This document discusses number systems, including the decimal, binary, and octal systems. It begins by introducing positional and non-positional number systems. The decimal system uses base 10 with digits 0-9, where the place value of each digit depends on its position. The binary system uses base 2 with digits 0-1. Conversions between decimal, binary, and octal systems are demonstrated through examples such as decimal to binary conversion by repeated division. Fractions are also converted between number systems. Finally, the octal system is introduced, which uses base 8 with digits 0-7.
Security Architecture and Design - CISSPSrishti Ahuja
Security Architecture and Design using CISSP guidelines, hardware and software security, kernel, virtualization, security models, ring model, security domains, BellLaPadula model, Biba model, Reading up and Writing down, Reading down and Writing up
Transactions and Concurrency Control in distributed systems. Transaction properties, classification, and transaction implementation. Flat, Nested, and Distributed transactions. Inconsistent Retrievals, Lost Update, Dirty Read, and Premature Writes Problem
Adbms 3 main characteristics of the database approachVaibhav Khanna
Self-describing nature of a database system:
A DBMS catalog stores the description of a particular database (e.g. data structures, types, and constraints)
The description is called meta-data.
This allows the DBMS software to work with different database applications.
Insulation between programs and data:
Called program-data independence.
Allows changing data structures and storage organization without having to change the DBMS access programs.
To Support Digital India, We are trying to enforce the security on the web and digital Information. This Slides provide you basic as well as advance knowledge of security model. Model covered in this slides are Chinese Wall, Clark-Wilson, Biba, Harrison-Ruzzo-Ullman Model, Bell-LaPadula Model etc.
Types of Access Control.
Security Architecture and Design - CISSPSrishti Ahuja
Security Architecture and Design using CISSP guidelines, hardware and software security, kernel, virtualization, security models, ring model, security domains, BellLaPadula model, Biba model, Reading up and Writing down, Reading down and Writing up
Transactions and Concurrency Control in distributed systems. Transaction properties, classification, and transaction implementation. Flat, Nested, and Distributed transactions. Inconsistent Retrievals, Lost Update, Dirty Read, and Premature Writes Problem
Adbms 3 main characteristics of the database approachVaibhav Khanna
Self-describing nature of a database system:
A DBMS catalog stores the description of a particular database (e.g. data structures, types, and constraints)
The description is called meta-data.
This allows the DBMS software to work with different database applications.
Insulation between programs and data:
Called program-data independence.
Allows changing data structures and storage organization without having to change the DBMS access programs.
To Support Digital India, We are trying to enforce the security on the web and digital Information. This Slides provide you basic as well as advance knowledge of security model. Model covered in this slides are Chinese Wall, Clark-Wilson, Biba, Harrison-Ruzzo-Ullman Model, Bell-LaPadula Model etc.
Types of Access Control.
1 Binary logic functions
2 Number System Conversion
Conversion for the fractional part
3 Other base number system to Decimal
Binary to Decimal Conversion
Convert Binary fraction to Decimal
4 Laws of Boolean Algebra
5 De Morgan's Theorem
6 Logic Gates
7 Examples
Number systems - Efficiency of number system, Decimal, Binary, Octal, Hexadecimalconversion
from one to another- Binary addition, subtraction, multiplication and division,
representation of signed numbers, addition and subtraction using 2’s complement and I’s
complement.
Binary codes - BCD code, Excess 3 code, Gray code, Alphanumeric code, Error detection
codes, Error correcting code.Deepak john,SJCET-Pala
Digital computers represent data by means of an easily identified symbol called a digit. The data may
contain digits, alphabets or special character, which are converted to bits, understandable by the computer.
In Digital Computer, data and instructions are stored in computer memory using binary code (or
machine code) represented by Binary digIT’s 1 and 0 called BIT’s.
The number system uses well-defined symbols called digits.
Number systems are classified into two types:
o Non-positional number system
o Positional number system
References:
"Digital Systems Principles And Application"
Sixth Edition, Ronald J. Tocci.
"Digital Systems Fundamentals"
P.W Chandana Prasad, Lau Siong Hoe,
Dr. Ashutosh Kumar Singh, Muhammad Suryanata.
