This document discusses different number systems including non-positional, positional, decimal, binary, octal, and hexadecimal systems. It provides examples of how to convert numbers between these bases using direct conversion methods or shortcuts. Key aspects covered include how the position and base of each digit determines its value in a number, converting a number to decimal and then to another base, and dividing binary, octal, or hexadecimal numbers into groups to convert to a different base number system.
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Number System
Decimal Number System
Binary Number System
Why Binary?
Octal Number System
Hexadecimal Number System
Relationship between Hexadecimal, Octal, Decimal, and Binary
Number Conversions
A power point presentation on number system which briefly explains the conversion of decimal to binary, binary to decimal, binary to octal, octal to decimal. Ping me at Twitter (https://twitter.com/rishabh_kanth), to Download this Presentation.
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Number System
Decimal Number System
Binary Number System
Why Binary?
Octal Number System
Hexadecimal Number System
Relationship between Hexadecimal, Octal, Decimal, and Binary
Number Conversions
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
The 10th Digital Learning Maths for IT sessions - The theme this time being the OCTAL number system which is used widely in computing circles - IP addressing being one.
Some straight forward conversion tasks for you!
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
The 10th Digital Learning Maths for IT sessions - The theme this time being the OCTAL number system which is used widely in computing circles - IP addressing being one.
Some straight forward conversion tasks for you!
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
Number System is a method of representing Numbers on the Number Line with the help of a set of Symbols and rules. These symbols range from 0-9 and are termed as digits. Number System is used to perform mathematical computations ranging from great scientific calculations to calculations like counting the number of Toys for a Kid or Number chocolates remaining in the box. Number Systems comprise of multiple types based on the base value for its digits.
What is the Number Line?
A Number line is a representation of Numbers with a fixed interval in between on a straight line. A Number line contains all the types of numbers like natural numbers, rationals, Integers, etc. Numbers on the number line increase while moving Left to Right and decrease while moving from right to left. Ends of a number line are not defined i.e., numbers on a number line range from infinity on the left side of the zero to infinity on the right side of the zero.
Positive Numbers: Numbers that are represented on the right side of the zero are termed as Positive Numbers. The value of these numbers increases on moving towards the right. Positive numbers are used for Addition between numbers. Example: 1, 2, 3, 4, …
Negative Numbers: Numbers that are represented on the left side of the zero are termed as Negative Numbers. The value of these numbers decreases on moving towards the left. Negative numbers are used for Subtraction between numbers. Example: -1, -2, -3, -4, …
Number and Its Types
A number is a value created by the combination of digits with the help of certain rules. These numbers are used to represent arithmetical quantities. A digit is a symbol from a set 10 symbols ranging from 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any combination of digits represents a Number. The size of a Number depends on the count of digits that are used for its creation.
For Example: 123, 124, 0.345, -16, 73, 9, etc.
Types of Numbers
Numbers are of various types depending upon the patterns of digits that are used for their creation. Various symbols and rules are also applied on Numbers which classifies them into a variety of different types:
Number and Its Types
1. Natural Numbers: Natural Numbers are the most basic type of Numbers that range from 1 to infinity. These numbers are also called Positive Numbers or Counting Numbers. Natural Numbers are represented by the symbol N.
Example: 1, 2, 3, 4, 5, 6, 7, and so on.
2. Whole Numbers: Whole Numbers are basically the Natural Numbers, but they also include ‘zero’. Whole numbers are represented by the symbol W.
Example: 0, 1, 2, 3, 4, and so on.
3. Integers: Integers are the collection of Whole Numbers plus the negative values of the Natural Numbers. Integers do not include fraction numbers i.e. they can’t be written in a/b form. The range of Integers is from the Infinity at the Negative end and Infinity at the Positive end, including zero. Integers are represented by the symbol Z.
Example: ...,-4, -3, -2, -1, 0, 1, 2, 3, 4,...
there are different number system such as binary, decimal, octal and hexadecimal. binary has 2 digits 0 & 1. decimal has 0 to 9 digits. octal has 0 to 7 digits. and hexadecimal number system has 0 to 9 digits and 10 to 15 are denoted by alphabets. such as A=10, B=11 etc.
Inductive programming incorporates all approaches which are concerned with learning programs or algorithms from incomplete (formal) specifications. Possible inputs in an IP system are a set of training inputs and corresponding outputs or an output evaluation function, describing the desired behavior of the intended program, traces or action sequences which describe the process of calculating specific outputs, constraints for the program to be induced concerning its time efficiency or its complexity, various kinds of background knowledge such as standard data types, predefined functions to be used, program schemes or templates describing the data flow of the intended program, heuristics for guiding the search for a solution or other biases.
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2. Learning Objectives
In this chapter you will learn about:
• Non-positional number system
• Positional number system
• Decimal number system
• Binary number system
• Octal number system
• Hexadecimal number system
(Continued on next slide)
3. Learning Objectives
(Continued from previous slide..)
