Computer System and Organization

1. Integer Addition And Substraction
•To execute arithmetic operations there is a separate
section called arithmetic processing unit in central
processing unit. The arithmetic instructions are
performed generally on binary or decimal data. Fixed-
point numbers are used to represent integers or
fractions. We can have signed or unsigned negative
numbers. Fixed-point addition is the simplest arithmetic
operation.
•

2. • In order to solve the computational problems, arithmetic
instructions are used in digital computers that manipulate
data. These instructions perform arithmetic calculations.
And these instructions perform a great activity in
processing data in a digital computer.
• As we already stated that with the four basic arithmetic
operations addition, subtraction, multiplication and
division, it is possible to derive other arithmetic operations
and solve scientific problems by means of numerical
analysis methods.
• A processor has an arithmetic processor(as a sub part of it)
that executes arithmetic operations. The data type,
assumed to reside in processor, registers during the
execution of an arithmetic instruction.
• Negative numbers may be in a signed magnitude or signed
complement representation. There are three ways of
representing negative fixed point - binary numbers signed
magnitude, signed 1’s complement or signed 2’s
complement. Most computers use the signed magnitude
representation for the mantissa.

3. • Addition and Subtraction :
• Addition and Subtraction with Signed –
Magnitude Data We designate the magnitude of
the two numbers by A and B. Where the signed
numbers are added or subtracted, we find that
there are eight different conditions to consider,
depending on the sign of the numbers and the
operation performed.
• These conditions are listed in the first column of
Table 4.1. The other columns in the table show
the actual operation to be performed with the
magnitude of the numbers. The last column is
needed to present a negative zero.

6. Ripple Carry Adder
• In digital electronics adding of two-bit binary numbers
can be possible by using half adder.
• And if the input sequence has a three-bit sequence,
then the addition process can be completed by using a
full adder. But if the numbers of bits are more in the
input sequence then the process can be completed by
using half adder.
• Because full adder cannot be able to complete the
addition operation. So these drawbacks can be
overcome by using “Ripple Carry Adder”.
• It’s a unique type of logic circuit used for adding the N-
bit numbers in digital operations. This article describes
an overview of what is ripple-carry-adder and its
operation.

7. • What is Ripple Carry Adder?
• A structure of multiple full adders is cascaded in a manner to
gives the results of the addition of an n bit binary sequence.
• This adder includes cascaded full adders in its structure so,
the carry will be generated at every full adder stage in a
ripple-carry adder circuit.
• These carry output at each full adder stage is forwarded to its
next full adder and there applied as a carry input to it. This
process continues up to its last full adder stage. So, each carry
output bit is rippled to the next stage of a full adder. By this
reason, it is named as “RIPPLE CARRY ADDER”.
• The most important feature of it is to add the input bit
sequences whether the sequence is 4 bit or 5 bit or any.
• “One of the most important point to be considered in this
carry adder is the final output is known only after the carry
outputs are generated by each full adder stage and forwarded
to its next stage. So there will be a delay to get the result with
using of this carry adder”.

8. • There are various types in ripple-carry adders.
They are:
• 4-bit ripple-carry adder
• 8-bit ripple-carry adder
• 16-bit ripple-carry adder

9. • 4-bit Ripple Carry Adder
• The below diagram represents the 4-bit ripple-
carry adder. In this adder, four full adders are
connected in cascade. Co is the carry input bit
and it is zero always. When this input carry
‘Co’ is applied to the two input sequences A1
A2 A3 A4 and B1 B2 B3 B4 then output
represented with S1 S2 S3 S4 and output carry
C4.
•