FYBSC IT Digital Electronics Unit I Chapter I Number System and Binary Arithm...Arti Parab Academics
Number System:
Analog System, digital system, numbering system, binary number
system, octal number system, hexadecimal number system, conversion
from one number system to another, floating point numbers, weighted
codes binary coded decimal, non-weighted codes Excess – 3 code, Gray
code, Alphanumeric codes – ASCII Code, EBCDIC, ISCII Code,
Hollerith Code, Morse Code, Teletypewriter (TTY), Error detection
and correction, Universal Product Code, Code conversion.
Numbering system, binary number system, octal number system, decimal number system, hexadecimal number system.
Code conversion, Conversion from one number system to another, floating point numbers
There are 10 kinds of people in the world—those who understand.docxchristalgrieg
There are 10 kinds of people in the world—those who understand binary and those who don’t.
—Anonymous
CHAPTER 2
Data Representation in Computer Systems
2.1 INTRODUCTION
The organization of any computer depends considerably on how it represents numbers, characters, and control
information. The converse is also true: Standards and conventions established over the years have determined certain
aspects of computer organization. This chapter describes the various ways in which computers can store and
manipulate numbers and characters. The ideas presented in the following sections form the basis for understanding
the organization and function of all types of digital systems.
The most basic unit of information in a digital computer is called a bit, which is a contraction of binary digit. In
the concrete sense, a bit is nothing more than a state of “on” or “off” (or “high” and “low”) within a computer circuit.
In 1964, the designers of the IBM System/360 mainframe computer established a convention of using groups of 8
bits as the basic unit of addressable computer storage. They called this collection of 8 bits a byte.
Computer words consist of two or more adjacent bytes that are sometimes addressed and almost always are
manipulated collectively. The word size represents the data size that is handled most efficiently by a particular
architecture. Words can be 16 bits, 32 bits, 64 bits, or any other size that makes sense in the context of a computer’s
organization (including sizes that are not multiples of eight). An 8-bit byte can be divided into two 4-bit halves called
nibbles (or nybbles). Because each bit of a byte has a value within a positional numbering system, the nibble
containing the least-valued binary digit is called the low-order nibble, and the other half the high-order nibble.
2.2 POSITIONAL NUMBERING SYSTEMS
At some point during the middle of the sixteenth century, Europe embraced the decimal (or base 10) numbering
system that the Arabs and Hindus had been using for nearly a millennium. Today, we take for granted that the
number 243 means two hundreds, plus four tens, plus three units. Notwithstanding the fact that zero means
“nothing,” virtually everyone knows that there is a substantial difference between having 1 of something and having
10 of something.
The general idea behind positional numbering systems is that a numeric value is represented through increasing
powers of a radix (or base). This is often referred to as a weighted numbering system because each position is
weighted by a power of the radix.
The set of valid numerals for a positional numbering system is equal in size to the radix of that system. For
example, there are 10 digits in the decimal system, 0 through 9, and 3 digits for the ternary (base 3) system, 0, 1, and
2. The largest valid number in a radix system is one smaller than the radix, so 8 is not a valid numeral in any radix
system smaller than 9. To distinguish among numbers in d ...
Digital computer deals with numbers; it is essential to know what kind of numbers can be handled most easily when using these machines. We accustomed to work primarily with the decimal number system for numerical calculations, but there is some number of systems that are far better suited to the capabilities of digital computers. And there is a number system used to represents numerical data when using the computer.
