An Overview & Number Systems
Analogue versus Digital
Most of the quantities in nature that can be measured are continuous. Examples
include
• Intensity of light during the day: The intensity of light gradually increases as the
sun rises in the morning; it remains constant throughout the day and then gradually
decreases as the sun sets until it becomes completely dark. The change in the light
throughout the day is gradual and continuous. Even with a sudden change in weather
when the sun is obscured by a cloud the fall in the light intensity although very sharp
however is still continuous and is not abrupt.
• Rise and fall in temperature during a 24-hour period: The temperature also rises
and falls with the passage of time during the day and in the night. The change in
temperature is never abrupt but gradual and continuous.
• Velocity of a car travelling from A to B: The velocity of a car travelling from one
city to another varies in a continuous manner. Even if it abruptly accelerates or stops
suddenly, the change in velocity seemingly very sudden and abrupt is never abrupt in
reality. This can be confirmed by measuring the velocity in short time intervals of few
milliseconds.
The measurable values generally change over a continuous range having a minimum
and maximum value. The temperature values in a summer month change between 23 0C
to 45 0C. A car can travel at any velocity between 0 to 120 mph.
Digital representing of quantities
Digital quantities unlike Analogue quantities are not continuous but represent
quantities measured at discrete intervals. Consider the continuous signal as shown in the
figure 1.1.
To represent this signal digitally the signal is sampled at fixed and equal intervals.
The continuous signal is sampled at 15 fixed and equal intervals. Figure 1.2. The set of
values (1, 2, 4, 7, 18, 34, 25, 23, 35, 37, 29, 42, 41, 25 and 22) measured at the sampling
points represent the continuous signal. The 15 samples do not exactly represent the
original signal but only approximate the original continuous signal. This can be
confirmed by plotting the 15 sample points. Figure 1.3. The reconstructed signal from the
15 samples has sharp corners and edges in contrast to the original signal that has smooth
curves.
If the number of samples that are collected is reduced by half, the reconstructed signal
will be very different from the original. The reconstructed signal using 7 samples have
missing peak and dip at 34 0C and 23 0C respectively. Figure 1.4. The reason for the
difference between the original and the reconstructed signal is due to under-sampling. A
more accurate representation of the continuous signal is possible if the number of
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samples and sampling intervals are increased. If the sampling is increased to infinity the
number of values would still be discrete but they would be very close and closely match
the actual signal.
Figure 1.1 Continuous signal showing temperature varying with time
Figure 1.2 Sampling the Continuous Signal at 15 equal intervals
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
time
temperature 0C
1 2
4
7
34
25
23
37
29
42 41
25
22
18
35
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
time
temperature 0C
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Figure 1.3 Reconstructed Signal by plotting 15 sampled values
Figure 1.4 Reconstructed Signal by plotting 7 sampled values
Electronic Processing of Continuous and Digital Quantities
Electronic Processing of the continuous quantities or their Digital representation
requires that the continuous signals or the discrete values be converted and represented in
terms of voltages. There are basically two types of Electronic Processing Systems.
• Analogue Electronic Systems: These systems accept and process continuous signals
represented in the form continuous voltage or current signals. The continuous
1 2
4
7
18
34
25
23
35
37
29
42 41
25
22
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
samples
temperature 0C
0
5
10
15
20
25
30
35
40
45
1 3 5 7 9 11 13 15
samples
temperature 0C
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quantities are converted into continuous voltage or current signals by transducers. The
block diagram describes the processing by an Analogue Electronic System. Figure
1.5.
• Digital Electronic Systems: These systems accept and process discrete samples
representing the actual continuous signal. Analogue to Digital Converters are used to
sample the continuous voltage signals representing the original signal.
Do the Digital Electronic Systems use voltages to represent the discrete samples of
the continuous signal? This question can be answered by considering a very simple
example of a calculator which is a Digital Electronic System. Assume that a calculator is
calibrated to represents the number 1 by 1 millivolt (mV). Thus the number 39 is
represented by the calculator in terms of voltage as 39 mV. Calculators can also represent
large numbers such as 6.25 x 1018 (as in 1 Coulomb = 6.25 x 1018 electrons). The value in
terms of volts is 6.25 x 1015 volts! This voltage value can not be practically represented
by any electronic circuit. Thus Digital Systems do not use discrete samples represented as
voltage values.
