View and vote on the article here: CAAHP Digital Computer Electronics: Chapter 1
CAAHP Digital Computer Electronics: Chapter 1| Category | | | Summary | | | Body | ::-=Digital Computer Electronics=-::
::-=Chapter One: Number Systems and Codes=-::
Most textbooks on digital electronics and
computer operation are sometimes hard to understand, in this course I'm trying
to simplify everything as much as I can to make it easier on people willing to
learn and succeed in such a field. I'm dividing this course into chapters, each
chapter is dependent on the chapters before, so you have to keep up with me
chapter by chapter, ok? Let's start.
-=00. Opening=-
Modern computers don't work with decimal numbers. Instead, they process
''binary numbers'', groups of 0s and 1s. Why binary numbers? Because electronic
devices are most reliable when designed for two-state (binary) operation. In this
chapter we will discuss the concepts needed to understand computer operation
as well as number systems and how to convert from one system to another.
-=01. What are computers? How they work?=-
The word computer is misleading
as it suggests a machine that can only solve numerical problems, that's not
true. A computer is more than an automatic adding machine. It can play
games, translate languages and so on. To suggest this broad range of
applications, a computer is often referred to as a data processor.
Data may represent anything
( names, numbers, facts, .... ), but as we said earlier computers can only
process binary numbers so before data is processed it's coded in binary form
so that a computer can understand it.
Besides the data, there's a program
that consists of a set of instructions that tell the computer what to do.
Programs also must be coded in binary form before they can be executed.
Programs are often referred to as
software. On the other hand, the electronic, magnetic and mechanical
devices of a computer are known as hardware. Bear in mind that a
computer is a pile of "dumb" metal without software.
-=02. Decimal Number System=-
Everyone has seen an odometer (miles
indicator) in action. When a car is new its odometer starts with (0000),
After 1 mile the reading becomes (0001), Successive miles produce (0002,
0003, ...... , 0009) then when the units wheel turns from
9 back to 0, a tab on this wheel forces the tens wheel to
advance by 1 and the odometer reads (0010), this action is
called reset-and-carry.
The numbers on each odometer wheel are called
digits. The decimal number system uses ten digits, 0 through 9.
-=03. Binary Number System=-
Binary means two, The binary
number system uses only two digits, 0 and 1. All other digits
(2 through 9) are thrown away. Visualize an odometer whose
wheels have only two digits, 0 and 1. When each wheel turns it
displays 0, then 1, and the cycle repeats so in a binary
odometer (0000) stands for decimal 0, (0001) stands for
decimal 1, and (0010) stands for decimal 2. Remember
that after the cycle of the 0 and 1, the wheel resets and
carries causing the other wheels to change.
In binary numbers, 0 may be
expressed in various ways like ( OFF LED, low voltage, .... ). On the
other hand 1 may be expressed as ( ON LED, high voltage, .... ). Try to get familiar
with binary number system because it's the mainstay of digital design.
A binary digit is abbreviated to a
bit, a binary number like (1100) has 4 bits, every 4 bits
are called a nibble, and every 8 bits are called a byte.
Note that :
1 Kilobyte = 1024 bytes
1 Megabyte = 1024 Kilobytes
1 Gigabyte = 1024 Megabytes
1 Terabyte = 1024 Gigabytes
-=04. Hexadecimal Number System=-
Hexadecimal means 16, This means that
it uses 16 digits to represent all numbers. The digits are (0, 1, 2, 3,
4, 5, 6, 7, 8, 9, A, B, C, D, E, F). Following the same reset-and-carry
rule we can generate hexadecimal numbers, observe the following table.
__Equivalances Table__
||Decimal|Binary|Hexadecimal|Decimal|Binary|Hexadecimal||0|0000|0|8|1000|8||2|0001|1|9|1001|9||2|0010|2|10|1010|A||3|0011|3|11|1011|B||4|0100|4|12|1100|C||5|0101|5|13|1101|D||6|0110|6|14|1110|E||7|0111|7|15|1111|F||
-=05. Base or Radix=-
The base or
radix of a number system equals the number of digits it has.
