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.
This document provides an introduction to the concepts and techniques of Vedic mathematics. It discusses how Vedic mathematics was developed based on concepts from the Atharva Veda and simplified mathematical operations. The core of Vedic mathematics is 16 sutras or formulae and 13 sub-sutras developed by ancient Hindu scientists to solve problems in 2-3 steps. Examples of applications include addition, subtraction, multiplication, division, algebra, and calculus. Mastering Vedic mathematics techniques can help solve problems faster and more efficiently. The document provides background on the development and concepts of Vedic mathematics to help teachers understand and apply its methods.
Vedic mathematics is a unique system of mental calculations based on 16 sutras or formulas derived from Vedic texts. It allows calculations like multiplication, division, square roots, and more to be done very quickly in the head. Some key sutras include Ekadhikena Purvena for squaring numbers ending in 5, Nikhilam Navatashcaramam Dashatah for multiplying numbers near multiples of 10, and Urdhva Tiryagbhyam for general multiplication using a vertical and diagonal approach. Vedic maths aims to reduce the usual effort and time of calculations through ingenious principles like proportionality, symmetry, and approximation.
The document discusses the Fibonacci sequence and its properties. It begins by explaining how the Fibonacci sequence is defined, with each subsequent number being the sum of the previous two numbers. It then provides examples of calculating Fibonacci numbers. The document also discusses how the Fibonacci sequence appears in nature, such as the spiral patterns of sunflowers and pinecones. Finally, it notes that the ratio of adjacent Fibonacci numbers approaches the golden ratio, an interesting mathematical property.
The document discusses direct proportion and provides examples. Direct proportion means that when one quantity changes, the other changes by the same factor or ratio. The examples given are: 1) Newton's second law of motion where acceleration and force are directly proportional, 2) the cost of cans of soup increasing by the same factor as the number of cans purchased, and 3) the number of points scored from field goals being directly proportional to the number of field goals made. Real world problems are provided to demonstrate setting up function tables and graphs to represent direct proportions.
Vedic mathematics is a system of mental calculation techniques discovered in ancient Hindu texts between 1911-1918 by Sri Bharti Krishna Tirath. It is based on 16 sutras or word formulas that allow complex mathematical problems to be solved very quickly in the mind. Some examples of the sutras include vertically-crosswise multiplication and the use of complementary numbers. Vedic math was developed as a more efficient system than modern mathematics and helps improve concentration and problem solving abilities.
The document discusses binary arithmetic operations including addition, subtraction, multiplication, and division. It provides examples and step-by-step explanations of how to perform each operation in binary. For addition and subtraction, it explains the rules and concepts like carry bits and two's complement. For multiplication, it describes the shift-and-add method. And for division, it outlines the long division approach of shift-and-subtract in binary.
This document explains how to multiply 102 x 98 using Vedic math sutra number 2 in 5 seconds. It involves taking the base (100) and the differences between it and the numbers (102 +2, 98 -2). You multiply the differences and cross add to the base, getting 10004. But since this is not the correct answer, you write it as 10,000 - 4 to get the actual answer of 9996. The key steps are taking the differences, multiplying them, and cross adding to the base.
Binary arithmetic is essential for digital computers and systems. It involves adding, subtracting, multiplying, and dividing binary numbers using basic rules. Signed binary numbers represent positive and negative values using sign-magnitude, 1's complement, and 2's complement methods. Arithmetic operations on signed binary numbers follow rules for handling the sign bit and complement representations.
This document provides an introduction to the concepts and techniques of Vedic mathematics. It discusses how Vedic mathematics was developed based on concepts from the Atharva Veda and simplified mathematical operations. The core of Vedic mathematics is 16 sutras or formulae and 13 sub-sutras developed by ancient Hindu scientists to solve problems in 2-3 steps. Examples of applications include addition, subtraction, multiplication, division, algebra, and calculus. Mastering Vedic mathematics techniques can help solve problems faster and more efficiently. The document provides background on the development and concepts of Vedic mathematics to help teachers understand and apply its methods.
