Number System & Representation Short notes - Digital Circuits - Electronics and Communication

 Page 1


Number S ystem and Representation
Introduction
The number system is a fundamental concept in digital circuits, enabling the
representation and manipulation of data. V arious number systems, such as bi-
nary , decimal, octal, and hexadecimal, are used to encode information, and their
conversions are essential for digital system design and oper ation.
K ey Concepts
• Number S ystems : Different bases used to represent numbers, including
binary (base-2), decimal (base-10), octal (base-8), and hexadecimal (base-
16).
• Binary S ystem : The foundation of digital circuits, using only 0 and 1 to
represent log ic states (low/high).
• Conversion : Tr ansforming numbers between different bases for process-
ing and interfac ing in digital systems.
• Representation : Numbers can be expressed as unsigned, signed (e.g., two’ s
complement), fix ed-point, or floating-point.
Common Number S ystems
• Decimal ( Base-10) : Uses digits 0–9. Example: 123
10
.
• Binary (Ba se-2) : Uses digits 0 and 1. Example: 1101
2
= 13
10
.
• Octal ( Base-8) : Uses digits 0–7. Example: 15
8
= 13
10
.
• Hexadecimal (Base-16) : Uses digits 0–9 and A–F (A=10, B=11, ..., F=15). Ex-
ample: D
16
= 13
10
.
Conversion F ormulas
• Decimal to Binary : Repeatedly divide the decimal number b y 2, recording
remainders in reverse order . Example: F or 19
10
:
19÷2 = 9( remainder 1), 9÷2 = 4( remainder 1), 4÷2 = 2( remainder 0), 2÷2 = 1( remainder 0), 1÷2 = 0( remainder 1).
Result: 19
10
= 10011
2
.
• Binary to Decimal : Sum the product of each binary digit and its positional
weight (powers of 2). Example: F or 10011
2
:
1·2
4
+0·2
3
+0·2
2
+1·2
1
+1·2
0
= 16+0+0+2+1 = 19
10
.
• Decimal to Hexadecimal : Divide b y 16, record remainders (use A–F for
10–15). Example: F or 29
10
:
29÷16 = 1( remainder 13 = D), 1÷16 = 0( remainder 1).
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Page 2


Number S ystem and Representation
Introduction
The number system is a fundamental concept in digital circuits, enabling the
representation and manipulation of data. V arious number systems, such as bi-
nary , decimal, octal, and hexadecimal, are used to encode information, and their
conversions are essential for digital system design and oper ation.
K ey Concepts
• Number S ystems : Different bases used to represent numbers, including
binary (base-2), decimal (base-10), octal (base-8), and hexadecimal (base-
16).
• Binary S ystem : The foundation of digital circuits, using only 0 and 1 to
represent log ic states (low/high).
• Conversion : Tr ansforming numbers between different bases for process-
ing and interfac ing in digital systems.
• Representation : Numbers can be expressed as unsigned, signed (e.g., two’ s
complement), fix ed-point, or floating-point.
Common Number S ystems
• Decimal ( Base-10) : Uses digits 0–9. Example: 123
10
.
• Binary (Ba se-2) : Uses digits 0 and 1. Example: 1101
2
= 13
10
.
• Octal ( Base-8) : Uses digits 0–7. Example: 15
8
= 13
10
.
• Hexadecimal (Base-16) : Uses digits 0–9 and A–F (A=10, B=11, ..., F=15). Ex-
ample: D
16
= 13
10
.
Conversion F ormulas
• Decimal to Binary : Repeatedly divide the decimal number b y 2, recording
remainders in reverse order . Example: F or 19
10
:
19÷2 = 9( remainder 1), 9÷2 = 4( remainder 1), 4÷2 = 2( remainder 0), 2÷2 = 1( remainder 0), 1÷2 = 0( remainder 1).
Result: 19
10
= 10011
2
.
• Binary to Decimal : Sum the product of each binary digit and its positional
weight (powers of 2). Example: F or 10011
2
:
1·2
4
+0·2
3
+0·2
2
+1·2
1
+1·2
0
= 16+0+0+2+1 = 19
10
.
• Decimal to Hexadecimal : Divide b y 16, record remainders (use A–F for
10–15). Example: F or 29
10
:
29÷16 = 1( remainder 13 = D), 1÷16 = 0( remainder 1).
1
Result: 29
10
= 1D
16
.
• Hexadecimal to Decimal : Sum the product of each hex digit and its posi-
tional weigh t (powers of 16). Example: F or 1D
16
:
1·16
1
+13·16
0
= 16+13 = 29
10
.
Binary Representations
• Unsigned Binary : Represents non-negative integers. Example: 1010
2
=
10
10
.
• Signed Binar y : Includes formats lik e:
– Sign-Magnitude : Most significant bit indicates sign (0 = positive, 1 =
negative). Example: -5 = 1101
2
(4-bit, sign bit 1).
– One’ s Complement : Negative numbers are formed b y inverting all bits.
Example: F or-5 , start with 5 = 0101
2
, invert to 1010
2
.
– Two’ s Complement : Negative numbers are formed b y inverting bits
and adding 1. Example: F or-5 in 4-bit:
5 = 0101
2
, invert = 1010
2
, add 1 = 1011
2
.
• Fixed-Point : Allocates a fixed number of bits for the fr actional part to rep-
resent decimals.
• Floating-Point : Represents numbers as sign× mantissa×2
exponent
, following
standards lik e I EEE 754.
Applications
• Performing binary arithmetic in digital circuits (e.g., adders, multipliers).
• Encoding data in memory , registers, and processors.
• F acilitating conversions for human-machine interfaces (e.g., displa ying dec-
imal values fr om binary data).
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