Figure 1.5 Analogue Electronic System processing continuous quantities
Digital Systems and Digital Values
Digital systems are designed to work with two voltage values. A +5 volts represents a
logic high state or logic 1 state and 0 volts represents a logic low state or logic 0 state.
The Digital Systems which are based on two voltage values or two states can easily
represent any two values. For example,
• The numbers '0' and '1'
• The state of a switch 'on' or 'off'
• The colour 'black' and 'white'
• The temperature 'hot' and 'cold'
• An object 'moving' or 'stationary'
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Representing two values or two states is not very practical, as many naturally
occurring phenomenons have values or state that are more than two. For example,
numbers have widely varying ranges, a colour palette might have 64 different shades of
the colour red, the temperature of boiling water at room temperature varies from 30 0C to
100 0C. Digital Systems are based on the Binary Number system which allows more than
two or multiple values to be represented very conveniently.
Binary Number System
The Binary Number System unlike the Decimal number system is based on two
values. Each digit or bit in Binary Number system can represent only two values, a '0'
and a '1'. A single digit of the Decimal Number system represents 10 values, 0, 1, 2 to 9.
The Binary Number System can be used to represent more than two values by combining
binary digits or bits. In a Decimal Number System a single digit can represent 10
different values (0 to 9), representing more than 10 values requires a combination of two
digits which allows up to 100 values to be represented (0 to 99). A Combination of
Binary Numbers is used to represent different quantities.
• Represent Colours: A palette of four colours red, blue, green and yellow can be
represented by a combination of two digital values 00, 01, 10 and 11 respectively.
• Representing Temperature: An analogue value such as 39oC can be represented in a
digital format by a combination of 0s and 1s. Thus 39 is 100111 in digital form.
Any quantity such as the intensity of light, temperature, velocity, colour etc. can be
represented through digital values. The number of digits (0s and 1s) that represents a
quantity is proportional to the range of values that are to be represented. For example, to
represent a palette of eight colours a combination of three digits is used. Representing a
temperature range of 00 C to 1000 C requires a combination of up to seven digits.
Digital Systems uses the Binary Number System to represent two or multiple
values, stores and processes the binary values in terms of 5 volts and 0 volts. Thus the
number 39 represented in binary as 100111 is stored electronically in as +5 v, 0v, 0v, +5
v, +5 v and +5 v.
Advantages of working in the Digital Domain
Handling information digitally offers several advantages. Some of the merits of a
digital system are spelled out. Details of some these aspects will be discussed and studied
in the Digital Logic Design course. Other aspects will be covered in several other
courses.
• Storing and processing data in the digital domain is more efficient: Computers
are very efficient in processing massive amounts of information and data. Computers
process information that is represented digitally in the form of Binary Numbers. A
Digital CD stores large number of video and audio clips. Sam number of audio and
video clips if stored in analogue form will require a number of video and audio
cassettes.
• Transmission of data in the digital form is more efficient and reliable: Modern
information transmission techniques are relying more on digital transmission due to
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its reliability as it is less prone to errors. Even if errors occur during the transmission
methods exist which allow for quick detection and correction of errors.
• Detecting and Correcting errors in digital data is easier: Coding Theory is an area
which deals with implementing digital codes that allow for detection and correction
of multi-bit errors. In the Digital Logic Design course a simple method to detect
single bit errors using the Parity bit will be considered.
• Data can be easily and precisely reproduced: The picture quality and the sound
quality of digital videos are far more superior to those of analogue videos. The reason
being that the digital video stored as digital numbers can be exactly reproduced where
as analogue video is stored as a continuous signal can not be reproduced with exact
precision.