Decimal number system has a base of 10, binary number
system has a base of 2 and hexadecimal number system has
a base of 16. A subscript attached to the number
indicates the base of that number.
Example:
3010 = 111102 = 1E16
This means that : 30 decimal = 11110 binary = 1E hexadecimal
How these terms are equal?
Continue to the next section.
06. -=Code Conversion=-
Decimal Weights
The decimal number
system is an example of positional notation; each digit
position has a weight or value. Here the weights are
units, tens, hundreds and so on. Have you ever noticed that
the sum of numbers multiplied by their weights gives you the
total amount being represented?
5
7
0
4
103
102
101
100
(5 X 103) + (7 X 102) + (0 X 101) + (4 X 100) = 5704
Binary to Decimal
Conversion
Positional notation is
also used with binary numbers, since only two digits are
used, the weights are powers of 2 instead of 10
in the decimal number system so the weights would be
1, 2, 4, 8 and so on. Notice that the weights
theory is the same meaning that we start with the lowest
weight at the right and move to left direction just like the
decimal number system, also the decimal equivalent of the
binary number would be calculated through calculating the
sum of the numbers multiplied by their weights.
1
1
0
1
23
22
21
20
(1 X 23) + (1 X 22) + (0 X 21) + (1 X 20) = 13
There's a streamlined way
to convert a binary number to its decimal equivalent, just
visualize the weights in your mind then calculate the sum of
weights that are represented as 1s and discard
weights represented as 0s.
Hexadecimal to Decimal
Conversion
As we said before the
hexadecimal number system also has weights but here the
weights are powers of 16, weights can be used easily
to convert hexadecimal numbers to decimal numbers.
C
7
2
F
163
162 161
160
(12 X 163) + (7 X 162) + (2 X 161) + (15 X 160) = 50991
Note that each
hexadecimal digit has been transformed first to its decimal
equivalent to be able to calculate the shown equation so you
must be able to visualize Equivalences Table stated
earlier.
Decimal to Binary
Conversion
You can convert from
decimal number system to binary number system using
successive division by 2, writing down each quotient
and its remainder. The remainders are the binary equivalent
of the decimal number. Let's go through an example.
Answer is : (1310 =11012).
Another
streamlined way to quickly convert from decimal to
binary is to visualize the binary weights in your mind
and split the decimal number to sum of binary weights
and then assign 1 to every weight you found in
this process and 0 to weights you discarded.
Decimal to Hexadecimal
Conversion
We can also use
successive division by 16 to convert from decimal
numbers to hexadecimal numbers as follows.
Answer is : (249710= 9AF16).
Binary to Hexadecimal,
Hexadecimal to Binary Conversion
Conversion between
binary and hexadecimal is so easy and is based on
Equivalences Table that was described earlier, all
you have to do is to split the binary number into
nibbles ( 4-bits each ) and convert the nibbles to
hexadecimal through the table and vice versa.
7 F
0111 1111
1101 0010
D 2
-=07. BCD Numbers=-
BCD is an abbreviation for
(Binary Coded Decimal). In this number system we convert each
decimal digit to its corresponding nibble as follows.
4 1 9 6
0100 0001 1001 0110
After completing this chapter
you should be able to recognize various number systems and convert
between them quickly, so take your time to get familiar to all
number systems and conversion methods mentioned here because
everything discussed later will be dependent on them, I hope that
was a good start, I will publish the upcoming chapters one be one as
soon as I finish them.
Next chapter will be about logic
gates and how we can design simple digital circuits.
--------------------------------------------------------------------------------
broadphase@gmail.com
2005-06-01
Copyright 2005 [http://caahp.com|CyberArmy Advanced Hardware Projects (CAAHP)]
CAAHP is a part of [http://university.cyberarmy.net/|CyberArmy University (CAU)]
-----
{BACKLINKS(info=>hits|user,exclude=>SandBox,include_self=>0,noheader=>0)}{BACKLINKS}
|
|
This article was imported from the CyberArmy University site. (original author: PixieLuv)
There are no replies to this post yet.
|