DOCSLIB.ORG

V : Number Systems and Set Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
V : Number Systems and Set Theory V : Number systems and set theory Any reasonable framework for mathematics should include the fundamental number systems which arise in the subject: 1. The natural numbers N (also known as the nonnegative integers). 2. The (signed) integers Z obtained by adjoining negative numbers to N. 3. The rational numbers Q obtained by adjoining reciprocals of nonzero integers to Z. 4. The real numbers R, which should include fundamental constructions like nth roots of positive rational numbers for an arbitrary integer n > 1, and also –1 –2 –k ⋅ ⋅ ⋅ all “infinite decimals” of the form b1 10 + b2 10 + … + bk 10 + … where each bi belongs to {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Up to this point we have tacitly assumed that such number systems are at our disposal. However, in both the naïve and axiomatic approaches to set theory it is eventually necessary to say more about them. The naïve approach. In naïve set theory it is necessary to do two things. First, one must describe the properties that the set – theoretic versions of these number systems should satisfy. Second, something should be said to justify our describing such systems as THE natural numbers, THE integers, THE rational numbers, and THE real numbers. This usage suggests that we have completely unambiguous descriptions of the number systems in terms of their algebraic and other properties. One way of stating this is that any system satisfying all the conditions for one of the standard systems N, Z, Q or R should be the same as N, Z, Q or R for all mathematical purposes, with some explicit means for mechanical translation from the given system to the appropriate standard model. In less formal terms, it we have any systems X which satisfy all the fundamental properties of one of the systems N, Z, Q or R, then X is essentially a mathematical clone of the appropriate number system. There are good theoretical and philosophical reasons for asking such questions about the essential uniqueness of the number systems, but these question also have some important practical implications for the development of mathematics. If there would be two systems that satisfy the basic properties of N, Z, Q or R but differ from a standard model in some significant fashion, then clearly we might get different versions of mathematics depending upon which example is chosen. To illustrate this, suppose we decided to develop a version of the real numbers in which infinite base 10 “decimal expansions” are replaced by expansions with some other number base, say 16 (to conform with the internal arithmetic of some computer) or 60 (as in Babylonian mathematics). We expect that everything should work the same regardless of the 70 numerical base we choose for expressing quantities, but at some point it is necessary to confirm that our expectation is fulfilled. Later in this unit we shall describe precisely the notion of a mathematical clone. For the time being we note that examples of this concept have already been encountered in Section IV.6 when we talked about whether two partially ordered sets have the same order type. Given two such partially ordered sets, the 1 – 1 order preserving correspondence from one to another can be viewed as a formal mathematical way of saying that either of the partially ordered sets is a clone of the other. Our coverage in this unit will mainly concern the first item described in the naïve approach; namely, the formal properties of the number systems and the mathematical statements of their uniqueness properties. Later in these notes (and largely for reference purposes) we shall explain why the basic properties describe these number systems in a totally unambiguous manner. The axiomatic approach. In axiomatic set theory it is necessary to assume the existence of systems with the given properties and to prove these properties describe them unambiguously (the latter proceeds exactly the same as in naïve set theory). One new issue in the axiomatic approach is the goal of keeping the basic assumptions for set theory as simple as possible. Assuming the existence of four separate but clearly interrelated number systems is a convenient first step, but at some point it is natural to ask if we really need to make such a long list of assumptions in order to set everything up. Aside from possible aesthetic considerations, there is the practical consideration that long lists of assumptions raise questions whether there might be some logical inconsistency; after all, the whole idea of a proof by contradiction is that one makes so many assumptions that the conclusions end up contradicting each other, and it would undermine