• Digital systems are easy to design and implement: Digital Systems are based on
two-state Binary Number System. Consequently the Digital Circuitry is based on the
two-voltage states, performing very simple operations. Complex Microprocessors are
implemented using simple digital circuits. Several simple Digital Systems will be
discussed in the Digital Logic course.
• Digital circuits occupy small space: Digital circuits are based on two logical states.
Electronic circuitry that implements the two states is very simple. Due to the
simplicity of the circuitry it can be easily implemented in a very small area. The PC
motherboard having an area of approximately 1 sq.ft has most of the circuitry of a
powerful computer. A memory chip small enough to be held in the palm of a hand is
able to store an entire collection of books.
Information Processing by a Digital System
A Digital system such as a computer not only handles numbers but all kinds of
information.
• Numbers: A computer is able to store and process all types of numbers, integers,
fractions etc. and is able to perform different kinds of arithmetic operations on the
numbers.
• Text: A computer in a news reporting room is used to write and edit news reports. A
Mathematician uses a computer to write mathematical equations explaining the
dissipation of heat by a heat sink. The computer is able to store and process text and
symbols.
• Drawings, Diagrams and Pictures: A computer can store very conveniently
complex engineering drawings and diagrams. It allows real life still pictures or videos
to be processed and edited.
• Music and Sound: Musicians and Composers uses\ a computer to work on a new
compositions. Computers understand spoken commands.
A Digital System (computer) is capable of handling different types of information,
which is represented in the form of Binary Numbers. The different types of information
use different standards and binary formats. For example, computers use the Binary
number system to represent numbers. Characters used in writing text are represented
through yet another standard known as ASCII which allows alphabets, punctuation marks
and numbers to be represented through a combination of 0s and 1s.
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Digital Components and their internal working
Digital system process binary information electronically through specialized circuits
designed for handling digital information. These circuits as mentioned earlier operate
with two voltage values of +5 volts and 0 volts. These specialized electronic circuits are
known as Logic Gates and are considered to be the Basic Building Blocks of any Digital
circuit.
The commonly used Logic Gates are the AND gate, the OR gate and the Inverter or
NOT Gate. Other gates that are frequently used include NOR, NAND, XOR and XNOR.
Each of these gates is designed to perform a unique operation on the input information
which is known as a logical or Boolean operation.
Large and complex digital system such as a computer is built using combinations of
these basic Logic Gates. These basic building blocks are available in the form of
Integrated Circuit or ICs. These gates are implemented using standard CMOS and TTL
technologies that determine the operational characteristics of the gates such as the power
dissipation, operational voltages (3.3v or 5 v), frequency response etc.
Figure 1.6 Symbolic representations of logic gates.
Combinational Logic Circuits and Functional Devices
The logic gates which form the basic building blocks of a digital system are designed
to perform simple logic operations. A single logic gate is not of much use unless it is
connected with other gates to collectively act upon the input data. Different gates are
combined to build a circuit that is capable of performing some useful operation like
adding three numbers. Such circuits are known as Combinational Logic Circuits or
Combinational Circuits. An Adder Combinational Circuit that is able to add two single
bit binary numbers and give a single bit Sum and Carry output is shown. Figure 1.7.
Implementing large digital system by connecting together logic gates is very tedious
and time consuming; the circuit implemented occupies large space, are power hungry,
slow and are difficult to troubleshoot.
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Figure 1.7 1-bit Full-Adder Combinational Circuit
Digital circuits to perform specific functions are available as Integrated Circuits for
use. Implementing a Digital system in terms of these dedicated functional units makes
the system more economical and reliable. Thus an adder circuit does not have to be
implemented by connecting various gates, a standard Adder IC is available that can be
readily used. Other commonly used combinational functional devices are Comparators,
Decoders, Encoders, Multiplexers and Demultiplexers.
Sequential logic and implementation
Digital systems are used in vast variety of industrial applications and house hold
electronic gadgets. Many of these digital circuits generate an output that is not only
dependent on the current input but also some previously saved information which is used
by the digital circuit. Consider the example of a digital counter which is used by many
digital applications where a count value or the time of the day has to be displayed. The
digital counter which counts downwards from 10 to 0 is initialized to the value 10. When
the counter receives an external signal in the form of a pulse the counter decrements the
count value to 9. On receiving successive pulses the counter decrements the currently
stored count value by one, until the counter has been decremented to 0. On reaching the
count value zero, the counter could switch off a washing machine, a microwave oven or
switch on an air-conditioning system.
