1.Digital Electronics overview & Number Systems.pptx

Outlines
Introduction : An Overview of Digital Electronics
Analog Quantity vs Digital Quantity
Advantages of Digital System
Number System Overviews
Decimal Number Systems
Binary Number Systems
Conversions Between Binary Numbers & Decimal Number

Digital Electronics: An overview
 Signals are used to carry information from one device to another.
There are two different forms of signals: Analog and Digital.
 Analog is a continuous wave that keeps on changing over a time
period (e.g. human voice).
 Digital: signal is discrete in nature. Electronic devices are
classified as being either Analog or Digital.

Digital Electronics: An overview
• Digital Electronics/Digital circuits:
circuits that operate on Digital signals.
• Digital Electronics use binary strings: 0
and 1 to send signals through a system.

Analog Quantity Vs Digital Quantity
 An analog quantity is one having continuous values.
 A digital quantity is one having a discrete set of values.
 Most things that can be measured quantitatively occur in
nature in analog form.
 For example, the air temperature changes over a continuous
range of values This is an Example of Analog Quantity
 Mobile Phones, Computers are Digital Signals.

Advantages of Digital System
1. Its components are Highly Reliable (higher accuracy).
2. It is Low Cost.
3. It is Small in Size.
4. It is Light Weight.
5. It Consumes Less power.
6. It is Faster in Operation.
7. Easier to Design.
8. Noise Immunity.
9. Easier Storage of Data.

Number Systems
Characteristics of Number Systems:
1.The number of independent digits used in the
number system.
2.The place values of the different digits
constituting the number and
3.The maximum numbers that can be written with
the given number of digits.

Number Systems…
• Among the three characteristic parameters, the most
fundamental is the number of independent digits or symbols
used in the number system.
• It is known as the Radix or Base of the number system.
Decimal Binary Octal Hex
Base 10 Base 2 Base 8 Base 16

Decimal Number Systems…
• The place values of different digits in the integer part of
the number are given by r0, r1, r2, r3 and so on, starting
with the digit adjacent to the radix point.
• For the fractional part, these are r−1, r−2, r−3 and so on,
again starting with the digit next to the radix point.
• Here, r is the radix of the number system.
• Also, maximum numbers that can be written with n
digits in a given number system are equal to rn.

Decimal Number System
• The place values of different digits in a mixed decimal
number, starting from the decimal point, are 100, 101,
102 and so on (for the integer part) and 10−1, 10−2, 10−3
and so on (for the fractional part).
• The value or magnitude of a given decimal number can
be expressed as the sum of the various digits multiplied
by their place values or weights.

Decimal Number System…
Example: Express the integer and fractional part of the
decimal number 3586.265 .
Solution
The integer part (3586) can be expressed as:
3586 = 6×100 +8×101+5×102 +3×103= 6+80+500+3000
= 3586
The fractional part can be expressed as
265 =2×10−1+6×10−2 +5×10−3 = 0.2+0.06+0. 005 = 0.265.

Example
 Express the decimal number 47 as a sum of the values of each
digit.
Solution
 The digit 4 has a weight of 10, which is 101 , as indicated by its
position.
 The digit 7 has a weight of 1, which is 100 , as indicated by its
position. 47 = (4 * 101 ) + (7 * 100 ) = (4 * 10) + (7 * 1) = 40 + 7
 Related Problem: Determine the value of each digit in 939

Binary Number System
• Starting from the binary point, the place values of
different digits in a mixed binary number are 20, 21, 22
and so on (for the integer part) and 2−1, 2−2, 2−3 and so on
(for the fractional part).
Advantages:
• Logic operations are the backbone of any digital
computer, although solving a problem on computer could
involve an Arithmetic operation too.

• The introduction of the mathematics of logic by George Boole
laid the foundation for the modern digital computer. He reduced
the mathematics of logic to a binary notation of ‘0’ and ‘1’.
• As the mathematics of logic was well established and had proved
itself to be quite useful in solving all kinds of logical problem, and
also as the mathematics of logic (also known as Boolean algebra)
had been reduced to a binary notation, the binary number
system had a clear edge over other number systems for use in
computer systems.

• Yet another significant advantage of this number
system was that all kinds of data could be
conveniently represented in terms of 0s and 1s.
• Also, basic electronic devices used for hardware
implementation could be conveniently and efficiently
operated in two distinctly different modes.
• Lastly, the circuits required for performing arithmetic
operations such as addition, subtraction,
multiplication, division, etc., become a simple affair
when the data involved are represented in the form
of 0s and 1s.

Octal Number System…
• The octal number system has a eight distinct/independent digits
(0, 1, 2, 3, 4, 5, 6 and 7). The next 10 numbers that follow ‘7’, for
example, would be 10, 11, 12,13, 14, 15, 16, 17, 20 and 21.
• In fact, if we omit all the numbers containing the digits 8 or 9, or
both, from the decimal number system, we end up with an octal
number system.
• The place values for the different digits in the octal number
system are 80, 81, 82 and so on (for the integer part) and 8−1, 8−2,
8−3 and so on (for the fractional part).

