Computer Arithmetic

1
Lecture 13

• Computer Arithmetic

2
Single Bit Arithmetic

• We can add two bits according to the following
  truth table

Designing with minterms gives us the Boolean
equations SUM A' B A B'
CARRY A B
3
Exclusive Or

• The truth table for sum is an exclusive or
• SUM A' B A B' A?B
• It is possible to make exclusive or gates very
  simply at the transistor level, so we will use
  them in our designs

4
Exclusive Nor

• The Exclusive Nor is also easy to manufacture

5
EOR Simplification Rules

• We can add the exclusive or and nor as
  simplification rules in Boolean Algebra
• A'B AB' A?B
• AB A'B' (A?B)
• (proof by truth table)
• But we cannot spot them easily on the Karnaugh
  map

6
The Half Adder

• A one bit adder, called a half adder is
  represented by the following equation and
  circuit
• CARRY AB SUM A?B

7
The full adder

• To add numbers of more than one bit we need to
  include a carry in

8
Equations of the full adder

• SUM A'B'C A'BC' AB'C' ABC
• A'(B'CBC') A(B'C'BC)
• A'(B?C) A(B?C)'
• A?B?C
• CARRY A'BC AB'C ABC' ABC
• C(A'BAB') AB
• C(A?B) AB

9
Circuit of the full adder
10
Building an n-bit adder
11
Ripple through carry

• The previous circuit is called the ripple through
  carry adder.
• There are faster circuits which are designed to
  propagate the carry faster. In the books you will
  find many others, e.g. the look ahead carry
  adder.
• We will not discuss these further here.

12
The serial adder bits arrive "least significant
first"
13
Subtraction

• Subtraction is done by the normal borrow and
  payback method

14
Problem Break

• Evaluate 11001 - 10110 using the truth table
  below

15
Subtractor circuit

• The minterm method can be used to determine a
  circuit for the subtractor.
• We will not bother with the algebra, but must
  note that the simplified equations are
• DIFFERENCE A?B?P (again!)
• BORROW BP A'(BP)

16
One bit full subtractor
17
Two's complement subtractor

• It is more usual to use a two's complement
  subtractor in hardware.
• A - B A TwosComplement(B)
• The two's complement is formed by flipping each
  bit of a number and adding one.

18
Notice the use of carry-in for increment
19
Multiplication

• Consider how we multiply two digit numbers
• A1A0B1B0
• A1B1102 A1B010 A0B110 A0B0
• or, for binary numbers
• A1A0B1B0
• A1B122 A1B02 A0B12 A0B0

20
Multiplication

• For binary digits, the AND operator is the same
  as multiply, so
• A1A0B1B0
• A1B122 A1B02 A0B12 A0B0
• And multiplying by 2 can be replaced by shifting

21
The 2 bit multiplier
22
A four bit multiplier

• Essentially a four bit multiplier works just like
  a base four multiplier.
• If we write two four bit numbers as PQ and RS,
  where P,Q,R and S are 2 bit numbers then
• PQRS
• PR42 PS4 QR4 QS
• The products can be computed using the two bit
  multiplier just designed.

23
The four bit multiplier
24
The eight bit multiplier

• The same principal can be used to make an eight
  bit multiplier from four four bit multipliers.
• Multipliers of size 2n can be designed using four
  multipliers of size 2n-1

25
Division

• There are hardware circuits for divide, but they
  are quite complex.
• We will not discuss them on this course

26
The ALU

• In practice, arithmetic circuits are bundled
  together in a multi-function package called the
• Arithmetic and Logic Unit (ALU)
• We will look at a commercial ALU next lecture,
  but for now we illustrated the principal by
  designing an add/subtract circuit.

27
Add/ Subtract circuit