 = - 2
110 = - 1
111 = - 0
2’s complement
000 = +0
001 = +1
010 = +2
011 = +3
100 = - 4
101 = - 3
110 = - 2
111 = - 1
(Preferred)

13. 2’s Complement Numbers 1 2 3 -1 -2 -3 -4 000 001 010 011 100 101 1101 2 3 -1 -2 -3 -4
000
001
010
011
100
101
110
111

14. 2’s Complement n-bit Numbers
Range: - 2n-1 through 2n-1- 1
Unique zero:
Expansion of bit length: stretch the left-most bit all the way, e.g., is still – 3.
Overflow rule: If two numbers with the same sign bit (both positive or both negative) are added, the overflow occurs if and only if the result has the opposite sign.
Subtraction rule: for A – B, add – B to A.

15. Converting 2’s Compliment to Decimal
an-1an a1a0 = -2n-1an-1 + Σ 2i ai
i=0
8-bit conversion box
Example
= = -3

16. 32-bit Integers
What is the range of integers (positive and negative) that can be represented?
Positive integers: 0 to 2,147,483,647
Negative integers: - 1 to - 2,147,483,648
What are the binary representations of the extreme positive and negative integers?
= = 2,147,483,647
= = - 2,147,483,648
What is the binary representation of zero?

17. Addition Adding bits: Adding integers: carry 1 1 0
0 + 0 =
0 + 1 =
1 + 0 =
1 + 1 = (1) 0
Adding integers:
carry
two = 7ten
two = 6ten
= (1) 1 (1)0 (0)1 two = 13ten

18. Subtraction Direct subtraction Two’s complement subtraction
two = 7ten
two = 6ten
= two = 1ten
two = 7ten
two = - 6ten
= (1) 0 (1) 0 (0)1 two = 1ten

19. Overflow: An Error -2-3 = -5 110 = -2 3+2 = 5 011 = 3
Examples: Addition of 3-bit integers (range - 4 to +3)
-2-3 = = -2
= -3
= = 3 (error)
3+2 = = 3
010 = 2
= 101 = -3 (error)
Overflow rule: If two numbers with the same sign bit (both positive or both negative) are added, the overflow occurs if and only if the result has the opposite sign.
000
111
001 1 -1
010 2
110 -2 -3 3
011
101 -4
100

20. Adders

21. Addition of Unsigned Numbers – Half Adder

22. Addition and Subtraction of Signed Numbersx y
Carry-in c
Sum s
Carry-out c i i i i i +1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 s = x y c + x y c + x y c + x y c = x Å y i i i i i i i i i i i i i i i Å c i c = y c x c x y + + i +1 i i i i i i E
xample: X 7 1 1 1 x
Carry-out i
Carry-in + Y =
+ 6 = + 1 1 y 1 1 c i c i +1 i Z 13 1 1 1 s i
Legend for stage i
Logic specification for a stage of binary addition.

23. Addition and Subtraction of Signed Numbers
A full adder (FA) y i c x i i x y s i c i i c i + 1 i c i x x y i i i y i c
Full adder c i + 1 (F A) i s i
(a) Logic f
or a single stage

24. Addition and Subtraction of Signed Numbers
n-bit ripple-carry adder x y x y x y n - 1 n - 1 1 1 c c n - 1 1 c n F A F A F A c s s s n - 1 1
Most significant bit
Least significant bit
(MSB) position
(LSB) position
(b) An n
-bit r
ipple-carr
y adder

25. Addition and Subtraction of Signed Numbers
kn-bit ripple-carry adder x y x y x y x y x y k n - 1 k n - 1 2 n - 1 2 n - 1 n n n - 1 n - 1 c n -
bit n -
bit n n -
bit c c k n
adder
adder
adder s s s s s s k n - 1 ( k - 1 ) n 2 n - 1 n n - 1
(c) Cascade of k n-bit adders
. Logic for addition of binary vectors.

26. Addition and Subtraction of Signed Numbers
Addition/subtraction logic unit