Introduction
Phases of CPM and PERT
Some Important Definitions
Project management or representation by a network diagram
Types of activities
Types of events
Common Errors
Rules of network construction
Numbering the events
Time analysis
Determination of Floats and Slack times
Critical activity and Critical path
2 Critical Path Method - CPM
3 Program Evaluation and Review Technique - PERT
Introduction to LPP
Components of Linear Programming Problem
Basic Assumption in LPP
Examples of LPP
2 Formulation of LPP
Steps for Mathematical Formulation of LPP’s
Examples on Formulation of LPP
3 Basic Definitions
4 Graphical Method for solving LPP
5 Examples on Graphical method for solving LPP
1 Introduction
2 Types of events
3 Classical definition of probability
4 Examples on probability
5 Conditional probability
6 Bayes theorem
7 Random variables and Probability distributions
Sampling theory is a study of relationships existing between a population and samples drawn from the population. Sampling theory is applicable only to random samples. For this purpose the population or a universe may be defined as an aggregate of items possessing a common trait or traits. In other words, a universe is the complete group of items about which knowledge is sought. The universe may be finite or infinite. Infinite universe is one which has a definite and certain number of items, but when the number of items is uncertain and infinite, the universe is said to be an infinite universe. Similarly, the universe may be hypothetical or existent. In the former case the universe in fact does not exist and we can only imagin the items constituting it. Tossing of a coin or throwing a dice are examples of hypothetical universe. Existent universe is a universe of concrete objects i.e., the universe where the items constituting it really exist. On the other hand, the term sample refers to that part of the universe which is selected for the purpose of investigation. The theory of sampling studies the relationships that exist between the universe and the sample or samples drawn from it.
The main problem of sampling theory is the problem of relationship between a parameter and a statistic. The theory of sampling is concerned with estimating the properties of the population from those of the sample and also with gauging the precision of the estimate. This sort of movement from particular (sample) towards general (universe) is what is known as statistical induction or statistical inference. In more clear terms “from the sample we attempt to draw inference concerning the universe. In order to be able to follow this inductive method, we first follow a deductive argument which is that we imagine a population or universe (finite or infinite) and investigate the behavior of the samples drawn from this universe applying the laws of probability.” The methodology dealing with all this is known as sampling theory.
Control is a system for measuring and checking or inspecting a phenomenon. It suggests when to inspect, how often to inspect and how much to inspect. Control ascertains quality characteristics of an item, compares the same with prescribed quality characteristics of an item, compares the same with prescribed quality standards and separates defective item from non-defective ones.
Statistical Quality Control (SQC) is the term used to describe the set of statistical tools used by quality professionals.
SQC is used to analyze the quality problems and solve them. Statistical quality control refers to the use of statistical methods in the monitoring and maintaining of the quality of products and services.
Variation in manufactured products is inevitable; it is a fact of nature and industrial life. Even when a production process is well designed or carefully maintained, no two products are identical.
The difference between any two products could be very large, moderate, very small or even undetectable depending on the sources of variation.
For example, the weight of a particular model of automobile varies from unit to unit, the weight of packets of milk may differ very slightly from each other, and the length of refills of ball pens, the diameter of cricket balls may also be different and so on.
The existence of variation in products affects quality. So the aim of SQC is to trace the sources of such variation and try to eliminate them as far as possible.
The Statistical Inference is the process of drawing conclusions about on underlying population based on a sample or subset of the data.
In most cases, it is not practical to obtain all the measurements in a given population.
The statistical inference is deals with decision problems. There are two types of decision problems as mentioned below:
(i) Problems of estimation and
(ii) Test of hypotheses
In the problem of estimation, we must determine the value of parameter(s), while in test of hypothesis we must decide whether to accept or reject a specific value(s) of a parameter(s).
Decision theory as the name would imply is concerned with the process of making decisions. The extension to statistical decision theory includes decision making in the presence of statistical knowledge which provides some information where there is uncertainty. The elements of decision theory are quite logical and even perhaps intuitive. The classical approach to decision theory facilitates the use of sample information in making inferences about the unknown quantities. Other relevant information includes that of the possible consequences which is quantified by loss and the prior information which arises from statistical investigation. The use of Bayesian analysis in statistical decision theory is natural. Their unification provides a foundational framework for building and solving decision problems. The basic ideas of decision theory and of decision theoretic methods lend themselves to a variety of applications and computational and analytic advances.
The purpose of the book is to present the current techniques of operations research in such a way that they can be readily comprehended by the average business student taking an introductory course in operations research. Several OR teachers and teachers from management schools suggested that we should bring out a separate volume on OR with a view to meet the requirements of OR courses, which can also be used by the practising managers. The book can be used for one semester/term introductory course in operations research. Instructors may like to decide the appropriate sequencing of major topics covered.