Vedic mathematics is a unique system of mental calculations based on 16 sutras or formulas derived from Vedic texts. It allows calculations like multiplication, division, square roots, and more to be done very quickly in the head. Some key sutras include Ekadhikena Purvena for squaring numbers ending in 5, Nikhilam Navatashcaramam Dashatah for multiplying numbers near multiples of 10, and Urdhva Tiryagbhyam for general multiplication using a vertical and diagonal approach. Vedic maths aims to reduce the usual effort and time of calculations through ingenious principles like proportionality, symmetry, and approximation.
The document discusses the Fibonacci sequence and its properties. It begins by explaining how the Fibonacci sequence is defined, with each subsequent number being the sum of the previous two numbers. It then provides examples of calculating Fibonacci numbers. The document also discusses how the Fibonacci sequence appears in nature, such as the spiral patterns of sunflowers and pinecones. Finally, it notes that the ratio of adjacent Fibonacci numbers approaches the golden ratio, an interesting mathematical property.
The document discusses direct proportion and provides examples. Direct proportion means that when one quantity changes, the other changes by the same factor or ratio. The examples given are: 1) Newton's second law of motion where acceleration and force are directly proportional, 2) the cost of cans of soup increasing by the same factor as the number of cans purchased, and 3) the number of points scored from field goals being directly proportional to the number of field goals made. Real world problems are provided to demonstrate setting up function tables and graphs to represent direct proportions.
Vedic mathematics is a system of mental calculation techniques discovered in ancient Hindu texts between 1911-1918 by Sri Bharti Krishna Tirath. It is based on 16 sutras or word formulas that allow complex mathematical problems to be solved very quickly in the mind. Some examples of the sutras include vertically-crosswise multiplication and the use of complementary numbers. Vedic math was developed as a more efficient system than modern mathematics and helps improve concentration and problem solving abilities.
The document discusses binary arithmetic operations including addition, subtraction, multiplication, and division. It provides examples and step-by-step explanations of how to perform each operation in binary. For addition and subtraction, it explains the rules and concepts like carry bits and two's complement. For multiplication, it describes the shift-and-add method. And for division, it outlines the long division approach of shift-and-subtract in binary.
This document explains how to multiply 102 x 98 using Vedic math sutra number 2 in 5 seconds. It involves taking the base (100) and the differences between it and the numbers (102 +2, 98 -2). You multiply the differences and cross add to the base, getting 10004. But since this is not the correct answer, you write it as 10,000 - 4 to get the actual answer of 9996. The key steps are taking the differences, multiplying them, and cross adding to the base.
Binary arithmetic is essential for digital computers and systems. It involves adding, subtracting, multiplying, and dividing binary numbers using basic rules. Signed binary numbers represent positive and negative values using sign-magnitude, 1's complement, and 2's complement methods. Arithmetic operations on signed binary numbers follow rules for handling the sign bit and complement representations.
This document is an introduction to Gordon Rockmaker's book "101 Short Cuts in Math Anyone Can Do". It explains that short cuts are tricks that can save time and effort in calculations by cutting through unnecessary steps. The introduction also defines important mathematical terms like digits, integers, and place value positions for numbers. It aims to prepare the reader to understand and apply the short cuts presented in the book.
In this ppt , you will learn about the evolution of number systems, decimal, binary and hexadecimal and why hexadecima is the most important form of number systems when working with microcontroller programming.
The document discusses convergence of sequences and power series. It defines convergence of a sequence and states that the limit of a convergent sequence is unique. It also discusses Taylor series and Laurent series, stating that if a function f(z) is analytic inside a circle C with center z0, its Taylor series representation about z0 will converge to f(z) for all z inside C. Similarly, if f(z) is analytic in an annular region bounded by two concentric circles, its Laurent series will represent f(z) in that region.
