### lec10

```Chapter 7
Arithmetic Operations and Circuits
1
7-4 Hexadecimal Arithmetic
• 4 binary bits represent a single hexadecimal
digit
• Addition
– Add the digits in decimal
– If sum is less than 16, convert to hexadecimal
– Is sum is more than 16, subtract 16, convert to
hexadecimal and carry 1 to the next-moresignificant column
23
Example 7-12
Hexadecimal Arithmetic
• Subtraction
– When you borrow, the borrower increases by 16
– See example 7-15
24
Example 7-15
25
7-5 BCD Arithmetic
• Group 4 binary digits to get combinations
for 10 decimal digits
• Range of valid numbers 0000 to 1001
• Addition
– Add as regular binary numbers
– If sum is greater than 9 or if carry out
generated:
• Add 6 (0110) saving any carry out
26
7-6 Arithmetic Circuits
• Only two inputs are of concern in the LSB
column.
• More significant columns must include the
carry-in from the previous column as a third
input.
27
Arithmetic Circuits
• The addition of the third input (Cin) is shown
in the truth table below.
27
Arithmetic Circuits
• Half-Adder
– No carry in (LSB column)
– The 0 output is HIGH when A or B, but not
both, is high.
• Exclusive-OR function
– Cout is high when A and B are high.
• AND function
28
Arithmetic Circuits
• The half-adder can also be implemented
using NOR gates and one AND gate.
– The NOR output is Ex-OR.
– The AND output is the carry.
28
Arithmetic Circuits
• Full-Adder
– Provides for a carry input
– The 1 output is high when the 3-bit input is
odd.
• Even parity generator
– Cout is high when any two inputs are high.
• 3 AND gates and an OR
29
Arithmetic Circuits
• Full-adder sum from an even-parity
generator
32
Arithmetic Circuits
• Full-adder carry out function
33
Arithmetic Circuits
• Logic diagram of a complete full-adder
34
Arithmetic Circuits
• Block diagrams of a half-adder (HA) and a
full adder (FA).
35
Arithmetic Circuits
• Block diagram of a 4-bit binary adder
36
7-7 Four-Bit Full-Adder ICs
• Four full-adders in a single package
• Will add two 4-bit binary words plus one
carry input bit.
37
Four-Bit Full-Adder ICs
• Functional diagram
of the 7483
• Note that some
manufacturers label
inputs A0B0 to A1B3
• The carry-out is
internally connected
to the carry-in of the
next full-adder.
38
Four-Bit Full-Adder ICs
• Logic diagram for the 7483.
39
Four-Bit Full-Adder ICs
• Logic symbol for the 7483
39
Four-Bit Full-Adder ICs
• Fast-look-ahead carry
– Evaluates 4 low-order inputs
– High-order bits added at same time
– Eliminates waiting for propagation ripple
40
7-9 System Design Application
• Two’s-Complement Adder/Subtractor Circuit
41
System Design Application
• BCD Adder Circuit
42
7-10 Arithmetic/Logic Units
• The ALU is a multipurpose device
• Available in LSI
package
• 74181 (TTL)
• 74HC181 (CMOS)
• Mode Control input
– Arithmetic (M = L)
– Logic (M = H)
44
Arithmetic/Logic Units
• Function
Select selects
specific
function to be
performed
45
Summary
• The binary arithmetic functions of addition,
subtraction, multiplication, and division can
be performed bit-by-bit using several of the
same rules of regular base 10 arithmetic.
• The two’s-complement representation of
binary numbers is commonly used by
computer systems for representing positive
and negative numbers.
48
Summary
• Two’s-complement arithmetic simplifies the
process of subtraction of binary numbers.
• Hexadecimal addition and subtraction is
often required for determining computer
memory space and locations.
• When performing BCD addition a
correction must be made for sums greater
than 9 or when a carry to the next more
significant digit occurs.
49
Summary
• Binary adders can be built using simple
combinational logic circuits.
• A half-adder is required for addition of the
least significant bits
• A full-adder is required for addition of the
more significant bits.
50
Summary
• Multibit full-adder ICs are commonly used for
binary addition and two’s-complement
arithmetic.
• Arithmetic/logic units are multipurpose ICs
capable of providing several different
arithmetic and logic functions.
• The logic circuits for adders can be described
in VHDL using integer arithmetic.
51
Summary
• The Quartus II software provides 7400-series
macrofunctions and a Library of
Parameterized Modules (LPMs) to ease in
the design of complex digital systems.
• Conditional assignments can be made using
the IF-THEN-ELSE VHDL statements.
51
```