11. • Working of 4-bit Ripple Carry Adder
• Let’s take an example of two input sequences 0101 and 1010. These
are representing the A4 A3 A2 A1 and B4 B3 B2 B1.
• As per this adder concept, input carry is 0.
• When Ao & Bo are applied at 1st full adder along with input carry 0.
• Here A1 =1 ; B1=0 ; Cin=0
• Sum (S1) and carry (C1) will be generated as per the Sum and Carry
equations of this adder. As per its theory, the output equation for
the Sum = A1⊕B1⊕Cin and Carry = A1B1⊕B1Cin⊕CinA1
• As per this equation, for 1st full adder S1 =1 and Carry output i.e.,
C1=0.
• Same like for next input bits A2 and B2, output S2 = 1 and C2 = 0.
Here the important point is the second stage full adder gets input
carry i.e., C1 which is the output carry of initial stage full adder.
• Like this will get the final output sequence (S4 S3 S2 S1) = (1 1 1 1)
and Output carry C4 = 0
• This is the addition process for 4-bit input sequences when it’s
applied to this carry adder.

12. • Carry Look Ahead Adder
• A digital computer must contain circuits which can perform
arithmetic operations such as addition, subtraction,
multiplication, and division. Among these, addition and
subtraction are the basic operations whereas multiplication
and division are the repeated addition and subtraction
respectively.
• To perform these operations ‘Adder circuits’ are implemented
using basic logic gates. Adder circuits are evolved as Half-
adder, Full-adder, Ripple-carry Adder, and Carry Look-ahead
Adder.
• Among these Carry Look-ahead Adder is the faster adder
circuit. It reduces the propagation delay, which occurs during
addition, by using more complex hardware circuitry. It is
designed by transforming the ripple-carry Adder circuit such
that the carry logic of the adder is changed into two-level
logic.
•

13. • 4-Bit Carry Look-ahead Adder
• In parallel adders, carry output of each full adder
is given as a carry input to the next higher-order
state. Hence, these adders it is not possible to
produce carry and sum outputs of any state
unless a carry input is available for that state.
• So, for computation to occur, the circuit has to
wait until the carry bit propagated to all states.
This induces carry propagation delay in the
circuit.
•

14. • Consider the 4-bit ripple carry adder circuit above.
Here the sum S3 can be produced as soon as the inputs
A3 and B3 are given.
• But carry C3 cannot be computed until the carry bit C2
is applied whereas C2 depends on C1. Therefore to
produce final steady-state results, carry must
propagate through all the states. This increases the
carry propagation delay of the circuit.
• The propagation delay of the adder is calculated as
“the propagation delay of each gate times the number
of stages in the circuit”. For the computation of a large
number of bits, more stages have to be added, which
makes the delay much worse. Hence, to solve this
situation, Carry Look-ahead Adder was introduced.
• To understand the functioning of a Carry Look-ahead
Adder, a 4-bit Carry Look-ahead Adder is described
below.

16. • In this adder, the carry input at any stage of
the adder is independent of the carry bits
generated at the independent stages.
• Here the output of any stage is dependent
only on the bits which are added in the
previous stages and the carry input provided
at the beginning stage.
• Hence, the circuit at any stage does not have
to wait for the generation of carry-bit from the
previous stage and carry bit can be evaluated
at any instant of time.

17. • Shift-and-Add Multiplication
• Shift-and-add multiplication is similar to the
multiplication performed by paper and pencil.
• This method adds the multiplicand X to itself Y
times, where Y denotes the multiplier.
• To multiply two numbers by paper and pencil,
the algorithm is to take the digits of the multiplier
one at a time from right to left, multiplying the
multiplicand by a single digit of the multiplier and
placing the intermediate product in the
appropriate positions to the left of the earlier
results.
• As an example, consider the multiplication of two
unsigned 4-bit numbers, 8 (1000) and 9 (1001).

18. • Multiplicand 1000 ×
Multiplier 1001
• Product 1001000
• In the case of binary multiplication, since the
digits are 0 and 1, each step of the
multiplication is simple. If the multiplier digit
is 1, a copy of the multiplicand (1 ×
multiplicand) is placed in the proper positions;
if the multiplier digit is 0, a number of 0 digits
(0 × multiplicand) are placed in the proper
positions.