This book will be useful to the students of management, OR, industrial and production engineering, computer sciences, chartered and cost-accountancy, economics and commerce. The approach taken here is to illustrate the practical use of OR techniques and therefore, at places complicated mathematical proofs have been avoided. To enhance the understanding of the application of OR techniques, illustrations have been drawn from real life situations. The problems given at the end of each chapter have been designed to strengthen the student's understanding of the subject matter. Our long teaching experience indicates that an individual's comprehension of the various quantitative methods is improved immeasurably by working through and understanding the solutions to the problems.
It is not possible for us to thank individually all those who have contributed to the case histories. Our colleagues and many people have contributed to these studies and we gratefully acknowledge their help. Without their support and cooperation this book could not have been brought out. Our special thanks are due to Dr. K. H. Atkotiya who have assisted me in editing the case studies. we wish to express my sincere thanks to Mr. Chandraprakash Shah making available all facilities needed for this job. We express my gratitude to my parents who have been a constant source of Inspiration.
We Strongly believe that the road to improvement is never-ending. Suggestions and criticism of the books will be very much appreciated and most gratefully acknowledged.
THIS PRESENTATION COVERED FOLLOWING TOPICS IN MATRIX ALGEBRA
1. Introduction
2. Elementary Matrix Operations
3. Gauss elimination and Gauss- Jordan elimination methods
4. Rank of a matrix
5. Inverse of a matrix
6. Solution of Linear Simultaneous Equations
7. Orthogonal, Symmetric, Skew-symmetric, Harmitian, Skew-
Harmitian, Normal and Unitary matrices and their elementary
Properties.
8. Eigen Values and Eigen Vectors of a matrix
9. Cayley-Hamilton theorem (Without proof) and regarded
Examples
This presentation is covered the following 5 - measure topics of statistics :
1. Introduction to statistics
2. Measure of central tendency
3. Measure of Dispersion
4. Correlation and Regression
5. Random Variable and Probability distributions
and is useful for all students who studied in any branch of mathematics as well as statistics.
This presentation covered the following topics :
1. Random experiments
2. Sample space
3. Events and their probability
4. random variable probability distribution
5. t - Test
6. paired t - Test
7. F- Test
8. Comparison of results of above tests
and is useful for B.Sc , M.Sc mathematics and statistics students.
This presentation covered the following topics:
1. Definition of Correlation and Regression
2. Meaning of Correlation and Regression
3. Types of Correlation and Regression
4. Karl Pearson's methods of correlation
5. Bivariate Grouped data method
6. Spearman's Rank correlation Method
7. Scattered diagram method
8. Interpretation of correlation coefficient
9. Lines of Regression
10. regression Equations
11. Difference between correlation and regression
12. Related examples
This presentation covered the following topics :
1. Variance
2. Standard Deviation
3. Meaning and Types of Skewness
4. Related Examples
and is useful for B.Sc & M.Sc students.
This presentation covered following topics :
1. Introduction
2. Arithmetic Progression (AP)
3. Sum of Series in AP
4. Arithmetic and Geometric Mean
5. Geometric Progression (GP)
6. Sum of Series in GP
7. Relation Between AM, GM and HM
and is useful for B.Com and BBA students.
The presentation is covered the following topics :
1.Introduction
2.Finite Differences
(a) Forward Differences
(b) Backward Differences
(c) Central Differences
3.Interpolation for equal intervals
(a) Newton Forward and Backward Interpolation Formula
(b) Gauss Forward and Backward Interpolation Formula
(c)Stirling’s Interpolation Formula
4.Interpolation for unequal intervals
(a) Lagrange’s Interpolation Formula
5.Inverse interpolation
6.Relation between the operators
7.Newton Divided Difference Interpolation Formula
and is useful for Engineering and B.Sc students.
