Topic:
NUMBER SYSTEM
Objectives :
At the end of topic, students will be able to:
1. Identify type of number system used
2. Differentiate one number system from the other
3. Convert from one number system to another number system
4. Perform addition and subtraction using the different number system
Digital systems are designed to store, process, and communicate information in digital form.
They are found in a wide range of applications, including process control, communication
systems, digital instruments, and consumer products. The digital computer, more commonly
called the computer, is an example of a typical digital system.
A computer manipulates information in digital, or more precisely, binary form. A binary number
has only two discrete values — zero or one. Each of these discrete values is represented by the
OFF and ON status of an electronic switch called a transistor. All computers, therefore, only
understand binary numbers. Any decimal number (base 10, with ten digits from 0 to 9) can be
represented by a binary number (base 2, with digits 0 and 1).
The basic blocks of a computer are the central processing unit (CPU), the memory, and the
input/output (I/O). The CPU of the computer is basically the same as the brain of a human.
Computer memory is conceptually similar to human memory. A question asked to a human is
analogous to entering a program into the computer using an input device such as the keyboard,
and answering the question by the human is similar in concept to outputting the result required
by the program to a computer output device such as the printer. The main difference is that
human beings can think independently, whereas computers can only answer questions that they
are programmed for. (https://www.oreilly.com/library/view/fundamentals-ofdigital/9781118969304/9781118969304c01.xhtml )
Digital as well as Analog System, both are used to transmit signals from one place to another
like audio/video. Digital system uses binary format as 0 and 1 whereas analog system uses
electronic pulses with varying magnitude to send data.
Following are some of the important differences between Digital System and Analog System.
Sr. No.
Key
Digital System
Analog System
1
Signal Type
Digital System uses discrete
signals as on/off representing
binary format. Off is 0, On is 1.
Analog System uses continous signals
with varying magnitude.
2
Wave Type
Digital System uses square
waves.
Analog system uses sine waves.
3
Technology
Digital system first transform the
analog waves to limited set of
numbers and then record them as
digital square waves.
Analog systems records the physical
waveforms as they are originally
generated.
4
Transmission
Digital transmission is easy and
can be made noise proof with no
loss at all.
Analog systems are affected badly by
noise during transmission.
5
Flexibility
Digital system hardware can be
easily modulated as per the
requirements.
Analog system's hardwares are not
flexible.
6
Bandwidth
Digital transmission needs more
bandwidth to carry same
information.
Analog tranmission requires less
bandwidth.
7
Memory
Digital data is stored in form of
bits.
Analog data is stored in form of
waveform signals.
8
Power
requirement
Digital system needs low power
as compare to its analog
counterpart.
Analog systems consume more power
than digital systems.
9
Best suited
for
Digital system are good for
computing and digital electronics.
Analog systems are good for
audio/video recordings.
10
Cost
Digital system are costly.
Analog systems are cheap.
11
Example
Digital system are: Computer, CD,
DVD.
Analog systems are: Analog
electronics, voice radio using AM
frequency.
(source: https://www.tutorialspoint.com/differences-between-digital-and-analog-system)
In the modern world of electronics, the term Digital is generally associated with a computer
because the term Digital is derived from the way computers perform operation, by counting
digits. For many years, the application of digital electronics was only in the computer system.
But now-a-days, digital electronics is used in many other applications. Following are some of
the examples in which Digital electronics is heavily used.
●
Industrial process control
●
Military system
Television
Communication system
Medical equipment
Radar
Navigation
●
●
●
●
●
Signal
Signal can be defined as a physical quantity, which contains some information. It is a function
of one or more than one independent variables. Signals are of two types.
●
Analog Signal
●
Digital Signal
Analog Signal
An analog signal is defined as the signal having continuous values. Analog signal can have
infinite number of different values. In real world scenario, most of the things observed in nature
are analog. Examples of the analog signals are following.
●
●
●
●
●
●
●
Temperature
Pressure
Distance
Sound
Voltage
Current
Power
Graphical representation of Analog Signal (Temperature)
The circuits that process the analog signals are called as analog circuits or system. Examples
of the analog system are following.