Hexadecimal Number System…
• The basic digits of hexadecimal number are 0, 1, 2, 3, 4,
5, 6, 7, 8, 9, A, B, C, D, E and F.
• The place values or weights of different digits in a
mixed hexadecimal number are 160, 161, 162 and so on
(for the integer part) and 16−1,16−2, 16−3 and so on (for
the fractional part).
• The decimal equivalent of A, B, C, D, E and F are 10, 11,
12, 13, 14 and 15 respectively, for obvious reasons

Why Hexadecimal Number?
• The hexadecimal number system provides a condensed way of
representing large binary numbers stored and processed inside
the computer.
• One such example is in representing addresses of different
memory locations. Let us assume that a machine has 64K of
memory. Such a memory has 64K (= 216 = 65 536) memory
locations and needs 65 536 different addresses.
• These addresses can be designated as 0 to 65 535 in the decimal
number system and 00000000 00000000 to 11111111 11111111
in the binary number system.

Why Hexadecimal Number?
• The decimal number system is not used in computers
and the binary notation here appears too cumbersome
and inconvenient to handle.
• In the hexadecimal number system, 65 536 different
addresses can be expressed with four digits from 0000 to
FFFF.
• Similarly, the contents of the memory when represented
in hexadecimal form are very convenient to handle.

Class Activity
Example 1.1: Consider an arbitrary number system with
the independent digits as 0, 1 and X. What is the radix of
this number system? List the first 10 numbers in this
number system.
Solution:
• The radix of the proposed number system is 3. • The
first 10 numbers in this number system would be 0, 1, X,
10, 11, 1X, X0, X1, XX and 100

• Every number system uses positional notation.
• Each position is power of 2, which begins at 0 and
increase by 1.
positional values for an 8-bit binary number, all bits are on
• The value of a binary number is based on the presence
of 1 bits and their positional value.
• So the value of the given binary number is: 1 + 2 + 4 +
8 +16 + 32 + 64 + 128 = 255, which is same as 28 - 1.

Binary Numbers
• Digits are 1 & 0
1 = true /On or High
0 = false /Off or Low
• MSB – Most Significant Bit
• LSB – Least Significant Bit
• Bit numbering:

Translating Binary to Decimal
• Weighted positional notation shows how to calculate
the decimal value of each binary bit:
• D = binary digit
• binary 00001001 = decimal 9:(1 × 23) + (1 × 20) = 9

Binary-to-Decimal Conversion
 The decimal value of any binary number can be found by adding the
weights of all bits that are 1 and discarding the weights of all bits that are
0

Binary to Decimal (Fraction)
 The decimal equivalent of the binary number (1001.0101)2 is
determined as follows:
 The integer part = 1001
The decimal equivalent = 1 × 20+ 0 × 21 + 0 × 22 + 1 × 23= 1 + 0 + 0
+ 8 = 9
 The fractional part = .0101
Therefore, the decimal equivalent = 0 × 2−1 + 1 × 2−2 + 0 × 2−3 + 1
× 2−4 = 0 + 0.25 + 0 + 0.0625 = 0.3125.
• Therefore, the decimal equivalent of (1001.0101)2 = 9.3125

Decimal-to-Binary Conversion
 Sum-of-Weights Method
To get the binary number for a given decimal
integer number, find the binary weights that add up
to the decimal number.

Repeated Division-by-2 Method
 To get the binary number for a given decimal integer
number, divide the decimal number by 2 until the quotient
is 0. Remainders from bottom to top form the binary
number.

Translating Decimal into Binary
 Repeatedly divide the decimal integer by 2. Each remainder is
a binary digit in the translated value:
37 = 100101

Translating Decimal to Binary (Fraction)
find the binary equivalent of (13.375)10
The binary equivalent of
(13)10 is therefore (1101)2
The fractional part = .375
• 0.375 × 2 = 0.75 with a carry of 0
• 0.75 × 2 = 0.5 with a carry of 1
• 0.5 × 2 = 0 with a carry of 1
• The binary equivalent of (0.375)10 = (.011)2
• Therefore, the binary equivalent of (13.375)10 = (1101.011)2

Outlines
Hexadecimal Numbers System
Counting in Hexadecimal Number
Converting binary to hexadecimal
Converting Hexadecimal to Binary
Hexadecimal-to-Decimal Conversion
Decimal-to-Hexadecimal Conversion

Hexadecimal Numbers
 The Hexadecimal number system has a base of sixteen;
that is, it is composed of 16 numeric and alphabetic
characters.
 The hexadecimal number system consists of digits 0–9
and letters A–F.
 Ten numeric digits and six alphabetic characters make up
the hexadecimal number system. The use of letters A, B, C,
D, E, and F to represent numbers may seem strange at
first, but keep in mind that any number system is only a set

Counting in Hexadecimal Number
 Converting a binary number to hexadecimal is a
straightforward procedure.
 Simply break the binary number into 4-bit groups,
starting at the right-most bit and replace each 4-bit
group with the equivalent hexadecimal symbol.
 The maximum 3-digit hexadecimal number is FFF16.
 The maximum 4-digit hexadecimal number is FFFF16.