The presentation on Numerical Methods covered the following topics :
1. Introduction
2. Bisection Method with proof
3. False Position method with proof
4. Successive Approximation method
5. Newton Raphson (N-R)Method
6. Iterative Formulae for finding qth root, square
root and reciprocal of positive number N, Using N-R method
7. Secant Method
8. Power Method
and this is useful for engineering and B,Sc students.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
1. Unit - III: Number System
Dr. Tushar Bhatt
Assistant Professor in Mathematics
Department of Science and Technology
Faculty of Engineering and Technology
Atmiya University
Rajkot - 360005
Dr. Tushar Bhatt Unit - III: Number System 1 / 52
2. Table of Content
1 Introduction
2 Decimal Number System (Base 10)
3 Binary Number System (Base 2)
4 Conversation - I: Decimal to Binary
5 Conversation - II: Decimal fraction to Binary
6 Conversation - III: Binary to Decimal
7 Conversation - IV: Binary fraction to Decimal
8 Octal Number System (Base 8)
9 Conversation - V: Octal to Decimal
10 Conversation - VI: Octal fraction to Decimal
11 Conversation - VII: Decimal to Octal
12 Conversation - VIII: Decimal fraction to Octal
13 Conversation - IX: Octal to Binary
14 Conversation - X: Octal fraction to Binary
Dr. Tushar Bhatt Unit - III: Number System 2 / 52
3. 1. Introduction
A digital computer manipulates discrete elements of data and that these
elements are represented in the binary forms. Operands used for calculations
can be expressed in the binary number system.
Other discrete elements including the decimal digits, are represented in bi-
nary codes. Data processing is carried out by means of binary logic elements
using binary signals. Quantities are stored in binary storage elements.
The purpose of this unit is to introduce the various binary concepts as a
frame of reference for further study in the succeeding units.
Mainly there are two types of number systems as given below:
1 Non-positional Number system.
2 Positional number system.
Dr. Tushar Bhatt Unit - III: Number System 3 / 52
4. 1.1 Non-positional Number system
In this number system
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.
It is difficult to perform arithmetic with such a number system.
Dr. Tushar Bhatt Unit - III: Number System 4 / 52
5. 1.2 Positional number system
In this number system
Use only a few symbols called digits.
These symbols represent different values depending on the position
they occupy in the number.
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.
Radix or base = total number of digits in the number system.
The maximum value of a single digit is always equal to one less than
the value of the base.
Dr. Tushar Bhatt Unit - III: Number System 5 / 52
6. 1.3 Categories of positional number systems
There are four categories of positional number system defined as follows:
1 Decimal Number System.
2 Binary Number System.
3 Octal Number System.
4 Hexadecimal Binary Number System.
Dr. Tushar Bhatt Unit - III: Number System 6 / 52
7. 2. Decimal Number System (Base 10)
Decimal number system have ten digits represented by
0,1,2,3,4,5,6,7,8 and 9. So, the base or radix of such system is 10.
In this system the successive position to the left of the decimal point
represent units, tens, hundreds, thousands etc.
For example, if we consider a decimal number 257, then the digit
representations are,
Position Position Position
Hundred Tens Ones
2 5 7
The weight of each digit of a number depends on its relative position within
the number.
Dr. Tushar Bhatt Unit - III: Number System 7 / 52
8. 2.1 Decimal Number System (Base 10) - Example
Ex - 1 : The weight of each digit of the decimal number 6472
6472 = 6000 + 400 + 70 + 2
= (6 × 103
) + (4 × 102
) + (7 × 101
) + (2 × 100
)
∴ The weight of digits from right hand side are
Weight of 1st digit = 2 × 100.
Weight of 2nd digit = 7 × 101.
Weight of 3rd digit = 4 × 102.
Weight of 4th digit = 6 × 103.
The above expressions can be written in general forms as the weight of nth
digit of the number from the right hand side
= nth
digit × 10n−1
Dr. Tushar Bhatt Unit - III: Number System 8 / 52
9. 3. Binary Number System (Base 2)
Only two digits 0 and 1 are used to represent a binary number system. So
the base or radix of binary system is two (2). The digits 0 and 1 are called
bits (Binary Digits). In this number system the value of the digit will be
two times greater than its predecessor. Thus the value of the places are
< ... < −32 < −16 < −8 < −4 < −2 < −1 < ...
The weight of each binary bit depends on its relative position within the
number. It is explained by the following example:
The weight of bits of the binary number 10110 is:
10110 = (1 × 24
) + (0 × 23
) + (1 × 22
) + (1 × 21
) + (0 × 20
)
= 16 + 0 + 4 + 2 + 0
= 22(decimal number)
Dr. Tushar Bhatt Unit - III: Number System 9 / 52
10. 3. Binary Number System (Base 2)
The weight of each bit of a binary no. depends on its relative pointer
within the number and explained from right hand side
∴ The weight of digits from right hand side are
Weight of 1st bit = 0 × 20.
