Binary Addition and Subtraction

Tom Kelliher, CS 220

Oct. 10, 2001

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Assignment

Read 3.10

From Last Time

Decoders, encoders, muxes.

Outline

1. Binary Adders

2. Unsigned binary subtraction; complements.

Coming Up

Combined signed addition/subtraction.

Binary Adders

1. Two half-adders and an OR gate give us a full binary adder. In the text, note how an earlier computed XOR is used to eliminate a gate from the carry equation.

2. Full binary adder: three inputs, two outputs.

3. Ripple carry adder: example of reuse and divide and conquer.
   1. Wire together n full binary adders in order to add two n bit numbers.

      Wiring example.

   2. Running time of a ripple carry adder.

      Running time of a full binary adder is O. Ripple carry adder?

      Excessive!!!

4. An O (!!!) adder.
   1. Important equations (briefly explain):

      Carry generate at bit position i: .

      Carry propagate at position i: .

   2. Carry-in is .

      .

      .

      .

      Etc.

   3. What's the circuit depth of ?

   4. What's wrong with this picture?

5. Carry lookahead addition.
   1. A different type of divide and conquer adder using a tree hierarchy to compute and distribute carry information.

   2. What's the height of a binary tree?

      (CLA is not binary, more like quad, but that's ok.)

   3. What's the running time of a CLA?

Unsigned Binary Subtraction

Unsigned here means we can use a minus sign. Realistic?

Let A = 110101 and B = 011010. Compute A - B and B - A.

A - B: fine. B - A: borrow out of msb.

1. Actual value computed: .

2. We want .

3. So, compute .

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?

Adding to subtract:

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.