The counter stores or remembers the previous count value. The new count value is
determined by the previously stored count value and the new input which it receives in
the form of a pulse (a binary 1). The diagram of the counter circuit is shown in the figure.
Figure 1.8.
Digital circuits that generate a new output on the basis of some previously stored
information and the new input are known as Sequential circuits. Sequential circuits are a
combination of Combinational circuits and a memory element which is able to store some
previous information. Sequential circuits are a very important part of digital systems.
Most digital systems have sequential logic in addition to the combinational logic. An
A
B
Σ
Cout
Cin
P
G
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example of sequential circuits is counters such as the down-counter which generates a
new decremented output value based on the previous stored value and an external input.
The storage element or the memory element which is an essential part of a sequential
circuit is implemented a flip-flop using a very simple digital circuit known as a flip-flop.
Figure 1.8 A Counter Sequential Circuit
Programmable Logic Devices (PLDs)
The modern trend in implementing specialized dedicated digital systems is through
configurable hardware; hardware which can be programmed by the end user. A digital
controller for a washing machine can be implemented by connecting together pieces of
combinational and sequential functional units. These implementations are reliable
however they occupy considerable space. The implementation time also increases. A
general purpose circuit that can be programmed to perform a certain task like controlling
a washing machine reduces the implementation cost and time.
Cost is incurred on implementing a digital controller for a washing machine which
requires that an inventory of all its components such as its logic circuits, functional
devices and the counter circuits be maintained. The implementation time is significantly
high as all the circuit components have to be placed on a circuit board and connected
together. If there is a change in the controller circuit the entire circuit board has to be
redesigned. A PLD based washing machine controller does not require a large inventory
of components to be maintained. Most of the functionality of the controller circuit is
implemented within a single PLD integrated circuit thereby considerably reducing the
circuit size. Changes in the controller design can be readily implemented by
programming the PLD.
Programmable Logic Devices can be used to implement Combinational and
Sequential Digital Circuits.
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Memory
Memory plays a very important role in Digital systems. A research article being
edited by a scientist on a computer is stored electronically in the digital memory whilst
changes are being made to the document. Once the document has be finalized and stored
on some media for subsequent printing the memory can be reused to work on some other
document. Memory also needs to store information permanently even when the electrical
power is turned off. Permanent memories usually contain essential information required
for operating the digital system. This important information is provided by the
manufacturer of a digital system.
Memory is organized to allow large amounts of data storage and quick access.
Memory (ROM) which permanently stores data allows data to be read only. The Memory
does not allow writing of data. Volatile memory (RAM) does not store information
permanently. If the power supplied to the RAM circuitry is turned off, the contents of the
RAM are permanently lost and can not be recovered when power is restored. RAM
allows reading and writing of data. Both RAM and ROM are an essential part of a digital
system.
Analogue to Digital and Digital to Analogue conversion and
Interfacing
Real-world quantities as mention earlier are continuous in nature and have widely
varying ranges. Processing of real-world information can be efficiently and reliably done
in the digital domain. This requires real-world quantities to be read and converted into
equivalent digital values which can be processed by a digital system. In most cases the
processed output needs to be converted back into real-world quantities. Thus two
conversions are required, one from the real-world to the digital domain and then back
from the digital domain to the real-world.