Weight of 2nd bit = 1 × 21.
Weight of 3rd bit = 1 × 22.
Weight of 4th bit = 0 × 23.
Weight of 5th bit = 1 × 24.
The above expressions can be written in general forms as the weight of nth
digit of the number from the right hand side
= nth
bit × 2n−1
Dr. Tushar Bhatt Unit - III: Number System 10 / 52
11. 3.1 How to convert Decimal number to Binary number?
Divide the given decimal number by 2 and note down the remainder.
Now, divide the obtained quotient by 2, and note the remainder again.
Repeat the above steps until you get 0 as the quotient.
Now, write the remainders in such a way that the last remainder is
written first, followed by the rest in the reverse order.
This can also be understood in another way which states that the
Least Significant Bit (LSB) of the binary number is at the top and
the Most Significant Bit (MSB) is at the bottom. This number is the
binary value of the given decimal number.
Dr. Tushar Bhatt Unit - III: Number System 11 / 52
12. 3.2 Decimal to Binary Conversation of numbers 0 to 9
Decimal Number Binary Number
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
Dr. Tushar Bhatt Unit - III: Number System 12 / 52
13. 3.3 Decimal to Binary Conversation - Example
Dr. Tushar Bhatt Unit - III: Number System 13 / 52
15. 3.3 Decimal to Binary Conversation - Examples
Ex - 1: Convert (75)10 to its binary equivalent.
Solution:
Remainder
2 75
2 37 1
2 18 1
2 9 0
2 4 1
2 2 0
1 0
1
Now arrange remainders upward ↑. That is binary representation of given
number 75.
i.e.
(75)10 = (1001011)2
Dr. Tushar Bhatt Unit - III: Number System 15 / 52
16. 3.4 Decimal to Binary Conversation - Examples
Ex - 2: Convert (174)10 to binary.
Solution:
Division by 2 Quotient Remainder
174 ÷ 2 87 0 (LSB)
87 ÷ 2 43 1
43 ÷ 2 21 1
21 ÷ 2 10 1
10 ÷ 2 5 0
5 ÷ 2 2 1
2 ÷ 2 1 0
1 ÷ 2 0 1 (MSB)
After noting the remainders, we write them in the reverse order such that
the Most Significant Bit (MSB) is written first, and the Least Significant
Bit is written in the end. Hence, the binary equivalent for the given
decimal number (174)10 is (10101110)2.
Dr. Tushar Bhatt Unit - III: Number System 16 / 52
18. 3.5 Convert Decimal fraction to Binary number - Steps
For this conversations, we will consider following three steps:
1 Convert the integral part of decimal to binary equivalent.
Divide the decimal number by 2 and store remainders in array.
Divide the quotient by 2.
Repeat step 2 until we get the quotient equal to zero.
Equivalent binary number would be reverse of all remainders of step 1.
2 Convert the fractional part of decimal to binary equivalent
Multiply the fractional decimal number by 2
Integral part of resultant decimal number will be first digit of fraction
binary number.
Repeat step 1 using only fractional part of decimal number and then
step 2.
If any fraction will have more and more fractions after multiply by 2
then fix the step size according to given instruction.
In the fraction part must arrange the integer part in the downward
direction ↓.
3 Combine both integral and fractional part of binary number.
Dr. Tushar Bhatt Unit - III: Number System 18 / 52
19. 3.6 Convert Decimal fraction to Binary number - Examples
Ex - 1: Convert (17.75)10 to binary.
Solution:
Step - 1: Convert the integral part of decimal to binary equivalent.
Division by 2 Quotient Remainder
17 ÷ 2 8 1 (LSB)
8 ÷ 2 4 0
4 ÷ 2 2 0
2 ÷ 2 1 0
1 ÷ 2 0 1 (MSB)
Step - 2: Convert the fractional part of decimal to binary equivalent.
