Computer Fundamentals:
Pradeep
K.
Sinha
&
Priti
Sinha
Computer Fundamentals:
Pradeep
K.
Sinha
&
Priti
Sinha
Slide 1/40
Chapter 3: Number Systems
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Number Systems: Understanding Binary, Octal, and Hexadecimal Number Systems, Lecture notes of Information Technology
A study guide on number systems, specifically focusing on binary, octal, and hexadecimal number systems. It covers the basics of positional and non-positional number systems, the characteristics and uses of each system, and methods for converting numbers between different bases. It also includes examples and key phrases for further understanding.
Typology: Lecture notes
2018/2019
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In this chapter you will learn about:
Ā§ Non-positional number system
Ā§ Positional number system
Ā§ Decimal number system
Ā§ Binary number system
Ā§ Octal number system
Ā§ Hexadecimal number system
Learning Objectives Learning Objectives
(Continued on next slide)
Two types of number systems are:
Ā§ Non-positional number systems
Ā§ Positional number systems
Number Systems Number Systems
Ā§ Characteristics
Ā§ Use symbols such as I for 1, II for 2, III for 3, IIII
for 4, IIIII for 5, etc
Ā§ Each symbol represents the same value regardless
of its position in the number
Ā§ The symbols are simply added to find out the value
of a particular number
Ā§ Difficulty
Ā§ It is difficult to perform arithmetic with such a
number system
Non-positional Number Systems Non-positional Number Systems
Ā§ The value of each digit is determined by:
1. The digit itself
2. The position of the digit in the number
3. The base of the number system
( base = total number of digits in the number
system)
Ā§ The maximum value of a single digit is
always equal to one less than the value of
the base
Positional Number Systems Positional Number Systems
(Continued from previous slide..)
Characteristics
Ā§ A positional number system
Ā§ Has 10 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7,
8, 9). Hence, its base = 10
Ā§ The maximum value of a single digit is 9 (one
less than the value of the base)
Ā§ Each position of a digit represents a specific
power of the base (10)
Ā§ We use this number system in our day-to-day
life
Decimal Number System Decimal Number System
(Continued on next slide)
Characteristics
Ā§ A positional number system
Ā§ Has only 2 symbols or digits (0 and 1). Hence its
base = 2
Ā§ The maximum value of a single digit is 1 (one less
than the value of the base)
Ā§ Each position of a digit represents a specific power
of the base (2)
Ā§ This number system is used in computers
Binary Number System Binary Number System
(Continued on next slide)
Example
101012 = (1 x 2^4 ) + (0 x 2^3 ) + (1 x 2^2 ) + (0 x 2^1 ) x (1 x 2^0 ) = 16 + 0 + 4 + 0 + 1 = (^2110)
Binary Number System Binary Number System
(Continued from previous slide..)
Ā§ Bit stands for bi nary digi t
Ā§ A bit in computer terminology means either a 0 or a 1
Ā§ A binary number consisting of n bits is called an n-bit
number
Bit Bit
Characteristics
Ā§ A positional number system
Ā§ Has total 8 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7).
Hence, its base = 8
Ā§ The maximum value of a single digit is 7 (one less
than the value of the base
Ā§ Each position of a digit represents a specific power of
the base (8)
Octal Number System Octal Number System
(Continued on next slide)
Characteristics
Ā§ A positional number system
Ā§ Has total 16 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7,
8, 9, A, B, C, D, E, F). Hence its base = 16
Ā§ The symbols A, B, C, D, E and F represent the
decimal values 10, 11, 12, 13, 14 and 15
respectively
Ā§ The maximum value of a single digit is 15 (one less
than the value of the base)
Hexadecimal Number System Hexadecimal Number System
(Continued on next slide)
Ā§ Each position of a digit represents a specific power
of the base (16)
Ā§ Since there are only 16 digits, 4 bits (
4
= 16) are
sufficient to represent any hexadecimal number in
binary
Example
1AF 16 = (1 x 16
2
) + (A x 16
1
) + (F x 16
0
= 1 x 256 + 10 x 16 + 15 x 1
Hexadecimal Number System Hexadecimal Number System
(Continued from previous slide..)
Example 4706
=?
47068 = 4 x 8 3
- 7 x 8 2
- 0 x 8 1
- 6 x 8 0 = 4 x 512 + 7 x 64 + 0 + 6 x 1 = 2048 + 448 + 0 + 6 = 2502 10 Common values multiplied by the corresponding digits Sum of these products (Continued from previous slide..) Converting a Number of Another Base to a Decimal Number Converting a Number of Another Base to a Decimal Number
Division-Remainder Method Step 1: Divide the decimal number to be converted by the value of the new base Step 2: Record the remainder from Step 1 as the rightmost digit (least significant digit) of the new base number Step 3: Divide the quotient of the previous divide by the new base Converting a Decimal Number to a Number of Another Base Converting a Decimal Number to a Number of Another Base (Continued on next slide)