Converting binary to hexadecimal
To convert a binary number to its hexadecimal
equivalent, break it into 4 consecutive groups )like
Somali banquet) , starting from the right, and
write those groups over the corresponding digits
of the hexadecimal number.
Example: Binary number 1000110011010001
is equivalent to Hexadecimal 8CD1

Translating Binary to Hexadecimal
 Example: Translate the following binary integer
000101101010011110010100 into hexadecimal
number:
 Each hexadecimal digit corresponds to 4 binary bits.

Converting Hexadecimal to Binary
To convert a hexadecimal number to
binary just write each hexadecimal digit
into its 4-digit binary equivalent.
Example: Hexadecimal number FAD8 is
equivalent to binary 1111101011011000.

Hexadecimal-to-Binary Conversion
 To convert from a hexadecimal number to a binary number,
reverse the process and replace each hexadecimal symbol with
the appropriate four bits.

Hexadecimal-to-Decimal Conversion
 One way to find the decimal equivalent of a hexadecimal
number is to first convert the hexadecimal number to binary
and then convert from binary to decimal.

Converting Hexadecimal to Decimal
 Multiply each digit by its corresponding power
of 16:
• Dec = (D3 × 163) + (D2 × 162) + (D1 × 161) + (D0 × 160)
• Hex =1234 equals (1 × 163) + (2 × 162) + (3 × 161) + (4
× 160), or decimal 4,660.
• Hex =3BA4 equals (3 × 163) + (11 * 162) + (10 × 161) +
(4 × 160), or decimal 15,268.

 Another way to convert a hexadecimal number to its decimal
equivalent is to multiply the decimal value of each hexadecimal
digit by its weight and then take the sum of these products.
 The weights of a hexadecimal number are increasing powers of 16
(from right to left).
 For a 4-digit hexadecimal number, the weights are
 163 162 161 160
 4096 256 16 1

Decimal-to-Hexadecimal Conversion

Converting Decimal to Hexadecimal
Decimal 422 = 1A6 Hexadecimal
Example: Convert the Decimal Number of 422
into Hexadecimal Number;

Outlines
Octal Numbers
Octal-to-Decimal Conversion
Decimal Number-to-Octal Conversion
Octal-to-Binary Conversion
Converting Hexadecimal to Octal
Number Representation into Binary
Sign-Bit Magnitude

Octal Numbers
 Like the hexadecimal number system, the octal
number system provides a convenient way to express
binary numbers and codes.
 However, it is used less frequently than Hexadecimal in
conjunction with computers and microprocessors to
express binary quantities for input and output
purposes.
The octal number system has a base of 8

Octal-to-Decimal Conversion
 Since the octal number system has a base of eight, each successive
digit position is an increasing power of eight, beginning in the right-
most column with 80.
 The evaluation of an octal number in terms of its decimal
equivalent is accomplished by multiplying each digit by its weight
and summing the products, as illustrated here for 2378 .
 Weight: 83 82 81 80
 Octal number: 2 3 7 4
 23748 = (2 * 83 ) + (3 * 82 ) + (7 * 81 ) + (4 * 80 )
= (2 * 512) + (3 * 64) + (7 * 8) + (4 * 1)
= 1024 + 192 + 56 + 4
= 127610

Converting Octal to Decimal
 Multiply each digit by its corresponding
power of 8:
 The decimal equivalent of the octal number
(137.21)8 is determined as follows:
 The integer part = 137 = 7 × 80 + 3 × 81 + 1 ×
82 = 7 + 24 + 64 = 95
• The fractional part = 0.21 = 2 × 8−1 + 1 × 8−2 =
0.25 + 0.015625 = 0.265625
 Therefore, (137.21)8= (95.265625)10

Decimal Number-to-Octal Conversion
 A method of converting a decimal number to an octal
number is the repeated division by-8 method, which is
similar to the method used in the conversion of decimal
numbers to binary or to hexadecimal. To show how it
works, let’s convert the decimal number 359 to octal.
 Each successive division by 8 yields a remainder that
becomes a digit in the equivalent octal number. The first
remainder generated is the least significant digit (LSD).

Converting Decimal to Octal
 The progressive Division in the case of the
Integer part and the progressive Multiplication
while working on the fractional part here are by
‘8’ which is the Radix or Base of the octal
Number system.

