Signed Binary Addition and Subtraction

Tom Kelliher, CS 220

Oct. 12, 2001

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Assignment

Read 3.13

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Decoders, encoders, muxes.

Outline

1. Complements.

2. Subtraction using 2's complement.

3. Signed numbers.

4. Combined 2's complement adder/subtractor.

Coming Up

VHDL.

Complements

Used for signed representations.

1. Diminished radix complement: 1's complement.
   1. The 1's complement of an n bit binary number A is .

   2. What's the bit representation of ? The one's complement of A? A plus its one's complement?

2. Radix complement: 2's complement.
   1. The 2's complement of an n bit binary number A is .

   2. 1's complement plus one.

      Two's complement of A? A plus its two's complement?

Subtraction Using 2's Complement

Subtract by adding!

Adding works the same.

1. Denote the 2's complement of B as .

   Recall .

2. .

   Note we should get a carry out of the msb when we perform .

3. Work the two examples again.

Signed Numbers

Skip sign-magnitude representation.

1. The Complement (1's, 2's) of a number is its additive identity.

   Well, almost. What's the 1's complement of 0? In 1's complement, what does a number and its complement add to?

2. Msb is sign bit. Weight of sign bit. 2's complement: . 1's complement: .

   Bit patterns for: most positive number, most negative number, 1, -1.

3. Range:
   1. 2's complement

   2. 1's complement

Practice

For six bit numbers, what is the range of:

1. Unsigned integers.

2. 1's complement integers.

3. 2's complement integers.

In 1's and 2's complements, what are the representations of 15, -18, 27, -4, 33, -32, -35, 10?

A 2's Complement Combined Adder/Subtractor

1. 2's complement: invert bits, add one.

2. EXOR gate can be used as a conditional inverter.

3. We're not using for anything.

Computes A + B or A - B.