Double integral using polar coordinatesHarishRagav10
Maths is aan acknowledge for all humanity's stast of technology and many resources now in all country maths is the father of technology is a wondering..maths comes into play a vital role in all form of chemistry physics and all other subjects...in our day to day life maths playa a role everything and everywhere
The document discusses square roots and how to estimate them to varying degrees of precision. It defines the principal root as the positive square root of a number. It provides examples of perfect squares and their roots. It explains that irrational numbers are those that cannot be expressed as a ratio of integers, like π. The document then gives steps to estimate square roots to the nearest tenths or hundredths place by considering the closest perfect squares before and after the given number.
This document introduces number systems and provides examples of converting between different number systems. It discusses decimal, binary, octal, and hexadecimal number systems. Conversion between these systems can be done directly by dividing and taking remainders or via shortcuts by grouping digits. Understanding number systems is important for IT professionals as computers use binary to represent all data internally.
Karnaugh maps are a graphical technique used to simplify Boolean logic equations. They represent truth tables in a two-dimensional layout where physically adjacent cells imply logical adjacency. This adjacency allows common terms to be factored out to minimize logic expressions. Karnaugh maps are most commonly used to manually minimize logic with up to four variables into sum-of-products or product-of-sums form.
Vedic mathematics is a system of mathematics consisting of 16 sutras or aphorisms obtained from ancient Hindu scriptures called the Vedas. It was presented in the early 20th century by Bharati Krishna Tirthaji Maharaja, an Indian scholar. The sutras provide concise formulae for solving problems through unique techniques like vertically-and-crosswise calculations without needing multiplication tables beyond 5x5. Some examples include techniques for squaring numbers and multiplying multi-digit numbers mentally through a carry-over method. Vedic mathematics was applied in areas like astronomy, astrology and constructing calendars.
The document discusses the binary number system, explaining that it uses only two digits, 0 and 1, with each column in a binary number being twice the value of the previous column. It covers converting between binary and decimal numbers, with examples, and explains that binary is important because computers represent information using electronic switches that can be either on or off, corresponding to 1 and 0.
1) Srinivasa Ramanujan was one of India's greatest mathematical geniuses who made substantial contributions to analytical number theory, elliptic functions, and infinite series.
2) He was mostly self-taught and showed extraordinary talent from a young age, mastering advanced mathematical concepts from books he received.
3) Ramanujan struggled for recognition in India but eventually his work was brought to the attention of the English mathematician G.H. Hardy, who helped arrange for Ramanujan to travel to Cambridge University in 1914 where he spent five productive years collaborating before falling ill and returning to India, where he passed away in 1920.
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
Translating english sentence to mathematical sentenceselle155401
This document provides verbal phrases and their mathematical translations. It includes an assignment to translate two additional phrases: 1) Five added to three times a number is less than 26. 2) Thrice the difference of x and one is less than 1.
Vedic mathematics is a system of mathematics that was rediscovered from ancient Hindu scriptures called the Vedas between 1911-1918. It is based on 16 sutras or word-formulas and 13 sub-sutras that describe how the mind naturally works. The Vedic system is more coherent and unified than modern mathematics, with techniques that are easy to understand and relate to one another. It allows complex problems to be solved quickly through intuitive and direct methods.
The document discusses the binary number system. It begins by defining number systems and the decimal system. It then introduces the binary number system which has a base of 2 and uses only the digits 0 and 1. It shows how to write binary numbers and provides a table to demonstrate counting and place values in the binary system. The document explains two methods for converting between decimal and binary numbers - the division method to convert decimals to binary, and the expansion method to convert binary to decimal. It includes examples and practice problems for students to convert numbers between the two number systems.
The document provides steps to convert a binary number to decimal. It shows converting the binary number 11111011 to decimal. The steps are: 1) Count the binary digits from right to left and assign powers of 2 to each position. 2) Multiply each 1 by its corresponding power of 2 and add the results to get the decimal number. Converting 11111011 to decimal is 123.
This document provides an overview of database concepts including creating, altering, and dropping databases and tables. It discusses data definition language (DDL) commands like CREATE, ALTER, DROP as well as data manipulation language (DML) commands like INSERT, SELECT, UPDATE, DELETE. It also covers database constraints, joins, functions for aggregation, strings, numbers, dates and more. The document is an introduction to core SQL concepts for a course on data management and database design.