everything if such contradictions could be derived from the axioms for set theory itself. We shall address some of these issues in the final unit of the notes. Some more specific objectives Much of this unit is devoted to summarizing familiar properties of the four basic number systems, so we shall indicate some points that are less elementary and particularly important. In Section 1 the most significant new item is the statement of the Peano Axioms for the natural numbers, and in Section 2 the discussion of finite induction and recursive definitions in the framework of set theory is one of the main topics in the unit. The formulas for counting the numbers of elements in various finite sets in Section 3 start with familiar ideas, and they give systematic rules that are important both for their own sake and for the remaining units of the course. Finally, the description of the real numbers in Section 4 is fundamentally important. Although this description is fairly concise, it contains everything that is needed to justify the standard facts about real numbers and to develop calculus in a mathematically rigorous fashion. The latter development is covered in subsequent courses. Although the justification of the usual expansions for real numbers is also somewhat peripheral to the present course, for the sake of completeness we shall explain how our formal description of the real numbers yields their familiar properties which are used in everyday work, both inside and outside of mathematics. 71 V . 1 : The natural numbers and integers (Halmos, §§ 11 – 13; Lipschutz, §§ 2.1, 2.7 – 2.9) In many respects the positive integers form the most basic number system in all of the mathematical sciences. Some reasons for this are historical or philosophical, but logical considerations are particularly important for the systematic development of mathematics. Clearly we would like our descriptions of number systems to summarize their basic algebraic properties concise but understandable. In particular, it simplifies things considerably if we can say that addition, subtraction and multiplication are always defined. Since the positive integers are not closed under subtraction, clearly they do not fulfill this condition. Therefore we shall begin by describing the integers, and we shall view the positive integers as a subset of the integers with certain special properties. The important algebraic properties of the integers split naturally into three classes, two of which are fairly general and one of which is more focused. Basic rules for addition and multiplication. Formally, these are the conditions defining an abstract type of mathematical system known as a commutative ring with unit. FIRST AXIOM GROUP FOR THE INTEGERS. The integers are a set Z, and they have binary operations A : Z × Z → Z, normally expressed in the form A(u, v) = u + v, and M : Z × Z → Z, normally expressed in the form M(u, v) = u v or u ⋅ v or u × v, which satisfy the following algebraic conditions: 1. (Associative Laws). For all a, b, c in Z, (a + b) + c = a + (b + c) and (a b) c = a (b c). 2. (Commutative Laws). For all a, b in Z, a + b = b + a and a b = b a. 3. (Distributive Law). For all a, b, c in Z, a (b + c) = a b + a c. 4. (Existence of 0 and 1). There are distinct elements 0, 1 in Z such that for all a we have a + 0 = a, a × 0 = 0 and a × 1 = a. 5. (Existence of negatives or additive inverses). For each a in Z there is an element – a in Z such that a + (– a) = 0. Notational footnote: The notation Z for the integers has become fairly standard in mathematical writings, and it is apparently derived from the German word for numbers (Zahlen) and/or cyclic (zyklisch). We shall need the following basic consequences of the preceding algebraic conditions: Proposition 1. If a belongs to a system satisfying the properties listed above, then we have (– a) (– b) = a b. 72 Proof. The following are special cases of the axioms: 0 = a 0 = a [b + (– b)] = a b + a (– b) 0 = 0 (– b) = [a + (– a)] (– b) = a (– b) + (– a) (– b) The preceding results also show that a b = – [a (– b)] = (– a) (– b). Basic rules for ordering. When combined with the previous conditions, these yield a type of mathematical system known as an ordered integral domain. SECOND AXIOM GROUP FOR THE INTEGERS. There is a linear ordering on Z such that the following hold: 1. If a > 0 and b > 0, then a + b > 0 and a b > 0. 2. For all a, b in Z, we have a > b if and only if a – b > 0. Well – ordering of positive elements. This is the assumption that the set N of nonnegative elements in Z, often called the natural numbers, is well – ordered with respect to the standard linear ordering. WELL - ORDERING AXIOM FOR THE POSITIVE INTEGERS. The set N of all x in Z such that x ≥ 0 is well – ordered.