Convert a number’s base
• Another base to decimal base
• Decimal base to another base
• Some base to another base
Shortcut methods for converting
• Binary to octal number
• Octal to binary number
• Binary to hexadecimal number
• Hexadecimal to binary number
Fractional numbers in binary number
•
•
system•
4. Number Systems
Two types of number systems are:
Non-positional number systems•
Positional number systems•
5. Non-positional Number Systems
Characteristics
• Use symbols such as I for 1, II for 2, III for 3, IIII
for 4, IIIII for 5, etc
• Each symbol represents the same value regardless
of its position in the number
• The symbols are simply added to find out the value
of a particular number
•
Difficulty•
It is difficult to perform
number system
arithmetic with such a•
6. Positional Number Systems
Characteristics•
Use only a few symbols called digits•
These symbols represent different values depending•
on the position they occupy in the number
(Continued on next slide)
7. Positional Number Systems
(Continued from previous slide..)
The value of each digit is determined by:•
1.
2.
3.
The digit itself
The position of the digit in the number
The base of the number system
(base =
system)
total number of digits in the number
The maximum value of a single digit is
always equal to one less than the value of
the base
•
8. Decimal Number System
Characteristics
A positional number system
Has 10 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7,
•
•
8, 9). Hence, its base = 10
The maximum value of a single digit is 9 (one
less than the value of the base)
•
Each position of a digit represents a specific•
power of the
We use this
life
base (10)
number system in our day-to-day•
(Continued on next slide)
9. Decimal Number System
(Continued from previous slide..)
Example
258610 = (2 x 103) + (5 x 102) 101) x 100)+ (8 x + (6
= 2000 + 500 + 80 + 6
10. Binary Number System
Characteristics
A positional number system
Has only 2 symbols or digits (0 and 1). Hence its
base = 2
The maximum value of a single digit is 1 (one less
than the value of the base)
Each position of a digit represents a specific power
of the base (2)
•
•
•
•
This number system is used in computers•
(Continued on next slide)
11. Binary Number System
(Continued from previous slide..)
Example
101012 = (1 x 24) + (0 x 23) + (1 x 22) + (0 x 21) x (1 x 20)
= 16 + 0 + 4 + 0 + 1
= 2110
12. Representing Numbers in Different Number
Systems
In order to be specific about which number system we
theare referring to, it is a common practice to indicate
base as a subscript. Thus, we write:
101012 = 2110
13. Bit
• Bit stands for binary digit
• A bit in computer terminology means either a 0 or a 1
• A binary
number
number consisting of n bits is called an n-bit
14. Octal Number System
Characteristics
A positional number system
Has total 8 symbols or digits
Hence, its base = 8
•
• (0, 1, 2, 3, 4, 5, 6, 7).
The maximum value of a single digit is 7 (one less
than the value of the base
•
Each position of a digit represents a specific power of•
the base (8)
(Continued on next slide)
15. Octal Number System
(Continued from previous slide..)
• Since there are only 8 digits, 3 bits (23 = 8) are
sufficient to represent any octal number in binary
Example
20578
=
= (2 x 83) + (0 x 82) + (5 x 81) + (7 x 80)
1024 + 0 + 40 + 7
= 107110
16. Hexadecimal Number System
Characteristics
A positional number system
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
The symbols A, B, C, D, E and F represent the
decimal values 10, 11, 12, 13, 14 and 15
respectively
•
The maximum
than the value
value of a single
of the base)
digit is 15 (one less•
(Continued on next slide)
17. Hexadecimal Number System
(Continued from previous slide..)
• Each position of a digit represents a specific power
of the base (16)
• Since there are only 16 digits, 4 bits (24 = 16) are
sufficient to represent any hexadecimal number
binary
in
Example
1AF16 =
=
=
=
(1 x 162) + (A x 161) + (F x 160)
1 x 256 + 10 x 16 + 15 x 1
256 + 160 + 15
43110
18. Converting a Number of Another Base to a
Decimal Number
Method
Step 1: Determine the column (positional) value of
each digit
Step 2: Multiply the obtained column values by the
digits in the corresponding columns
Step 3: Calculate the sum of these products
(Continued on next slide)
19. Converting a Number
Decimal Number
(Continued from previous slide..)
of Another Base to a
Example
47068 = ?10
Common
values
multiplied
by the
corresponding
digits
47068 = 4 x 83 + 7 x 82 + 0 x 81 + 6 x 80
=
=
=
4 x 512 + 7
2048 + 448
250210
x 64 + 0 + 6 x 1
+ 0 + 6 Sum of these
products
20. Converting a Decimal Number to a Number of
Another Base
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
therightmost
new base
digit (least significant digit) of
number
Step 3: Divide the quotient of the previous divide by the
new base
(Continued on next slide)
21. Converting a Decimal Number to a Number of
Another Base
(Continued from previous slide..)