Fractional part × 2 Result Integer part of result
0.75 × 2 1.50 1 (LSB)
0.50 × 2 1.00 1(MSB)
Step - 3: Combine both integral and fractional part of binary
number i.e. (10001.11)2
Dr. Tushar Bhatt Unit - III: Number System 19 / 52
20. 3.6 Convert Decimal fraction to Binary number - Examples
Ex - 2: Convert (89.25)10 to binary.
Solution: Step - 1: Convert the integral part
Division by 2 Quotient Remainder
89 ÷ 2 44 1 (LSB)
44 ÷ 2 22 0
22 ÷ 2 11 0
11 ÷ 2 5 1
5 ÷ 2 2 1
2 ÷ 2 1 0
1 ÷ 2 0 1 (MSB)
Step - 2: Convert the fractional part of decimal to binary equivalent.
Fractional part × 2 Result Integer part of result
0.25 × 2 0.50 0 (LSB)
0.50 × 2 1.00 1(MSB)
Step - 3: Combine both i.e. (1011001.01)2 .
Dr. Tushar Bhatt Unit - III: Number System 20 / 52
22. 3.7 How to convert Binary number to Decimal number?
We know that the binary to decimal number system conversion is the
process of changing from a binary (base 2) to a decimal number
system (base 10 number system).
To convert a binary number system to a decimal number system,
follow the procedures below.
Step - 1: Multiply each digit of the specified binary number by the
exponents of the base starting with the right most digit (i.e.,
20, 21, 22, and so on).
Step - 2: As we move right to left, the exponents should increase by
one, such that the exponents begin with 0.
Step - 3: Simplify and find the sum of each of the product values
obtained in the previous steps.
Dr. Tushar Bhatt Unit - III: Number System 22 / 52
23. 3.8 Binary to Decimal conversation examples
Ex - 1: Convert (11110)2 into a decimal number system.
Solution:
11110 = (1 × 24
) + (1 × 23
) + (1 × 22
) + (1 × 21
) + (0 × 20
)
= (16) + (8) + (4) + (2) + (1)
= 30
∴ (11110)2 = (30)10.
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24. 3.8 Binary to Decimal conversation examples
Ex - 2: Convert (0110)2 into a decimal number system.
Solution:
0110 = (0 × 23
) + (1 × 22
) + (1 × 21
) + (0 × 20
)
= (0) + (4) + (2) + (0)
= 6
∴ (0110)2 = (6)10.
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25. 3.8 Binary to Decimal conversation examples
Ex - 3: Convert (1110110)2 into a decimal number system.
Solution:
1110110 = (1 × 26
) + (1 × 25
) + (1 × 24
) + (0 × 23
)
+ (1 × 22
) + (1 × 21
) + (0 × 20
)
= (64) + (32) + (16) + (0) + (4) + (2) + (0)
= 118
∴ (1110110)2 = (118)10.
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26. 3.8 Binary to Decimal conversation examples
Ex - 4: The binary number (1111)2 is equal to (x)10.Then find the
value of x.
Solution:
1111 = (1 × 23
) + (1 × 22
) + (1 × 21
) + (1 × 20
)
= (8) + (4) + (2) + (1)
= 15
∴ (1111)2 = (15)10.
∴ x = 15.
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28. 3.9 Fraction Binary number to Decimal number
conversation
Ex - 5: Convert (1110.1010)2 into a decimal number system.
Solution:
Here first we have to separate the integral and fraction part
Integral Part: 1110
1110 = (1 × 23) + (1 × 22) + (1 × 21) + (0 × 20) = 14
Fractional Part: .1010
.1010 = (1 × 2−1) + (0 × 2−2) + (1 × 2−3) + (0 × 2−4) = 1
2 + 0 + 1
8 + 0 =
0.5 + 0.125 = 0.625
Now
(1110.1010)2 = Integral part + Fractional part = 14 + 0.625 = 14.625
.
∴
(1110.1010)2 = (14.625)10
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29. 4. Octal Number System (Base 8)
A commonly used positional number system is the Octal Number
System. This system has eight (8) digit representations as
0,1,2,3,4,5,6 and 7. The base or radix of this system is 8.
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).
Since there are only 8 digits, 3 bits (23 = 8) are sufficient to represent
any octal number in binary.