●
Filter
●
Amplifiers
Television receiver
Motor speed controller
●
●
Disadvantage of Analog Systems
●
Less accuracy
● Less versatility
● More noise effect
● More distortion
● More effect of weather
Digital Signal
A digital signal is defined as the signal which has only a finite number of distinct values.
Digital signals are not continuous signals. In the digital electronic calculator, the input is given
with the help of switches. This input is converted into electrical signal which have two discrete
values or levels. One of these may be called low level and another is called high level. The
signal will always be one of the two levels. This type of signal is called digital signal. Examples
of the digital signal are following.
●
●
●
Binary Signal
Octal Signal
Hexadecimal Signal
Graphical representation of the Digital Signal (Binary)
The circuits that process the digital signals are called digital systems or digital circuits.
Examples of the digital systems are following.
●
●
Registers
Flip-flop
●
●
Counters
Microprocessors
Advantage of Digital Systems
●
More accuracy
●
More versatility
● Less distortion
● Easy communicate
● Possible storage of information
Comparison of Analog and Digital Signal
S.N.
Analog Signal
Digital Signal
1
Analog signal has infinite values.
Digital signal has a finite number of values.
2
Analog signal has a continuous nature.
Digital signal has a discrete nature.
3
Analog signal is generated by
transducers and signal generators.
Digital signal is generated by A to D
converter.
4
Example of analog signal − sine wave,
triangular waves.
Example of digital signal − binary signal.
Source: https://www.tutorialspoint.com/computer_logical_organization/overview.htm
Number systems
The technique to represent and work with numbers is called number system. Decimal
number system is the most common number system. Other popular number systems
include binary number system, octal number system, hexadecimal number system, etc.
Decimal Number System
Decimal number system is a base 10 number system having 10 digits from 0 to 9. This means
that any numerical quantity can be represented using these 10 digits. Decimal number system
is also a positional value system. This means that the value of digits will depend on its
position.
The weightage of each position can be represented as follows −
In digital systems, instructions are given through electric signals; variation is done by varying
the voltage of the signal. Having 10 different voltages to implement decimal number system in
digital equipment is difficult. So, many number systems that are easier to implement digitally
have been developed. Let’s look at them in detail.
Binary Number System
The easiest way to vary instructions through electric signals is two-state system – on and off.
On is represented as 1 and off as 0, though 0 is not actually no signal but signal at a lower
voltage. The number system having just these two digits – 0 and 1 – is called binary number
system.
Each binary digit is also called a bit. Binary number system is also positional value system,
where each digit has a value expressed in powers of 2, as displayed here.
In any binary number, the rightmost digit is called least significant bit (LSB) and leftmost digit
is called most significant bit (MSB).
Computer memory is measured in terms of how many bits it can store. Here is a chart for
memory capacity conversion.
●
●
●
●
●
●
●
●
1 byte (B) = 8 bits
1 Kilobytes (KB) = 1024 B
1 Megabyte (MB) = 1024 KB
1 Gigabyte (GB) = 1024 MB
1 Terabyte (TB) = 1024 GB
1 Exabyte (EB) = 1024 PB
1 Zettabyte = 1024 EB
1 Yottabyte (YB) = 1024 ZB
Octal Number System
Octal number system has eight digits – 0, 1, 2, 3, 4, 5, 6 and 7. Octal number system is also
a positional value system with where each digit has its value expressed in powers of 8, as
shown here −
Hexadecimal Number System
Hexadecimal number system has 16 symbols – 0 to 9 and A to F where A is equal to 10, B is
equal to 11 and so on till F. Hexadecimal number system is also a positional value system with
where each digit has its value expressed in powers of 16, as shown here −
Number System Relationship
The following table depicts the relationship between decimal, binary, octal and hexadecimal
number systems.