Modern digitally controlled industrial units extensively use Analogue to Digital (A/D)
and Digital to Analogue (D/A) converters to covert quantities represented as an analogue
voltage into an equivalent digital representation and vice versa. Consider the example of
an industrial controller that controls a chemical reaction vessel which is being heated to
expedite the chemical reaction. Figure 1.9. Temperature of the vessel is monitored to
control the chemical reaction. As the temperature of the vessel rises the heat has to be
reduced by a proportional level. An electronic temperature sensor (transducer) converts
the temperature into an equivalent voltage value. This voltage value is continuous and
proportion to the temperature. The voltage representing the temperature is converted into
a digital representation which is fed to a digital controller that generates a digital value
corresponding to the desired amount of heat. The digitized output representing the heat is
converted back to a voltage value which is continuous and is used to control a valve that
regulates the heat. An A/D converter converts the analogue voltage value representing the
temperature into a corresponding digital value for processing. A D/A converter converts
back the digital heat value to its corresponding continuous value for regulating the heater.
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Figure 1.9 Digitally Controlled Industrial Heater Unit
A/D and D/A converters are an important aspect of digital systems. These devices
serve as a bridge between the real and digital world allow the two to communicate and
interact together.
Number Systems and Codes
Decimal Number System
The decimal number system has ten unique digits 0, 1, 2, 3… 9. Using these single
digits, ten different values can be represented. Values greater than ten can be represented
by using the same digits in different combinations. Thus ten is represented by the number
10, two hundred seventy five is represented by 275 etc. Thus same set of numbers 0,1
2… 9 are repeated in a specific order to represent larger numbers.
The decimal number system is a positional number system as the position of a digit
represents its true magnitude. For example, 2 is less than 7, however 2 in 275 represents
200, whereas 7 represents 70. The left most digit has the highest weight and the right
most digit has the lowest weight. 275 can be written in the form of an expression in terms
of the base value of the number system and weights.
2 x 102 + 7 x 101 + 5 x 100 = 200 + 70 + 5 = 275
where, 10 represents the base or radix
102, 101, 100 represent the weights 100, 10 and 1 of the numbers 2, 7 and 5
Fractions in Decimal Number System
In a Decimal Number System the fraction part is separated from the Integer part by a
decimal point. The Integer part of a number is written on the left hand side of the decimal
point. The Fraction part is written on the right side of the decimal point. The digits of the
Digital
Controller
Transducer
A/D
Converter
D/A
Converter
Vessel
Heater
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Integer part on the left hand side of the decimal point have weights 100, 101, 102 etc.
respectively starting from the digit to the immediate left of the decimal point and moving
away from the decimal point towards the most significant digit on the left hand side.
Fractions in decimal number system are also represented in terms of the base value of the
number system and weights. The weights of the fraction part are represented by 10-1, 10-2,
10-3 etc. The weights decrease by a factor of 10 moving right of the decimal point. The
number 382.91 in terms of the base number and weights is represented as
3 x 102 + 8 x 101 + 2 x 100 + 9 x 10-1 + 1 x 10-2 = 300 + 80 + 2 + 0.9 + 0.01 = 382.91
Caveman number system
A number system discovered by archaeologists in a prehistoric cave indicates that
the caveman used a number system that has 5 distinct shapes Σ, Δ, >, Ω and ↑.
Furthermore it has been determined that the symbols Σ to ↑ represents the decimal
equivalents 0 to 5 respectively.
Centuries ago a caveman returning after a successful hunting expedition records
his successful hunt on the cave wall by carving out the numbers Δ↑. What does the
number Δ↑ represent? The table 1.1 indicates that the Caveman numbers Δ↑ represents
decimal number 9.
Decimal Number Caveman Number Decimal Number Caveman Number
0 Σ 10 >Σ
1 Δ 11 >Δ
2 > 12 >>
3 Ω 13 >Ω
4 ↑ 14 >↑
5 ΔΣ 15 ΩΣ
6 ΔΔ 16 ΩΔ
7 Δ> 17 Ω>
8 ΔΩ 18 ΩΩ
9 Δ↑ 19 Ω↑
20 ↑Σ
Table 1.1 Decimal equivalents of the Caveman Numbers
The Caveman is using a Base-5 number system. A Base-5 number system has five
unique symbols representing numbers 0 to 4. To represent numbers larger than 4, a
combination of 2, 3, 4 or more combinations of Caveman numbers are used. Therefore, to
represent the decimal number 5, a two number combination of the Caveman number
system is used. The most significant digit is Δ which is equivalent to decimal 1. The least
significant digit is Σ which is equivalent to decimal 0. The five combinations of
Caveman numbers having the most significant digit Δ, represent decimal values 5 to 9
respectively. This is similar to the Decimal Number system, where a 2-digit combination
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of numbers is used to represent values greater than 9. The most significant digit is set to 1
and the least significant digit varies from 0 to 9 to represent the next 10 values after the
largest single decimal number digit 9.
