Number Representation

Tom Kelliher, CS 220

Sept. 29, 1997

Announcements

Review on Wednesday.

Assignment

Read Chapter 4.

Number Representation Selection

Numerous ways of representing a value. Selection criteria:

1. Ease of implementing: addition, negation (subtraction), comparison.

2. Symmetry.

3. Speed.

4. Hardware requirements.

One of the ``holy wars'' in architecture:

1. Memory is byte addressable.

2. A word is multiple bytes/bits.

3. How are the bytes/bits numbered?

4. Little endian.

5. Big endian.

Integer Representations

Bit numbering:

For examples, assume b = 8.

Unsigned

Unsigned value of an binary integer:

Maximum, minimum values and representations?

Sign-and-Magnitude Representation

Msb is the sign bit:

1. Has no weight.

2. If 0, the number is ``positive.''

3. If 1, the number is ``negative.''

4. Value:

   

5. Problems:
   1. Multiple representations of 0.

   2. Hard to build good adders.

6. Minimum, maximum values and representations?

7. Symmetric range.

8. Sign extension. Why is this important?

1's-Complement Representation

Msb is the sign bit:

1. Has negative weight: .

2. If 0, the number is ``positive.''

3. If 1, the number is ``negative.''

4. One's complement:

   

5. Value:

   

   How did we get the weight for ?

6. Problems:
   1. Multiple representations of 0.

   2. Wrap-around carrys.

7. Minimum, maximum values and representations?

8. Symmetric range.

9. Sign extension.

2's-Complement Representation

Msb is the sign bit:

1. Has negative weight: .

2. If 0, the number is ``positive.''

3. If 1, the number is ``negative.''

4. Two's complement:

   

   One's complement plus one.

5. Value:

   

6. Problems:
   1. Asymmetric range.

7. Minimum, maximum values and representations?

8. Sign extension. Why does this work?

Bias Representation

1. Add a bias to a value to get the representation.

2. Bias 127: add 127 to value.

3. value:

   

   where B is the bias.

4. Minimum, maximum values and representations?

5. Used for exponent representation in floating point.

Example Conversions

Represent each of the following in sign-magnitude, one's complement, two's complement, and bias 127 form: 1, -1, 127, -127, 128, -128, 45, -45.

Character Representation

1. ASCII.

2. One character per byte.

3. Packed formats --- importance of shifts.