Fibonacci Sequence .
The Fibonacci Sequence is a definite pattern that can begin with either 0, 1 or 1, 1. The sequence is generated by adding the previous terms, so that 0 +1 equals 1, 1+1 equals 2, 2 + 1 equals 3, 2 + 3 equals 5, 5 + 3 equals 8, 8 + 5 equals 13, 13 + 8 equals 21, 21 + 13 equals 34, 34 + 21 equals 55, and so on. Fibonacci can be found in in every domain such as nature, art, infrastructure ,humans etc. .
The document discusses various properties of operations on rational numbers. It states that rational numbers are closed under addition, subtraction, multiplication and division. This means these operations on rational numbers always yield a rational number as the result. It also discusses properties like commutativity, associativity and distributivity that apply to some operations but not others. For example, addition and multiplication of rational numbers are commutative but subtraction and division are not.
This document contains a 20 question matrices worksheet for Class 12 students. It covers topics like addition and multiplication of matrices, inverse of matrices, determinant, rank of matrices, and solving systems of linear equations using matrices. The worksheet is from Sthitpragya Science Classes, an institute providing advanced mathematics coaching for engineering entrance exams like JEE and BITSAT in Gandhidham, Gujarat.
The document provides information about different number systems used in computers, including binary, octal, hexadecimal, and decimal. It explains the characteristics of each system such as the base and digits used. Methods for converting between number systems like binary to decimal and vice versa are presented. Shortcut methods for direct conversions between binary, octal, and hexadecimal are also described. Binary arithmetic and binary-coded decimal number representation are discussed.
The document discusses different number systems used in digital computers including binary, decimal, octal, and hexadecimal systems. It describes the characteristics of each system such as the base and digits used. Methods for converting between these different number systems are presented, including using division or grouping bits. The representation of signed integers as binary numbers is also covered, comparing sign-magnitude, one's complement, and two's complement representations. Binary addition is demonstrated with examples.
This document is an introduction to Gordon Rockmaker's book "101 Short Cuts in Math Anyone Can Do". It explains that short cuts are tricks that can save time and effort in calculations by cutting through unnecessary steps. The introduction also defines important mathematical terms like digits, integers, and place value positions for numbers. It aims to prepare the reader to understand and apply the short cuts presented in the book.
In this ppt , you will learn about the evolution of number systems, decimal, binary and hexadecimal and why hexadecima is the most important form of number systems when working with microcontroller programming.
The document discusses convergence of sequences and power series. It defines convergence of a sequence and states that the limit of a convergent sequence is unique. It also discusses Taylor series and Laurent series, stating that if a function f(z) is analytic inside a circle C with center z0, its Taylor series representation about z0 will converge to f(z) for all z inside C. Similarly, if f(z) is analytic in an annular region bounded by two concentric circles, its Laurent series will represent f(z) in that region.
Double integral using polar coordinatesHarishRagav10
Maths is aan acknowledge for all humanity's stast of technology and many resources now in all country maths is the father of technology is a wondering..maths comes into play a vital role in all form of chemistry physics and all other subjects...in our day to day life maths playa a role everything and everywhere
The document discusses square roots and how to estimate them to varying degrees of precision. It defines the principal root as the positive square root of a number. It provides examples of perfect squares and their roots. It explains that irrational numbers are those that cannot be expressed as a ratio of integers, like π. The document then gives steps to estimate square roots to the nearest tenths or hundredths place by considering the closest perfect squares before and after the given number.
This document introduces number systems and provides examples of converting between different number systems. It discusses decimal, binary, octal, and hexadecimal number systems. Conversion between these systems can be done directly by dividing and taking remainders or via shortcuts by grouping digits. Understanding number systems is important for IT professionals as computers use binary to represent all data internally.
