KL 2164
DIGITAL ELECTRONICS
ADDER / SUBSTRACTER
Pn. Wan Nurdiana Wan Ibrahim
nurdiana@eng.ukm.my

Contents
Half Adder
Full Adder
Subtracter
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Adder
The result of adding two binary digits could produce a carry value
Recall that 1 + 1 = 10
in base two
Half adder
A circuit that computes the sum of two bits
and produces the correct carry bit
Full Adder
A circuit that takes the carry-in value into account
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Contents
Half Adder
Full Adder
Subtracter
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Example of 1-bit Adder
Design a simple binary adder that adds two 1-bit binary
numbers, a and b, to give a 2-bit sum. The numeric
values for the adder inputs and outputs are as follows:
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Represent inputs to the adder by the logic
variables A and B and the 2-bit sum by the logic
variables X and Y, and the truth table:
Because a numeric value of 0 is represented by a logic
0 and a numeric value of 1 by a logic 1, the 0’s and 1’s
in the truth table are exactly the same as in the previous
table.
Boolean expression :
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Half Adder
 Circuit diagram representing a half adder
 Boolean expressions
sum = A B
carry = AB
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2-bits Binary Adder
Design an adder which adds two 2-bit binary
numbers to give a 3-bit binary sum. Find the truth
table for the circuit. The circuit has four inputs and
three outputs as shown:
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Contents
Half Adder
Full Adder
Subtracter
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Full Adder
The logic equations for the full adder
derived from the truth table are:
(X=A & Y = B)
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Implementation of Full Adder
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Design of Binary Adders and Subtracters
Design a parallel adder that adds two
4-bit unsigned binary numbers and a
carry input to give a 4-bit sum and a
carry output.
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Parallel Adder for 4-Bit Binary Numbers
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One approach would be to construct a truth
table with nine inputs and five outputs and
then derive and simplify the five output
equations.
A better method is to design a logic module
that adds two bits and a carry, and then
connect four of these modules together to
form a 4-bit adder.
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Parallel Adder Composed of Four Full
Adders
Example : 1011+ 1011 = ?
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One’s complement addition
 To add one’s complement numbers:
◦ First do unsigned addition on the numbers, including the sign bits.
◦ Then take the carry out and add it to the sum.
 Two examples:
0111 (+7) 0011 (+3)
+ 1011 + (-4) + 0010 + (+2)
1 0010 0 0101
0010 0101
+ 1 + 0
0011 (+3) 0101 (+5)
 This is simpler and more uniform than signed magnitude
addition.
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Two’s complement addition
 Negating a two’s complement number takes a bit of work, but
addition is much easier than with the other two systems.
 To find A + B, you just have to:
◦ Do unsigned addition on A and B, including their sign bits.
◦ Ignore any carry out.
 For example, to find 0111 + 1100, or (+7) + (-4):
◦ First add 0111 + 1100 as unsigned numbers:
01 1 1
+ 1 1 00
1 001 1
◦ Discard the carry out (1).
◦ The answer is 0011 (+3).
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Unsigned numbers overflow
 Carry-out can be used to detect overflow
 The largest number that we can represent with 4-bits
using unsigned numbers is 15
 Suppose that we are adding 4-bit numbers: 9 (1001) and
10 (1010).
1 001 (9)
+ 1 0 10 (10)
1 001 1 (19)
 The value 19 cannot be represented with 4-bits
 When operating with unsigned numbers, a carry-out of
1 can be used to indicate overflow
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Overflow for Signed Binary Numbers
An overflow has occurred if adding two numbers gives a negative result
or adding two negative numbers gives a positive result.
• Negative number  in compliment form
define an overflow signal, V = 1 if an overflow occurs.
V = A3′B3′S3 + A3B3S3′
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Contents
Half Adder
Full Adder
Subtracter
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Binary Subtracter Using Full Adders
Full Adders may be used to form A – B using the 2’s complement
representation for negative numbers. The 2’s complement of B can be
formed by first finding the 1’s complement and then adding 1.
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Parallel Subtracter
Alternatively, direct subtraction can be
accomplished by employing a full subtracter in a
manner analogous to a full adder.
d = difference
bi = borrow
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Consider xi = 0, yi = 1, and bi = 1:
Step 1 : (if b=1) X= Xi – 1= 0 – 1 Need to borrow from column i+1
 bi+1 =1 & adding 10 (210) to Xi
Step 2 : X = 10 – 1 = 1
Step 3 : X -Y = 1 – 1  d = 0
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Truth Table for Binary Full Subtracter
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