EET 1131 Unit 7 Arithmetic Operations and Circuits

EET 1131 Unit 7
Arithmetic Operations and Circuits



Read Kleitz, Chapter 7, skipping
Sections 7-4, 7-5, and 7-8.
Homework #7 and Lab #7 due next
week.
Quiz next week.
Binary Addition
The rules for binary addition are
0+0=0
Sum = 0, carry out = 0
0+1=1
Sum = 1, carry out = 0
1+0=1
Sum = 1, carry out = 0
1 + 1 = 10
Sum = 0, carry out = 1
When a carry in = 1 due to a previous result, the rules are
1 + 0 + 0 = 01
1 + 0 + 1 = 10
1 + 1 + 0 = 10
1 + 1 + 1 = 11
Sum = 1, carry out = 0
Sum = 0, carry out = 1
Sum = 0, carry out = 1
Sum = 1, carry out = 1
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Binary Addition
Add the binary numbers 00111 and 10101 and show
the equivalent decimal addition.
0111
00111
10101
7
21
11100 = 28
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Binary Subtraction, Multiplication, Division


The textbook also gives rules for doing binary
subtraction, multiplication, and division.
From a practical standpoint, though, it’s
easier to do one of the following:
 Either convert the numbers from binary to
decimal, then perform the arithmetic
operation, then convert back to binary.
 Or use Windows Calculator to perform
these operations directly on binary
numbers.
Half-Adder
Basic rules of binary addition are performed by a
half adder, which has two binary inputs (A and B)
and two binary outputs (Carry out and Sum).
The inputs and outputs can be summarized on a
truth table.
Inputs Outputs
A
0
0
1
1
B
0
1
0
1
Cout
0
0
0
1
S
0
1
1
0
The logic symbol and equivalent circuit are:
A
S
S
S
A
B
Cout
B
Cout
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Full-Adder
Inputs
By contrast, a full adder has three binary
inputs (A, B, and Carry in) and two binary
outputs (Carry out and Sum). The truth table
summarizes the operation.
A full-adder can be constructed from two
half adders as shown:
A
A
S
S
A
S
S
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
Outputs
Cin
0
1
0
1
0
1
0
1
Cout
0
0
0
1
0
1
1
1
S
0
1
1
0
1
0
0
1
Sum
S
B
B
Cout
B
A
Cout
B
Cin
Cin
Cout
S
Cout
Symbol for Full Adder
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Full-Adder
1
A
0
B
For the given inputs, determine
the intermediate and final outputs
of the full adder.
S
S 1
A
Cout 0
B
S
S
0
Cout
1
1
Sum
Cout
1
The first half-adder has inputs of 1 and 0;
therefore the Sum =1 and the Carry out = 0.
The second half-adder has inputs of 1 and 1; therefore the
Sum = 0 and the Carry out = 1.
The OR gate has inputs of 1 and 0, therefore the final carry
out = 1.
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Full-Adder
Notice that the result from the previous example can be
read directly on the truth table for a full adder.
Inputs
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
Outputs
Cin
0
1
0
1
0
1
0
1
Cout
0
0
0
1
0
1
1
1
S
0
1
1
0
1
0
0
1
1
A
0
B
1
S
S 1
A
Cout 0
B
S
S
0
Cout
1
Sum
Cout
1
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Convention for Writing Multi-Bit Numbers



We’ll use subscripts to refer to the
individual bits in a binary number.
The bit on the right-hand end, or least
significant bit (LSB), always gets the
smallest subscript, which may be
either 1 or 0.
Example: In a four-bit number A, the
bits are labeled either
A4A3A2A1
or
A3A2A1A0
Parallel Adders
Full adders are combined into parallel adders that can add binary
numbers with multiple bits. A 4-bit adder is shown.
A4 B4
A3 B3
A2 B2
A1 B1
C0
A B Cin
Cout
S
C4
S4
A B Cin
Cout
C3
S
S3
A B Cin
Cout
C2
S
S2
A B Cin
Cout
C1
S
S1
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Parallel Adders
The logic symbol for a 4-bit parallel adder is shown. This 4-bit adder
includes a Carry In (labeled C0) and a Carry Out (labeled C4).
Binary
number A
Binary
number B
Carry
In
1
2
3
4
1
2
3
4
C0
S
1
2
3
4
C4
4-bit
sum
Carry
Out
The 74283 is an example.
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Two Adder Chips