Karnaugh maps are a graphical technique used to simplify Boolean logic equations. They represent truth tables in a two-dimensional layout where physically adjacent cells imply logical adjacency. This adjacency allows common terms to be factored out to minimize logic expressions. Karnaugh maps are most commonly used to manually minimize logic with up to four variables into sum-of-products or product-of-sums form.
Vedic mathematics is a system of mathematics consisting of 16 sutras or aphorisms obtained from ancient Hindu scriptures called the Vedas. It was presented in the early 20th century by Bharati Krishna Tirthaji Maharaja, an Indian scholar. The sutras provide concise formulae for solving problems through unique techniques like vertically-and-crosswise calculations without needing multiplication tables beyond 5x5. Some examples include techniques for squaring numbers and multiplying multi-digit numbers mentally through a carry-over method. Vedic mathematics was applied in areas like astronomy, astrology and constructing calendars.
The document discusses the binary number system, explaining that it uses only two digits, 0 and 1, with each column in a binary number being twice the value of the previous column. It covers converting between binary and decimal numbers, with examples, and explains that binary is important because computers represent information using electronic switches that can be either on or off, corresponding to 1 and 0.
1) Srinivasa Ramanujan was one of India's greatest mathematical geniuses who made substantial contributions to analytical number theory, elliptic functions, and infinite series.
2) He was mostly self-taught and showed extraordinary talent from a young age, mastering advanced mathematical concepts from books he received.
3) Ramanujan struggled for recognition in India but eventually his work was brought to the attention of the English mathematician G.H. Hardy, who helped arrange for Ramanujan to travel to Cambridge University in 1914 where he spent five productive years collaborating before falling ill and returning to India, where he passed away in 1920.
Newton's Backward Interpolation explained with example. History of interpolation along with it's advantages and disadvantages. Applications of interpolation in computer sciences.
Translating english sentence to mathematical sentenceselle155401
This document provides verbal phrases and their mathematical translations. It includes an assignment to translate two additional phrases: 1) Five added to three times a number is less than 26. 2) Thrice the difference of x and one is less than 1.
Vedic mathematics is a system of mathematics that was rediscovered from ancient Hindu scriptures called the Vedas between 1911-1918. It is based on 16 sutras or word-formulas and 13 sub-sutras that describe how the mind naturally works. The Vedic system is more coherent and unified than modern mathematics, with techniques that are easy to understand and relate to one another. It allows complex problems to be solved quickly through intuitive and direct methods.
The document discusses the binary number system. It begins by defining number systems and the decimal system. It then introduces the binary number system which has a base of 2 and uses only the digits 0 and 1. It shows how to write binary numbers and provides a table to demonstrate counting and place values in the binary system. The document explains two methods for converting between decimal and binary numbers - the division method to convert decimals to binary, and the expansion method to convert binary to decimal. It includes examples and practice problems for students to convert numbers between the two number systems.
The document provides steps to convert a binary number to decimal. It shows converting the binary number 11111011 to decimal. The steps are: 1) Count the binary digits from right to left and assign powers of 2 to each position. 2) Multiply each 1 by its corresponding power of 2 and add the results to get the decimal number. Converting 11111011 to decimal is 123.
This document provides an overview of database concepts including creating, altering, and dropping databases and tables. It discusses data definition language (DDL) commands like CREATE, ALTER, DROP as well as data manipulation language (DML) commands like INSERT, SELECT, UPDATE, DELETE. It also covers database constraints, joins, functions for aggregation, strings, numbers, dates and more. The document is an introduction to core SQL concepts for a course on data management and database design.
Fibonacci Sequence .
The Fibonacci Sequence is a definite pattern that can begin with either 0, 1 or 1, 1. The sequence is generated by adding the previous terms, so that 0 +1 equals 1, 1+1 equals 2, 2 + 1 equals 3, 2 + 3 equals 5, 5 + 3 equals 8, 8 + 5 equals 13, 13 + 8 equals 21, 21 + 13 equals 34, 34 + 21 equals 55, and so on. Fibonacci can be found in in every domain such as nature, art, infrastructure ,humans etc. .