Load more

Recommended publications
  • Bits, Data Types, and Operations
    Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Agenda 1. Data Types 2. Unsigned & Signed Integers 3. Arithmetic Operations Chapter 2 4. Logical Operations 5. Shifting Bits, Data Types, 6. Hexadecimal & Octal Notation and Operations 7. Other Data Types COMPSCI210 S1C 2009 2 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1. Data Types Computer is a binary digital system. How do we represent data in a computer? Digital system: Binary (base two) system: At the lowest level, a computer is an electronic machine. • finite number of symbols • has two states: 0 and 1 • works by controlling the flow of electrons Easy to recognize two conditions: 1. presence of a voltage – we’ll call this state “1” 2. absence of a voltage – we’ll call this state “0” Basic unit of information is the binary digit, or bit. Values with more than two states require multiple bits. • A collection of two bits has four possible states: Could base state on value of voltage, 00, 01, 10, 11 but control and detection circuits more complex. • A collection of three bits has eight possible states: • compare turning on a light switch to 000, 001, 010, 011, 100, 101, 110, 111 measuring or regulating voltage • A collection of n bits has 2n possible states. COMPSCI210 S1C 2009 3 COMPSCI210 S1C 2009 4 1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
    [Show full text]
  • 1. Understanding Decimal, Binary, Octal and Hexadecimal Numbers
    Objectives: 1. Understanding decimal, binary, octal and hexadecimal numbers. 2. Counting in decimal, binary, octal and hexadecimal systems. 3. Convert a number from one number system to another system. 4. Advantage of octal and hexadecimal systems. 1. Understanding decimal, binary, octal and hexadecimal numbers Decimal number systems: Decimal numbers are made of decimal digits: (0,1,2,3,4,5,6,7,8,9 --------10-base system) The decimal system is a "positional-value system" in which the value of a digit depends on its position. Examples: 453→4 hundreds, 5 tens and 3 units. 4 is the most weight called "most significant digit" MSD. 3 carries the last weight called "least significant digit" LSD. number of items that a decimal number represent: 9261= (9× )+(2× )+(6× )+(1× ) The decimal fractions: 3267.317= (3× )+(2× )+(6× )+(7× )+ (3× ) + (6× ) + (1× ) Decimal point used to separate the integer and fractional part of the number. Formal notation→ . Decimal position values of powers of (10). Positional values "weights" 2 7 7 8 3 . 2 3 4 5 MSD LSD Binary numbers: . Base-2 system (0 or 1). We can represent any quantity that can be represented in decimal or other number systems using binary numbers. Binary number is also positional–value system (power of 2). Example: 1101.011 1 1 0 1 . 0 1 1 MSD LSD Notes: . To find the equivalent of binary numbers in decimal system , we simply take the sum of products of each digit value (0,1)and its positional value: Example: = (1× ) + (0× ) + (1× ) + (1× )+ (1× )+ (0× ) +(1× ) = 8 + 0 + 2 + 1 + + 0 + = In general, any number (decimal, binary, octal and hexadecimal) is simply the sum of products of each digit value and its positional value.
    [Show full text]
  • Number Systems and Number Representation Aarti Gupta
    Number Systems and Number Representation Aarti Gupta 1 For Your Amusement Question: Why do computer programmers confuse Christmas and Halloween? Answer: Because 25 Dec = 31 Oct -- http://www.electronicsweekly.com 2 Goals of this Lecture Help you learn (or refresh your memory) about: • The binary, hexadecimal, and octal number systems • Finite representation of unsigned integers • Finite representation of signed integers • Finite representation of rational numbers (if time) Why? • A power programmer must know number systems and data representation to fully understand C’s primitive data types Primitive values and the operations on them 3 Agenda Number Systems Finite representation of unsigned integers Finite representation of signed integers Finite representation of rational numbers (if time) 4 The Decimal Number System Name • “decem” (Latin) => ten Characteristics • Ten symbols • 0 1 2 3 4 5 6 7 8 9 • Positional • 2945 ≠ 2495 • 2945 = (2*103) + (9*102) + (4*101) + (5*100) (Most) people use the decimal number system Why? 5 The Binary Number System Name • “binarius” (Latin) => two Characteristics • Two symbols • 0 1 • Positional • 1010B ≠ 1100B Most (digital) computers use the binary number system Why? Terminology • Bit: a binary digit • Byte: (typically) 8 bits 6 Decimal-Binary Equivalence Decimal Binary Decimal Binary 0 0 16 10000 1 1 17 10001 2 10 18 10010 3 11 19 10011 4 100 20 10100 5 101 21 10101 6 110 22 10110 7 111 23 10111 8 1000 24 11000 9 1001 25 11001 10 1010 26 11010 11 1011 27 11011 12 1100 28 11100 13 1101 29 11101 14 1110 30 11110 15 1111 31 11111 ..