HEXADECIMAL
DECIMAL
OCTAL
BINARY
0
0
0
0000
1
1
1
0001
2
2
2
0010
3
3
3
0011
4
4
4
0100
5
5
5
0101
6
6
6
0110
7
7
7
0111
8
8
10
1000
9
9
11
1001
A
10
12
1010
B
11
13
1011
C
12
14
1100
D
13
15
1101
E
14
16
1110
F
15
17
1111
(https://www.tutorialspoint.com/basics_of_computers/basics_of_computers_number_system.ht
m)
Number base conversion
As you know decimal, binary, octal and hexadecimal number systems are positional value
number systems. To convert binary, octal and hexadecimal to decimal number, we just need to
add the product of each digit with its positional value. Here we are going to learn other
conversion among these number systems.
A positive number, regardless of its radix or base (r), is represented by a series of digits;
An An-1 An-2 …A2 A1 A0 . A-1 A-2 … A-m
Where:
Ai is greater than or equal to 0 and less than its radix
. radix point
To convert binary, octal and hexadecimal to decimal number, the general formula can be used;
(number)r = ∑𝑥=0
𝑥=𝑛
𝐴𝑥 ● 𝑟 𝑥 + ∑𝑥=−𝑚
𝑥=−1
𝐴𝑥 ● 𝑟 𝑥
The first summation is for the integer part, and the second summation is for the
fractional part
Example 1. Convert 3678 to decimal
Solution:
Using the general formula
3678
= A2●82 + A1●81 + A0●80
= 3●64 + 6●8 + 7●1; any number raise to the power of 0 is equal to 1
= 192 + 48 + 7
= 247
3678
= 24710
Example 2. Convert 0.8E3F16 to decimal
Solution:
Using the general formula
0.8E3F16 = A-1●16-1 + A-2●16-2+ A-3●16-3 + A-4●16-4
= 8●0.0625 + 14●0.00390625 + 3●0.0002441406+15●0.0000152588
= 0.5+0.0546875+0.0007324219+0.0002288818
= 0.5556488037
0.8E3F16 = 0.555648803710
Example 3. Convert 1100101.11012 to decimal
Solution:
Using the general formula
1100101.11012
= A6●26+ A5●25 + A4●24 + A3●23+ A2●22 + A1●21 + A0●20
+ A-1●2-1 + A-2●2-2+ A-3●2-3 + A-4●2-4
= 1●64+1●32+0●16+0●8+1●4+0●2+1●1
+1●.5+1●.25+0●.125+1●.0625
= 65+32+0+0+4+0+1+.5+.25+0+.0625
= 102.8125
1100101.11012
= 102.812510
To convert decimal to binary, octal or hexadecimal, two formulas will be used. One formula for
the integer part and another formula to convert the fractional part.
For the integer part, repeated division is used and for the fractional part, repeated multiplication
is used. Equivalent value is by combining the results.
Steps in converting the integer part
1.
2.
3.
4.
5.
Divide the value by the target base or radix.
Record the answer in two different columns (quotient and remainder).
Divide the quotient obtained in step 2.
Repeat steps 2 and 3 until quotient is zero.
The equivalent of the value is by reading the remainder’s column starting from
the bottom (most significant bit) to top (least significant bit).
Steps in converting the fractional part
1.
2.
3.
4.
Multiply the value by the target base or radix.
Record the answer in two different columns (integer part and fractional part).
Multiply the fractional part obtained in step 2.
Repeat steps 2 and 3 until product is equal to zero or if the desired number of
fractional digit has been reached.
5. The equivalent of the value is by reading the integer part’s column starting from
the top to bottom.
(assume that the desired number of fractional digits is 5)
Example 1. Convert 48610 to octal
Since the decimal number is a whole number, only one formula will be used;
converting of integer part. Since the target base is octal (8), the decimal value will
be divided by 8.
Quotient
Remainder
Step 1
486/8
Step 2
60
6
Step 3
60/8
Step 2
7
4
Step 3
7/8
Step 2
0
7
Since quotient is already equal to zero, stop the process.
Therefore, 48610 is equal to 7468
lsb
msb
Example 2. Convert 0.972510 to hexadecimal
Since the decimal number contain only fractional part, only one formula will be
used; conversion of the fractional part. Since the target base is hexadecimal
(16), the value will be multiplied by 16.