The document discusses various properties of operations on rational numbers. It states that rational numbers are closed under addition, subtraction, multiplication and division. This means these operations on rational numbers always yield a rational number as the result. It also discusses properties like commutativity, associativity and distributivity that apply to some operations but not others. For example, addition and multiplication of rational numbers are commutative but subtraction and division are not.
This document contains a 20 question matrices worksheet for Class 12 students. It covers topics like addition and multiplication of matrices, inverse of matrices, determinant, rank of matrices, and solving systems of linear equations using matrices. The worksheet is from Sthitpragya Science Classes, an institute providing advanced mathematics coaching for engineering entrance exams like JEE and BITSAT in Gandhidham, Gujarat.
The document provides information about different number systems used in computers, including binary, octal, hexadecimal, and decimal. It explains the characteristics of each system such as the base and digits used. Methods for converting between number systems like binary to decimal and vice versa are presented. Shortcut methods for direct conversions between binary, octal, and hexadecimal are also described. Binary arithmetic and binary-coded decimal number representation are discussed.
The document discusses different number systems used in digital computers including binary, decimal, octal, and hexadecimal systems. It describes the characteristics of each system such as the base and digits used. Methods for converting between these different number systems are presented, including using division or grouping bits. The representation of signed integers as binary numbers is also covered, comparing sign-magnitude, one's complement, and two's complement representations. Binary addition is demonstrated with examples.
The document discusses digital electronics topics including number systems, binary logic functions, and Boolean algebra. It covers converting between decimal, binary, octal, and hexadecimal number systems using long division. Examples are provided to illustrate converting decimal numbers to other bases, such as converting 29 to binary and 105 to hexadecimal. Boolean logic topics like logic gates and De Morgan's theorems are also listed in the overview.
The document discusses different numerical systems including:
- Decimal - Uses base 10 and includes numbers like 1, 13, 2028, 12.1, 3.14159.
- Binary - Uses the digits 0 and 1 and is the system used for computer processing.
- Octal - A base-8 system that uses digits 0-7.
- Hexadecimal - A base-16 system that uses digits 0-9 and letters A-F.
It then explains different methods for converting between these systems including long division, using powers of the base, and a doubling method. Carried and borrowed digits are also discussed for addition and subtraction in these alternate bases.
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
This document discusses different number systems used in digital computers and their conversions. It begins with an introduction to digital number systems and then describes the decimal, binary, octal and hexadecimal number systems. It explains how to represent integers and real numbers in binary. The document also covers number conversions between these systems using different methods like repeated division. Finally, it discusses various ways of representing integers in binary like sign-magnitude, one's complement and two's complement representations.
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
This document discusses data representation in computers. It covers different data types like characters, integers, and real numbers. It explains how numbers are represented in the decimal and binary systems, including number bases, place values, and conversions between decimal and binary. It also discusses coding systems for representing characters as binary codes, including ASCII and EBCDIC. Additional topics covered include binary-coded decimal, hexadecimal, octal systems, number complements, signed and unsigned integers, and floating-point representation of real numbers.
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.
This document discusses different number systems including binary, decimal, octal, and hexadecimal. It provides details on each system such as their number bases and allowed digits. The document also describes how to convert between these different number systems using methods like dividing numbers by the target base or grouping binary digits into sets of four for hexadecimal conversion. The goal is to understand representation of numbers in computing systems which commonly use binary and hexadecimal formats.
This document provides an overview of number systems used in digital electronics. It discusses decimal, binary, octal and hexadecimal number systems. It describes how to convert between these different number systems, including binary to decimal and decimal to binary conversions. Binary addition and subtraction are also covered. The document introduces signed binary numbers to represent positive and negative values. Overall, the document aims to explain the fundamental concepts of number representation in digital circuits and computers.