Converting Decimal to Octal
 Example: find the octal equivalent of (73.75)10
 The Octal Equivalent
of (73)10 = (111)8
• The fractional part = 0.75
• 0.75 × 8 = 0 with a carry of 6
• The octal equivalent of (0.75)10= (0.6)8
• Therefore, the octal equivalent of (73.75)10 = (111.6)8

 Because 8 = 23 and each octal digit can be represented
by a 3-bit binary number, it is very easy to convert from
octal to binary.
 Each octal digit is represented by three bits as shown in
Table 2–4.

 To convert an octal number to a binary number, simply
replace each octal digit with the appropriate three bits.

Converting Octal to Binary
 Example: find the binary equivalent of (374.26)8
 Solution
•The given octal number = (374.26)8
•The binary equivalent = (011 111 100.010 110)2=
(011111100.010110)2
•Any 0s on the extreme left of the integer part and extreme right of
the fractional part of the equivalent binary number should be
omitted.
 Therefore, (374.26)8 = (11111100.01011)2

Binary-to-Octal Conversion
 Conversion of a binary number to an octal number is the reverse
of the octal-to-binary conversion.
 The procedure is as follows: Start with the right-most group of
three bits and, moving from right to left, convert each 3-bit group
to the equivalent octal digit.
 If there are not three bits available for the left-most group, add
either one or two zeros to make a complete group.
 These leading zeros do not affect the value of the binary number.

Converting Binary to Octal
 A binary number can be converted into an
equivalent octal number by splitting the integer
and fractional parts into groups of three bits,
starting from the binary point on both sides.
 The 0s can be added to complete the outside
groups if needed.

Example
 find the octal equivalent of
(1110100.0100111)2
 Solution
 The given binary number = (1110100.0100111)2
 (1110100.0100111)2 = (1 110 100.010 011 1)2 =
(001 110 100.010 011 100)2 = (164.234)8

Converting Hexadecimal to Octal
 For Hexadecimal–octal conversion, the given hex
number is firstly converted into its Binary equivalent
which is further converted into its octal equivalent
(more convenient and straightforward).
 An alternative approach is firstly to convert the given
Hexadecimal number into its Decimal equivalent and
then convert the decimal number into an equivalent
octal number.

Converting Octal to Hexadecimal
 The octal number may first be converted into an
equivalent binary number and then the binary
number transformed into its hex equivalent.
 The other option is firstly to convert the given
octal number into its decimal equivalent and
then convert the decimal number into its hex
equivalent.

Examples
 find the octal equivalent of (2F.C4)16 and the
hex equivalent of (762.013)8

Three formats are used for binary number
representation of both positive and negative
decimal numbers:
 sign-bit magnitude,
 1’s complement and
 2’s complement methods.

 The left-most bit in a signed binary number is the sign bit, which tells you
whether the number is positive or negative. A 0 sign bit indicates a
positive number, and a 1 sign bit indicates a negative number.
 A signed binary number consists of both sign and magnitude information.
 The sign indicates whether a number is positive or negative, and the
magnitude is the value of the number.

Sign-Bit Magnitude
• The MSB represents the ‘sign’, with a ‘0’ denoting
a plus sign and a ‘1’ denoting a minus sign.
• In eight-bit representation, MSB represents the
sign, while the remaining seven bits represent the
magnitude.

Sign-Bit Magnitude
Example:+9 = 00001001, −9 = 10001001.
An n−bit binary representation can be used to represent
decimal numbers in the range of −(2
n−1
−1) to +(2
n−1
−1).
Eight-bit representation can be used to represent decimal
numbers in the range from −127 to +127 using the sign-bit
magnitude format.

1’s Complement
The positive numbers remain unchanged. The negative numbers of
the 1’s complement are obtained by flipping the digits of the positive
number.
Example:+9 represented as 00001001 and −9 represented as
11110110, which is the 1’s complement of 00001001.
An n−bit binary representation can be used to represent decimal
numbers in the range of −(2n−1−1) to +(2n−1−1). Eight-bit
representation can be used to represent decimal numbers in the
range from −127 to +127 using the sign-bit magnitude format.

Finding the 1’s Complement
Example,
+42 = 00101010
Sign-bit magnitude of -42 = 10101010
1’s complement of -42 = 11010101

Finding the 2’s Complement
 The positive numbers remain unchanged. The 2’s complement
of a binary number is found by adding 1 to the 1’s
complement.
 2’s complement = (1’s complement) + 1

2’s Complement
A negative binary value is expressed in two's
complement notation. According to this rule, to
convert a binary number to its negative value is to
reverse its bit values and add 1.
Example: +9 would be represented as 00001001, and
−9 would be written as 11110111.

1.Digital Electronics overview & Number Systems.pptx

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