74283 Four-bit binary adder
7483 is an older chip that is
functionally identical to the 74283, but
the pins are laid out differently
Cascading Parallel Adders
When we connect the outputs from one circuit to the
inputs of another identical circuit to expand the number of
bits being operated on, we say that the circuits are
cascaded together.
For example, you can cascade two 4-bit parallel adders to
add two 8-bit numbers. To do this, connect the lowerorder adder’s Carry Out to the higher-order adder’s Carry
In.
See textbook’s Figure 7-18 (next slide).
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Arithmetic Logic Unit (ALU)



Central to any computer system is its
ALU, which performs mathematical
and logical operations on data.
In modern systems, the ALU is
contained on the computer’s
microprocessor chip.
In older systems, the ALU was a
separate chip, such as the 74181.
How Many Logical Operations?

You already know how to perform
some logical operations on two input
bits, A and B. Examples:


X = AB
X = A+B


Question: How many possible logical
operations are there on two input bits?
How Many Logical Ops? (Continued)

A
B
0
0
0
1
1
0
1
1
Let’s list them all:
74181 ALU chip



Can perform 16 logical operations (bitby-bit) and 16 arithmetic operations
on two 4-bit input numbers.
Data Sheet: 74LS181
See next slide (Fig. 7-27 from
textbook) for logic symbol and
function table.
Figure 7.27
The 74181 ALU: (a) logic symbol; (b) function table.
Digital Electronics: A Practical Approach with VHDL, 9th Edition
William Kleitz
Copyright ©2012 by Pearson Education, Inc.
All rights reserved.
74181 ALU (Continued)

Caution: In the “Arithmetic
Operations” columns of the 74181
function tables, the + symbol always
means logical OR, not addition. The
word “PLUS” is used for addition.
74181 ALU (Continued)

Fourteen Input Pins:





A0 to A3 form one of the 4-bit inputs.
B0 to B3 form the other 4-bit input.
CN is the carry-in bit, used only during
arithmetic ops (ignored during logic ops).
M is the mode pin (arithmetic or logic).
S0 to S3 select the operation performed.
74181 ALU (Continued)

Eight Output Pins:




F0 to F3 form the 4-bit output.
CN+4 is carry-out bit, meaningful only for
arithmetic ops. (Ignore it for logic ops.)
A=B is comparison bit, meaningful only
when performing “A MINUS B” operation.
(We’ll ignore this.)
P and G are carry-look-ahead bits for
high-speed arithmetic, when 74181 is
used in conjunction with 74182 chip.
(We’ll ignore these.)
Terminology: One’s-Complement and
Two’s-Complement




The one’s-complement of a binary
number is the binary number that you
get when you invert each bit.
Example: What is the one’s-complement
of 00110101?
The two’s-complement of a binary
number is the binary number that you
get when you invert each bit and then
add 1 to the result.
What is the two’s-complement of
00110101?
Interpreting Strings of 1s and 0s



In digital systems we have nothing but
1s and 0s to represent all kinds of info:
text, numbers, images, music, etc.
To interpret a string of 1s and 0s, you
have to be told what kind of info it
represents.
Example:
What does 010000012 represent?
Possible answers:
The number 65.
The letter A (in ASCII code).
…
Unsigned versus Signed Binary Integers



Up to now, whenever we’ve worked
with binary numbers, we’ve assumed
they were unsigned binary integers.
In other words, we’ve assumed that the
numbers were all positive.
Sometimes we also need to be able to
represent negative integers, in which
case we’re dealing with signed binary
integers.
To do this, we’ll use one bit to indicate
the number’s sign (positive or
negative).
Range of Unsigned Binary Integers

