1. 2’s Complement Another system that lets us represent negative numbers
MSB is STILL the sign bit, but there is no negative zero
Negative numbers count backwards and wrap around
Calculating 2’s complement (Pos  Neg)
Flip the bits ( 01 and 10 )
Add 1

2. Example ( Pos  Neg ) 110  -110 00012  11112 Flip bits: 1110
110 
00012 
Flip bits: 1110
Add 1:
+0001
1111

3. Another Example ( Pos  Neg )
2510  
Flip bits:
Add 1:

4. Your Turn
Assuming an 8-bit restriction, what is in 2’s complement?
Flip bits
Add 1
Answer:

5. Your Turn
Assuming an 8-bit restriction, what is in 2’s complement?
Flip bits
Add 1
Answer:

6. Example ( Neg Pos ) -410  410 11002  01002 Flip bits: 0011
-410 
11002 
Flip bits: 0011
Add 1:
+0001
0100

7. Another Example ( Neg  Pos ) 
Flip bits:
Add 1:

8. Your Turn
Assuming 2’s complement, what is the decimal value of ?
Flip bits
Add 1
Answer: -710

9. Your Turn
Assuming 2’s complement, what is the decimal value of ?
Flip bits
Add 1
Answer: 22

10. 2’s Complement Chart Binary Decimal 0111 7 0110 6 0101 5 0100 4 0011 3
0010 2
0001 1
0000
1111 -1
1110 -2
1101 -3
1100 -4
1011 -5
1010 -6
1001 -7
1000 -8
Binary
Decimal 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -7 -8

11. SHORTCUT! Find the 1 on the farthest right
Flip all the bits to the left of the 1 (DO NOT FLIP THE 1)
Example:
4210  

12. Awesomeness of 2’s Complement
No more negative zero
Lower minimum value: -(2N-1)
So what’s the big deal?
Everything is addition
No need for special hardware to do subtraction

13. 2’s Complement Addition
Just like normal positive binary addition
You MUST restrict the number of bits
IGNORE any overflow bits
maintain bit-size restriction

14. Positive Addition Example
= 1610
Assuming 2’s complement

15. Negative Addition Example
= -1610
NOTE: We ignored the last overflow bit on the left!

16. Your Turn
Show the binary addition of = -17

17. Subtraction Example 1610 – 410 = 1610 + -410 = 1210 000100002
NOTE: We ignored the last overflow bit on the left!

18. Your Turn
Show the binary subtraction of 23 – 10 = 13

19. Overflow / Underflow Problem
Addition and bit-size restriction allow for possible overflow / underflow
Overflow – when the addition of two binary numbers yields a result that is greater than the maximum possible value
Underflow – when the addition/subtraction of two binary numbers yields a result that is less than the minimum possible value

20. Overflow Example Assume 4-bit restriction and 2’s complement
Maximum possible value: 24-1 – 1 = 7
= 910
 not good!

21. Underflow Example Assume 4-bit restriction and 2’s complement
Minimum possible value: -(24-1) = -8
= -1010
 not good!