Transcript Binary Number System - Nick Reeder`s Home Page

EET 1131
Digital Electronics
Professor Nick Reeder
Reminders



Please turn off cell phones.
No food or soft drinks in the
classroom.
Stow water bottles at floor level.
EET 1131 Unit 1
Number Systems and Codes
Read Kleitz, Chapter 1 (but you
can skip Sections 1-6 and 1-7).
 Homework #1 and Lab #1 due
next week.
 Quiz next week.

Analog versus Digital
Analog = continuous
 Digital = discrete
 Example:

An analog clock, whose hands move
smoothly and continuously.
 A digital clock, whose digits jump
from one value to the next.

Analog Quantities
Most natural quantities (such as temperature, pressure, light
intensity, …) are analog quantities that vary continuously.
Temperature
(°F)
100
95
90
85
80
75
70
Time of day
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
A .M .
P.M .
Digital systems can process, store, and transmit data more
efficiently but can only assign discrete values to each point.
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
The Digital Revolution


Recently, many types of devices have
been converted from analog to digital.
Examples:
Analog
Record albums
VHS tapes
Analog television

Digital
CDs
DVDs
Digital TV
In all of these digital devices, info is
stored and transmitted as long strings
of 1s and 0s.
Analog and Digital Systems
Many systems use a mix of analog and digital electronics to
take advantage of each technology. A typical CD player
accepts digital data from the CD drive and converts it to an
analog signal for amplification.
CD drive
10110011101
Digital data
Digital-to-analog
converter
Linear amplifier
Analog
reproduction
of music audio
signal
Speaker
Sound
waves
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Voltage



Voltage is a basic electrical quantity
that is important in all circuits (analog
or digital).
You can think of a circuit as being like
a plumbing system, with water flowing
through pipes.
On this analogy, voltage is like the
water pressure in the pipes. Its value
will vary at different points in the
circuit.
A Simple Circuit
A wire is like a water pipe. The amount of
electricity per second flowing through a wire is called
current, which is measured in amperes.
The voltage
(pressure)
at this point
is greater than
the voltage
at this point.
A voltage source is like
a water pump. Its
voltage rating (in volts)
tells you how strong it is.
Resistors are like partial blockages
in the pipe. They restrict the amount
of current that flows through the circuit.
Examples of Voltage Sources

Voltage is measured in volts (V).

Flashlight battery ____ V

Wall outlet ____ V
Trainer Power Supplies
Fixed +5 V supply:
In this course we’ll
always use this one.
No matter which one
of these you use, you
must also use the
GROUND connection.
Variable supplies,
controlled by the
knobs at left. You’ll
use these in
other courses.
Measuring Exact Voltage

In other courses you’ll use a voltmeter
or digital multimeter, like the one
shown, to measure the exact voltage
at a point in a circuit.
Measuring Digital HIGHS or LOWS


In this course we
usually don’t care
about exact voltage
values. We just
care whether the
voltage at a point is
“high” or “low.”
To measure this,
we use a logic
probe, such as
the one shown.
Binary Digits and Logic Levels
• Digital electronics uses circuits that have two states,
which are represented by two voltage ranges called
HIGH and LOW. We often represent a HIGH state by
the number 1, and a LOW state by the number 0.
VH(max)
5.0 V
HIGH
VH(min)
2.0 V
Invalid
VL(max)
0.8 V
LOW
VL(min)
• The voltages on the upper and
lower edges of these ranges
vary for different
technologies. The values
shown are for the TTL
technology.
0V
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Breadboarding Guidelines