Arranging unsigned 8-bit integers in
order from least to greatest would
give you a list that starts and ends like
this:
Binary
00000000
00000001
00000010

11111101
11111110
11111111
Decimal
0
1
2

253
254
255
Range of Unsigned Binary Integers

For unsigned integers with a fixed
number of bits n:



The least integer we can represent is 0.
The greatest integer we can represent is
2n-1.
Example: Using 8 bits,


The least integer is 000000002, which is
equal to decimal 0.
The greatest integer is 111111112, which is
equal to decimal 255 (=28-1).
Representing Signed Binary Integers

To represent both positive and negative
integers, we use the leftmost bit as a
sign bit, like this (for 8 bits):
Negative integers
(sign bit = 1)
Zero and positive integers
(sign bit = 0)
Binary
10000000
10000001
10000010

11111111
00000000
00000001

01111110
01111111
Decimal
-128
-127
-126

-1
0
1

126
127
Two’s-Complement Representation



The scheme shown on the previous
slides is called two’s-complement
representation.
This is how computers represent
signed integers, because this scheme
results in simple circuits for doing
binary arithmetic.
Other common names for it: two’s
complement notation or two’s
complement form.
Range of Signed Binary Integers


For signed integers with a fixed number of
bits n, the least integer we can represent
is −2n−1, and the greatest integer we can
represent is 2n−1−1.
Example: Using 8 bits,
 The least integer we
100000002, which is
−128 (= −27).
 The greatest integer
011111112, which is
127 (= 27-1).
can represent is
equal to decimal
we can represent is
equal to decimal
Steps for Converting from Decimal to
Two’s-Complement Form
1.
2.
If the integer is positive, convert it to
binary as we’ve always done.
If the integer is negative:
a)
b)
c)

Ignoring the sign, convert it to binary as
we’ve always done.
Invert each bit. (That is, change each 0
to 1, and change each 1 to 0.)
Add 1 to the result.
Examples


Convert 2310 to two’s complement form using
8 bits.
Convert −2310 to two’s complement form using
8 bits.
Steps for Converting from Two’sComplement Form to Decimal
1.
2.
If the sign bit = 0, the integer is
positive. Convert to decimal as we’ve
always done.
If the sign bit = 1, the integer is
negative. Follow these steps:
Invert each bit.
b) Add 1.
c) Convert the result to decimal as we’ve
always done.
d) Write a negative sign in front.
 Examples: Convert 01100010 to decimal.

Convert 11100010 to decimal.
a)
Adding Signed Numbers
Using two’s complement form for negative integers
simplifies addition and subtraction of signed numbers.
Rules for addition: Add the two signed numbers. Discard
any final carry out of the MSB. The result is in two’s
complement form.
Examples:
00011110 30
+ 00001111 + 15
00101101 45
00001110 14
+11101111 + -17
11111101 -3
11111111 -1
+ 11111000 + -8
111110111 -9
Discard carry
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Subtracting Signed Numbers
Rules for subtraction: Negate the number being subtracted,
and then add. Discard any final carry out of the MSB. The
result is in two’s complement form.
Us the same numbers as on previous slide, but subtract:
00011110 30
- 00001111 – 15
00001110
14
- 11101111 – -17
11111111 -1
- 11111000 – -8
Negate the number being subtracted, and then add:
00011110 30
+11110001 +-15
100001111 15
Discard carry
00001110 14
+ 00010001 +17
00011111 31
11111111 -1
+ 00001000 + 8
100000111 7
Discard carry
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
An Adder/Subtractor Circuit


By combining parallel adder chips (such as
the 74283 or the 4008) with a controlled
inverter, we can make a circuit that either
adds or subtracts, depending on the
value of a control input.
See next slide (Fig. 7-23 from textbook).
Figure 7.23
8-bit two’s-complement adder/subtractor illustrating the subtraction 42 – 23 = 19.
Digital Electronics: A Practical Approach with VHDL, 9th Edition
William Kleitz
Copyright ©2012 by Pearson Education, Inc.
All rights reserved.