The Caveman number Δ↑ can be written in expression form based on the Base
value 5 and weights 50, 51, 52 etc.
= Δ x 51 + ↑ x 50 = Δ x 5 + ↑ x 1
Replacing the Caveman numbers Δ and ↑ with equivalent decimal values in the
expression yields
= Δ x 51 + ↑ x 50 = 1 x 5 + 4 x 1 = 9
The number ΔΩ↑Σ in decimal is represented in expression form as
Δ x 53 + Ω x 52 + ↑ x 51 + Σ x 50 = Δ x 125 + Ω x 25 + ↑ x 5 + Σ x 1
Replacing the Caveman numbers with equivalent decimal values in the expression yields
= (1) x 125 + (3) x 25 + (4) x 5 + (0) x 1 = 125 + 75 + 20 + 0 = 220
Binary Number System
The Caveman Number system is a hypothetical number system introduced to
explain that number system other than the Decimal Number system can exist and can be
used to represent and count numbers. Digital systems use a Binary number system.
Binary as the name indicates is a Base-2 number system having only two numbers 0 and
1. The Binary digit 0 or 1 is known as a 'Bit'. Table 1.2
Decimal Number Binary Number Decimal Number Binary Number
0 0 10 1010
1 1 11 1011
2 10 12 1100
3 11 13 1101
4 100 14 1110
5 101 15 1111
6 110 16 10000
7 111 17 10001
8 1000 18 10010
9 1001 19 10011
20 10100
Table 1.2 Decimal equivalents of Binary Number System
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Counting in Binary Number system is similar to counting in Decimal or Caveman
Number systems. In a decimal Number system a value larger than 9 has to be represented
by 2, 3, 4 or more digits. In the Caveman Number System a value larger than 4 has to be
represented by 2, 3, 4 or more digits of the Caveman Number System. Similarly, in the
Binary Number System a Binary number larger than 1 has to be represented by 2, 3, 4 or
more binary digits.
Any binary number comprising of Binary 0 and 1 can be easily represented in
terms of its decimal equivalent by writing the Binary Number in the form of an
expression using the Base value 2 and weights 20, 21, 22 etc.
The number 100112 (the subscript 2 indicates that the number is a binary number
and not a decimal number ten thousand and eleven) can be rewritten in terms of the
expression
100112 = (1 x 24) + (0 x 23) + (0 x 22) + (1 x 21) + (1 x 20)
= (1 x 16) + (0 x 8) + (0 x 4) + (1 x 2) + (1 x 1)
= 16 + 0 + 0 + 2 + 1
= 19
Fractions in Binary Number System
In a Decimal number system the Integer part and the Fraction part of a number are
separated by a decimal point. In a Binary Number System the Integer part and the
Fraction part of a Binary Number can be similarly represented separated by a decimal
point. The Binary number 1011.1012 has an Integer part represented by 1011 and a
fraction part 101 separated by a decimal point. The subscript 2 indicates that the number
is a binary number and not a decimal number. The Binary number 1011.1012 can be
written in terms of an expression using the Base value 2 and weights 23, 22, 21, 20, 2-1, 2-2
and 2-3.
1011.1012 = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) + (1 x 2-1) + (0 x 2-2) + (1 x 2-3)
= (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1) + (1 x 1/2) + (0 x 1/4) + (1 x 1/8)
= 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125
= 11.625
Computers do handle numbers such as 11.625 that have an integer part and a
fraction part. However, it does not use the binary representation 1011.101. Such numbers
are represented and used in Floating-Point Numbers notation which will be discussed
latter.
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