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
This document introduces number systems such as binary, decimal, octal and hexadecimal. It discusses how numbers are represented in these different systems using place values that are powers of the respective base. Decimal uses base 10, binary uses base 2, octal uses base 8 and hexadecimal uses base 16. Conversion between these number systems involves expressing the number in its place value form and then summing the place values weighted by the respective digits. The document provides examples of converting between decimal, binary, octal and hexadecimal 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 ...
Physics investigatory project for class 12 logic gatesbiswanath dehuri
This document provides an overview of digital electronics and Boolean algebra. It discusses digital and analog signals, different number systems including binary, and basic logic gates. Boolean algebra rules are also covered, including commutative, associative, distributive, AND, and OR laws. Common digital applications are listed such as industrial controls, medical equipment, and communications systems. The key advantages of digital systems are accuracy, versatility, less noise and distortion.
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.
The document discusses different number systems used in computers such as binary, decimal, octal and hexadecimal. It provides examples and techniques for converting between these number systems. The key number systems covered are binary, which uses two digits (0 and 1), and is used in computers, decimal which uses 10 digits and is used in everyday life, octal which uses 8 digits, and hexadecimal which uses 16 digits and letters A-F. The document also discusses techniques for converting fractions between decimal and binary.
This document discusses different number systems including positional and non-positional. It covers the decimal, binary, octal, and hexadecimal number systems. Methods are provided for converting between these bases, including direct conversion as well as shortcut methods. Representing fractional numbers in binary and octal number systems is also explained with examples.
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
This document provides an introduction to error analysis in numerical techniques. It discusses approximate vs exact numbers, significant figures, rounding off numbers, different types of errors including absolute, relative and percentage errors. It also covers error in arithmetic operations due to inherent, truncation and rounding errors. A general error formula is presented to calculate the error in a function with multiple variables based on the errors in each variable. An example is given to calculate the maximum relative error in a numerical computation.
This document outlines lecture material on sampling techniques from Dr. Tushar Bhatt of Saurashtra University. It discusses various probability and non-probability sampling methods including simple random sampling, stratified sampling, cluster sampling, systematic sampling, and PPS sampling. For each method, it provides definitions, formulas, steps for implementation, and examples. The document is intended as teaching material, covering core concepts in sampling and how to select samples from different populations.
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
Numerical Integration and Numerical Solution of Ordinary Differential Equatio...Dr. Tushar J Bhatt
This document contains a numerical techniques unit on numerical integration and numerical solutions to ordinary differential equations. It covers Trapezoidal rule, Simpson's 1/3 rule, Simpson's 3/8 rule, and examples of applying these rules to calculate definite integrals numerically. The examples provided calculate the integrals of functions from 0 to π by dividing the interval into 10 equal parts and applying the Trapezoidal and Simpson's 1/3 rules.
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.
How Barcodes Can Be Leveraged Within Odoo 17Celine George
In this presentation, we will explore how barcodes can be leveraged within Odoo 17 to streamline our manufacturing processes. We will cover the configuration steps, how to utilize barcodes in different manufacturing scenarios, and the overall benefits of implementing this technology.
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
The chapter Lifelines of National Economy in Class 10 Geography focuses on the various modes of transportation and communication that play a vital role in the economic development of a country. These lifelines are crucial for the movement of goods, services, and people, thereby connecting different regions and promoting economic activities.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
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4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
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6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
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Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt
Number_System .pdf
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.
Dr. Tushar Bhatt Unit - III: Number System 23 / 52
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.
Dr. Tushar Bhatt Unit - III: Number System 24 / 52
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.
Dr. Tushar Bhatt Unit - III: Number System 25 / 52
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.
Dr. Tushar Bhatt Unit - III: Number System 26 / 52
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
Dr. Tushar Bhatt Unit - III: Number System 28 / 52
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.
Dr. Tushar Bhatt Unit - III: Number System 29 / 52
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.
Dr. Tushar Bhatt Unit - III: Number System 31 / 52
32. 4.2 Examples on Octal to Decimal Conversation
Ex - 1: Convert (140)8 to decimal number.
Solution:
Dr. Tushar Bhatt Unit - III: Number System 32 / 52
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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