Step 4: Record the remainder from Step 3 as the next
digit (to the left) of the new base number
Repeat Steps 3 and 4, recording remainders from right to
left, until the quotient becomes zero in Step 3
Note that the last remainder thus
new
obtained will be the most
significant digit (MSD) of the base number
(Continued on next slide)
22. Converting a Decimal
Another Base
(Continued from previous slide..)
Number to a Number of
Example
95210 = ?8
Solution:
9528 Remainder
s 0
7
6
1
Hence, 95210 = 16708
119
14
1
0
23. Converting a Number of Some Base to a Number
of Another Base
Method
Step 1: Convert the original number to a decimal
number (base 10)
Step 2: Convert the decimal number so obtained to
the new base number
(Continued on next slide)
24. Converting a Number
of Another Base
(Continued from previous slide..)
of Some Base to a Number
Example
5456 = ?4
Solution:
Step 1: Convert from base 6 to base 10
5456 = 5 x 62 + 4 x 61 + 5 x 60
= 5 x 36 + 4 x 6 + 5 x 1
= 180 + 24 + 5
= 20910
(Continued on next slide)
25. Converting a Number of Some
of Another Base
(Continued from previous slide..)
Base to a Number
Step 2: Convert 20910 to base 4
4 Remainders
1
0
1
3
Hence, 20910 = 31014
So, 5456 = 20910 = 31014
Thus, 5456 = 31014
209
52
13
3
0
26. Shortcut Method for Converting a Binary Number
to its Equivalent Octal Number
Method
Step 1: Divide the digits into groups of three starting
from the right
Step 2: Convert each group of three binary digits to
toone octal digit using the method of binary
decimal conversion
(Continued on next slide)
27. Shortcut Method for Converting a Binary
to its Equivalent Octal Number
(Continued from previous slide..)
Number
Example
11010102 = ?8
Step 1: Divide the binary digits into
from right
groups of 3 starting
001 101 010
Step 2: Convert each group into one octal digit
0012
1012
0102
=
=
=
0
1
0
x
x
x
22
22
22
+
+
+
0
0
1
x
x
x
21
21
21
+
+
+
1
1
0
x
x
x
20
20
20
=
=
=
1
5
2
Hence, 11010102 = 1528
28. Shortcut Method for Converting an Octal
Number to Its Equivalent Binary Number
Method
Step 1: Convert
number
decimal
each octal digit to a 3 digit binary
(the octal digits may be treated as
for this conversion)
Step 2: Combine all the resulting binary
single
groups
binary(of 3 digits each) into a
number
(Continued on next slide)
29. Shortcut Method for Converting an Octal
Number to Its Equivalent Binary Number
(Continued from previous slide..)
Example
5628
Step
= ?2
1: Convert each octal digit to 3 binary digits
58 = 1012, 68 = 1102, 28 = 0102
Step 2: Combine the binary groups
5628 = 101
5
110
6
010
2
Hence, 5628 = 1011100102
30. Shortcut Method for Converting a Binary
Number to its Equivalent Hexadecimal Number
Method
Step 1: Divide the binary digits into groups of four
starting from the right
Step 2: Combine each group of four binary digits to
one hexadecimal digit
(Continued on next slide)
31. Shortcut Method for Converting a Binary
Number to its Equivalent Hexadecimal Number
(Continued from previous slide..)
Example
1111012 = ?16
Step 1: Divide the binary digits into groups of four
starting from the right
0011 1101
Step 2: Convert each group into a hexadecimal digit
00112 = 0 x
11012 = 1 x
23
23
+ 0 x
+ 1 x
22
22
+ 1 x
+ 0 x
21
21
+
+
1
1
x
x
20
20
=
=
310
1310
=
=
316
D16
Hence, 1111012 = 3D16
32. Shortcut Method for Converting a Hexadecimal
Number to its Equivalent Binary Number
Method
Step 1: Convert the decimal equivalent of each
hexadecimal digit to a 4 digit binary
number
Step 2: Combine all the resulting binary groups
(of 4 digits each) in a single binary number
(Continued on next slide)
33. Shortcut Method for Converting a Hexadecimal
Number to its Equivalent Binary Number
(Continued from previous slide..)
Example
2AB16 = ?2
Convert each hexadecimal
binary number
Step 1: digit to a 4 digit
216
A16
B16
=
=
=
210
1010
1110
=
=
=
00102
10102
10112
34. Shortcut Method for Converting a Hexadecimal
Number to
(Continued from previous slide..)
its Equivalent Binary Number
Step 2: Combine the binary
1010
A
groups
1011
B
2AB16 = 0010
2
Hence, 2AB16 = 0010101010112
35. Key Words/Phrases
Base
Binary
Binary
Bit
•
•
•
•
•
•
•
•
Least Significant Digit (LSD)
Memory dump
Most Significant Digit (MSD)
Non-positional number
system
Number system
Octal number system
•
•
•
•
number
point
system
Decimal number system
Division-Remainder
Fractional numbers
technique •
•
•Hexadecimal number system Positional number system