1. Number Systems and Circuits for Addition – Negative Integers
Lecture 7
Section 1.5
Mon, Jan 29, 2007

2. Addition of Fixed-Length Integers
Represent 150 and 106 as 8-bit integers.
150 =
106 =
Express the sum as an 8-bit integer. =
Carry-out bit is thrown away.
Conclusions
= 0.
150 = –106.

3. Example: UnsignedInt.cpp

4. Two’s Complement
For binary numbers of fixed length n, the two’s complement of a is
2n – a
If a > 0, then –a is stored as the two’s complement of a.
The two’s complement of the two’s complement of a is a.

5. Finding the Two’s Complement
To find the two’s complement of an n-bit binary number:
Reverse each bit, including leading zeros.
Add 1 to the result.
Reversing each bit is equivalent to subtracting from 111…1 = 2n – 1.

6. It’s a Matter of Interpretation :
Signed
-128 to -1
Unsigned
0 to 255
Signed
0 to 127

7. It’s a Matter of Interpretation :
Signed
-128 to -1 +1
Unsigned
0 to 255 +1
Signed
0 to 127

8. It’s a Matter of Interpretation :
Signed
-128 to -1 –1
Unsigned
0 to 255 –1
Signed
0 to 127

9. Signed Integers Let P = {0, …, 127} Let N = {–128, …, –1}
High-order bit
Members of P have high-order bit = 0.
Members of N have high-order bit = 1.
Signed vs. unsigned
If n  P, then signed(n) = unsigned(n).
If n  N, then signed(n) = unsigned(n) – 256.

10. Some Special Values 0 = 00000000 127 = 01111111 –128 = 10000000
– 1 =
What value is ?
What value is ?
What value is ?
What value is ?

11. 2-Byte vs. 4-Byte Integers
Assign a short to an int.
Exactly what bits are copied to what locations?
The sign bit must be extended.
short s = -1;
int i = s;

12. Example: ShortVsInt.cpp