Integer
Step 1
0.9725x16
fraction
Step 2
15 (F)
0.56
msb
Step 3
0.56x16
Step 2
8
0.96
Step 3
0.96x16
Step 2
15 (F)
0.36
Step 3
0.36x16
Step 2
5
0.76
Step 3
0.76x16
Step 2
12 (C)
0.16
lsb
Since the number of digits has been reached without having a product of zero,
process can be stopped.
Therefore, 0.972510 is equal to 0.F8F5C16
Example 3. Convert 115.37510 to binary
The given decimal number consist of a whole number and a fractional part,
therefore two formulas will be used;
For the integer part
Quotient
Remainder
Step 1
115/2
Step 2
57
1
Step 3
57/2
Step 2
28
1
Step 3
28/2
Step 2
14
0
Step 3
14/2
Step 2
7
0
Step 3
7/2
Step 2
3
1
Step 3
3/2
Step 2
1
1
Step 3
1/2
Step 2
0
1
Since quotient is already equal to zero, stop the process.
Therefore, 11510 is equal to 11100112
lsb
msb
For the fractional part
Integer
fraction
Step 1
0.375x2
Step 2
0
0.75
Step 3
0.75x2
Step 2
1
0.5
Step 3
0.5x2
Step 2
1
0
Since the fraction is already equal to zero, stop the process.
Therefore, 0.37510 is equal to 0.0112
msb
lsb
The equivalent of 115.37510 in binary is by combining the result of the two
process, 11100112 + 0.0112 = 1110011.0112.
To convert binary to octal and hexadecimal, conversion is done by grouping.
To convert binary to octal, from the binary point to the most significant bit (integer part), group
the binary bits by three. If the last group does not consist of three bits, add zero(es) to the
leftmost bit (it will not change the value). And from the binary point to the least significant bit
(fractional part), again group the binary bits by three. If the last group does not consist of three
bits, add zero(es) to the rightmost bit (it will change the value). Do not start grouping of bits from
the most significant bit nor from the least significant bit, always start from the binary point. Then,
get the equivalent of each group (refer to the table with relationship between different number
system).
To convert binary to hexadecimal, instead of groups of three bits, group by four bits and follow
the same process as converting binary to octal.
Example 1. Convert 1101100.110012 to octal
From the binary point to the most significant bit, group by three. The last group
has only one bit, add two zeroes
001
101
1
100
5
4
From the binary point to the least significant bit, again, group by three. The last
group has only two bits, add one zero
110 010
6
2
Therefore, 1101100.110012 is equal to 154.628
Example 2. Convert 1011011.001012 to hexadecimal
From the binary point to the most significant bit, group by four. The last group
has only three bits, add one zero
0101
5
1011
11 (B)
From the binary point to the least significant bit, again, group by four. The last
group has only one bit, add three zeroes
0010
2
1000
8
Therefore, 1011011.001012 is equal to 5B.2816
To convert octal and hexadecimal to binary, conversion is done by representing digits by either
3 bits (for octal) and four bits (for hexadecimal). For the binary equivalent, refer to the table with
relationship between different number system.
Example 1. Convert 364.5028 to binary
By referring to the table; 3 = 011, 6=110, 4=100, 5=101, 0=000, and 2=010
Therefore, 364.5028 is equal to 11110100.101000012
Example 2. Convert FA3.E816 to binary
Again, by referring to the table; F=1111, A=1010, 3=0011, E=1110, and 8=1000
Therefore, FA3.E816 is equal to 111110100011.111012
In converting octal to hexadecimal or hexadecimal to octal, two methods can be used.
For the first method (assuming octal is to be converted to hexadecimal), the octal value must
first be converted to decimal, then the converted value in decimal will be converted to
hexadecimal.
Second method is to covert octal value to binary, then the converted value in binary will then be
converted to hexadecimal.
It is easier and accurate to use the second method.
Exercises:
Convert the following numbers
1.
2.
3.
4.
5.
175.67510 = ____________________ 8
2F8.0E516 = ____________________ 10
376.45138 = ____________________ 2
11101100.01102 = _______________ 16
5ADC.EB0516 = __________________ 8