When you build circuits in lab, I
expect you to follow the
breadboarding guidelines on the
course website.
Ones and Zeros
•
•
•
•
Digital devices (computers, iPods, cell
phones, …) store information
(numbers, text, images, music, …) as
strings of 1s and 0s.
Each 1 or 0 in such a string is called a
bit (short for binary digit).
Example of an 8-bit string: 01101100
A typical song in an MP3 file might
contain 40 million bits.
Number Systems and Codes
•
This week we’ll look mainly at how to
represent numbers using 1s and 0s,
and also (briefly) how to represent
text using 1s and 0s.
Binary Number System
•
•
When we represent numbers using 1s
and 0s, we’re using the binary
number system. This system is
fundamental to everything in digital
electronics, so you must learn it
thoroughly.
First, we’ll briefly review the decimal
number system that you’ve used for
most of your life.
Decimal Numbers
The position of each digit in a weighted number system is
assigned a weight based on the base or radix of the system.
The base of decimal numbers is ten, because only ten
symbols (0 through 9) are used to represent any number.
The column weights of decimal numbers are powers
of ten that increase from right to left beginning with 100 =1:
…105 104 103 102 101 100.
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Decimal Numbers
Decimal numbers can be expressed as the sum of the
products of each digit times the column value for that digit.
Thus, the number 9240 can be expressed as
(9 x 103) + (2 x 102) + (4 x 101) + (0 x 100)
or
9 x 1,000 + 2 x 100 + 4 x 10 + 0 x 1
Express the number 480 as the sum of values of each digit.
480 = (4 x 102) + (8 x 101) + (0 x 100)
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Binary Numbers
For digital systems, the binary number system is used.
Binary has a base of two and uses the digits 0 and 1 to
represent quantities.
The column weights of binary numbers are powers of
two that increase from right to left beginning with 20 =1:
…25 24 23 22 21 20.
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Engineering Humor
•
Here’s a good joke for your next
party:
There are 10 kinds of people in this
world—those who know understand binary
numbers and those who don’t.
Binary Numbers
A binary counting sequence for numbers
from zero to fifteen is shown.
Notice the pattern of zeros and ones in
each column.
Decimal
Number
Binary
Number
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Binary-to-Decimal Conversions
The decimal equivalent of a binary number can be
determined by adding the column values of all of the bits
that are 1 and discarding all of the bits that are 0.
Convert the binary number 100101 to decimal.
Start by writing the column weights; then add the
weights that correspond to each 1 in the number.
32 16 8 4 2 1
1 0 0 1 0 1
32
+4 +1
=
37
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Decimal-to-Binary Conversions (First Method)
You can convert a decimal whole number to binary by
reversing the procedure. Write the decimal weight of each
column and place 1’s in the columns that sum to the decimal
number.
Convert the decimal number 49 to binary.
The column weights double in each position to the
right. Write down column weights until the last
number is larger than the one you want to convert.
64 32 16 8 4 2 1.
0 1 1 0 0 0 1.
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Decimal-to-Binary Conversions (Second Method)
You can convert decimal to any other base by repeatedly
dividing by the base. For binary, repeatedly divide by 2:
Convert the decimal number 49 to binary by
repeatedly dividing by 2.
You can do this by “reverse division” and the
answer will read from left to right. Put quotients to
the left and remainders on top.
1 1 0 0 0 1 remainder
0 1 3 6 12 24 49 2
Answer:
Continue until the
last quotient is 0
Quotient
Decimal
number
base
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
The Hexadecimal and Octal Systems
•
•
•
•
•
•
We’ve looked at the decimal and
binary systems.
Hexadecimal (often called “hex”) and
octal are useful as shorthand systems
for writing large binary numbers.
Hex is a base-16 system.
Octal is a base-8 system.
Hex is very widely used.
Octal was popular 40 years ago, but is
not used much today.
Decimal Hexadecimal Binary
Hexadecimal Numbers
Hexadecimal uses sixteen characters to
represent numbers: the numbers 0
through 9 and the alphabetic characters
A through F.
Large binary numbers can
easily be converted to hexadecimal
by grouping bits 4 at a time and
writing the equivalent hex character.
Express 1001 0110 0000 11102 in
hexadecimal:
Group the binary number by 4-bits
starting from the right. Thus, 960E
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Decimal Hexadecimal Binary
Hexadecimal Numbers
Hexadecimal is a weighted number
system. The column weights are
powers of 16, which increase from
right to left.
Column weights
16 16 16 16 .
{4096
256 16 1
3
2
1
0
Express 1A2F16 in decimal.
Start by writing the column weights:
4096 256 16 1
1
A 2 F16
1(4096) + 10(256) +2(16) +15(1) = 670310
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
Binary and Hex Conversion
Games


You must memorize the binary and
hex codes for the numbers from 1
through 15.
To practice, play the Binary-Decimal
and Binary-Hex matching games on
my Games page.
Decimal
BCD
Binary coded decimal (BCD) is a
weighted code that is commonly
used in digital systems when it is
necessary to show decimal
numbers such as in clock displays.
The table illustrates the
difference between straight binary and
BCD. BCD represents each decimal
digit with a 4-bit code. Notice that the
codes 1010 through 1111 are not used in
BCD.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
BCD
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
0001 0000
0001 0001
0001 0010
0001 0011
0001 0100
0001 0101
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
BCD
You can think of BCD in terms of column weights in
groups of four bits. For an 8-bit BCD number, the column
weights are: 80 40 20 10 8 4 2 1.
What are the column weights for the BCD number
1000 0011 0101 1001?
8000 4000 2000 1000 800 400 200 100 80 40 20 10 8 4 2 1
Note that you could add the column weights where there is
a 1 to obtain the decimal number. For this case:
8000 + 200 +100 + 40 + 10 + 8 +1 = 835910
© 2009 Pearson Education, Upper Saddle River, NJ 07458. All Rights Reserved
The ASCII Code
•ASCII (American Standard Code for Information
Interchange) is a binary code for alphanumeric
characters.
•ASCII encodes 128 characters using 7-bits.
•See Table 1-5 on page 19 (next slide) for the
list of ASCII codes.
•To write the ASCII code for any character in the
table, write the 3 bits at the top of the
character’s column followed by the 4 bits at the
left end of the character’s row.
Digital Electronics: A Practical Approach with VHDL, 9th Edition
William Kleitz
Copyright ©2012 by Pearson Education, Inc.
All rights reserved.
The ASCII Code
•From this table we can see that the ASCII code
for the letter N is 100 1110. (Expressed in hex,
this is 4E.)
The ASCII Code
•The first 32 entries are control characters based
on teletype requirements. Some of these are
obsolete, but some (such as Backspace, Line
Feed, Form Feed), are still widely used.