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31. 4.1 Steps to Convert Octal to Decimal
Step - 1: Since an octal number only uses digits from 0 to 7, we first
arrange the octal number with the power of 8.
Step - 2: We evaluate all the power of 8 values such as 80 is 1, 81 is
8, etc., and write down the value of each octal number.
Step - 3: Once the value is obtained, we multiply each number.
Step - 4: Final step is to add the product of all the numbers to obtain
the decimal number.
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32. 4.2 Examples on Octal to Decimal Conversation
Ex - 1: Convert (140)8 to decimal number.
Solution:
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33. 4.2 Examples on Octal to Decimal Conversation
Ex - 2: Convert (2057)8 to decimal number.
Solution:
(2057)8 = (2 × 83
) + (0 × 82
) + (5 × 81
) + (7 × 80
)
= 1024 + 0 + 40 + 7
= 1071
∴ (2057)8 = (1071)10
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34. Conversation - VI
Octal fraction ⇒ Decimal
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35. 4.2 Examples on Octal to Decimal Conversation
Ex - 3: Convert (246.28)8 to decimal number.
Solution:
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37. 4.3 Decimal to Octal conversation
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38. 4.3 Examples on Decimal to Octal Conversation
Ex - 1: Convert (1792)10 into an octal number.
Solution:
Decimal Number Operation Quotient Remainder
1792 ÷8 224 0
224 ÷8 28 0
28 ÷8 3 4
3 ÷8 0 3
Now write off the remainder from bottom to top, we have
(1792)10 = (3400)8
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39. 4.3 Examples on Decimal to Octal Conversation
Ex - 2: Convert (127)10 into an octal number.
Solution:
Decimal Number Operation Quotient Remainder
127 ÷8 15 7
15 ÷8 1 7
1 ÷8 0 1
Now write off the remainder from bottom to top, we have
(127)10 = (177)8
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40. 4.3 Examples on Decimal to Octal Conversation
Ex - 3: Convert (100)10 into an octal number.
Solution:
Decimal Number Operation Quotient Remainder
100 ÷8 12 4
12 ÷8 1 4
1 ÷8 0 1
Now write off the remainder from bottom to top, we have
(100)10 = (144)8
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42. Steps to Convert Decimal fraction to Octal
1 First, we calculate the integer part of the decimal point by dividing
the octal base number i.e. 8 until the quotient is less than 8.
2 The second part is calculated on the fraction part of the decimal
number where the number is multiplied with the base number 8 until
the fractional part is equal to zero.
3 Here, once multiplied we keep the integer part separate and the
fractional part separate.
4 The final octal number is calculated by adding both the integer and
the fractional number.
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43. 4.3 Examples on Decimal fraction to Octal Conversation
Ex - 4: Convert decimal number 29.45 into octal form, follows
seven steps.
Solution:
Step 1: Separate the decimal number into two parts - the integer and the
fractional. So, 29.45 = 29 + 0.45.
Step 2: Convert the integer part of the number first. So, we begin with 29
first by dividing it by the base number 8 until the quotient is less than 8.
Decimal Number(IP) Operation Quotient Remainder
29 ÷8 3 5
3 ÷8 0 3
Hence, 29 is 35 in an octal number.
Step - 3: Once the integer octal number is obtained, we proceed to the
fractional part. So, 0.45 is multiplied by 8 (octal base number) where the
result is again divided into its integer part and fractional part. The number
is multiplied by 8 until the fractional part is equal to zero.
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44. 4.3 Examples on Decimal fraction to Octal conversation;
Ex - 4: Solu...
Decimal Number(FP) Operation Result Integer part Fractional part
0.45 ×8 3.6 3 0.6
0.6 ×8 4.8 4 0.8
0.8 ×8 6.4 6 0.4
0.4 ×8 3.2 3 0.2
0.2 ×8 1.6 1 0.6
0.6 ×8 4.8 4 0.8
0.8 ×8 6.4 6 0.4
Write all the integer part from top to bottom that derives the octal
number of the fractional number. Hence, 0.45 = 0.3463146.
Step 4: Add both the integer and the fractional part together to obtain
the octal number. Hence, 35 + 0.3463146 = 35.3463146.
∴ (29.45)10 = (35.3463146)8 .
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