    [Show full text]
  • 1 Evolution and Base Conversion
    Indian Institute of Information Technology Design and Manufacturing, Kancheepuram logo.png Chennai { 600 127, India Instructor An Autonomous Institute under MHRD, Govt of India N.Sadagopan http://www.iiitdm.ac.in Computational Engineering Objectives: • To learn to work with machines (computers, ATMs, Coffee vending machines). • To understand how machines think and work. • To design instructions which machines can understand so that a computational task can be performed. • To learn a language which machines can understand so that human-machine interaction can take place. • To understand the limitations of machines: what can be computed and what can not be computed using machines. Outcome: To program a computer for a given computational task involving one or more of arithmetic, algebraic, logical, relational operations. 1 Evolution and Base Conversion Why Machines ? We shall now discuss the importance of automation; human-centric approach (manual) versus machines. Although machines are designed by humans, for many practical reasons, machines are superior to humans as far as problem solving is concerned. We shall highlight below commonly observed features that make machines powerful. • Reliable, accurate and efficient. • Can do parallel tasks if trained. • Efficient while peforming computations with large numbers. If designed correctly, then there is no scope for error. • Good at micro-level analysis with precision. • Consistent It is important to highlight that not all problems can be solved using machines (computers). For example, (i) whether a person is happy (ii) whether a person is lying (not speaking the truth) (iii) reciting the natural number set (iv) singing a song in a particular raga. Computer: A computational machine In this lecture, we shall discuss a computational machine called computer.
    [Show full text]
  • Conversion from Decimal There Is a Simple Method That Allows Conversions from the Decimal to a Target Number System
    APPENDICES A Number Systems This appendix introduces background material on various number systems and representations. We start the appendix with a discussion of various number systems, including the binary and hexadecimal systems. When we use multiple number systems, we need to convert numbers from system to another We present details on how such number conversions are done. We then give details on integer representations. We cover both unsigned and signed integer representations. We close the appendix with a discussion of the floating-point numbers. Positional Number Systems The number systems that we discuss here are based on positional number systems. The decimal number system that we are already familiar with is an example of a positional number system. In contrast, the Roman numeral system is not a positional number system. Every positional number system has a radix or base, and an alphabet. The base is a positive number. For example, the decimal system is a base-10 system. The number of symbols in the alphabet is equal to the base of the number system. The alphabet of the decimal system is 0 through 9, a total of 10 symbols or digits. In this appendix, we discuss four number systems that are relevant in the context of computer systems and programming. These are the decimal (base-10), binary (base-2), octal (base-8), and hexadecimal (base-16) number systems. Our intention in including the familiar decimal system is to use it to explain some fundamental concepts of positional number systems. Computers internally use the binary system. The remaining two number systems—octal and hexadecimal—are used mainly for convenience to write a binary number even though they are number systems on their own.
    [Show full text]
  • Non-Power Positional Number Representation Systems, Bijective Numeration, and the Mesoamerican Discovery of Zero
    Non-Power Positional Number Representation Systems, Bijective Numeration, and the Mesoamerican Discovery of Zero Berenice Rojo-Garibaldia, Costanza Rangonib, Diego L. Gonz´alezb;c, and Julyan H. E. Cartwrightd;e a Posgrado en Ciencias del Mar y Limnolog´ıa, Universidad Nacional Aut´onomade M´exico, Av. Universidad 3000, Col. Copilco, Del. Coyoac´an,Cd.Mx. 04510, M´exico b Istituto per la Microelettronica e i Microsistemi, Area della Ricerca CNR di Bologna, 40129 Bologna, Italy c Dipartimento di Scienze Statistiche \Paolo Fortunati", Universit`adi Bologna, 40126 Bologna, Italy d Instituto Andaluz de Ciencias de la Tierra, CSIC{Universidad de Granada, 18100 Armilla, Granada, Spain e Instituto Carlos I de F´ısicaTe´oricay Computacional, Universidad de Granada, 18071 Granada, Spain Keywords: Zero | Maya | Pre-Columbian Mesoamerica | Number rep- resentation systems | Bijective numeration Abstract Pre-Columbian Mesoamerica was a fertile crescent for the development of number systems. A form of vigesimal system seems to have been present from the first Olmec civilization onwards, to which succeeding peoples made contributions. We discuss the Maya use of the representational redundancy present in their Long Count calendar, a non-power positional number representation system with multipliers 1, 20, 18× 20, :::, 18× arXiv:2005.10207v2 [math.HO] 23 Mar 2021 20n. We demonstrate that the Mesoamericans did not need to invent positional notation and discover zero at the same time because they were not afraid of using a number system in which the same number can be written in different ways. A Long Count number system with digits from 0 to 20 is seen later to pass to one using digits 0 to 19, which leads us to propose that even earlier there may have been an initial zeroless bijective numeration system whose digits ran from 1 to 20.
    [Show full text]
  • Number Systems: Binary and Hexadecimal Tom Brennan
    Number Systems: Binary and Hexadecimal Tom Brennan Base 10 The base 10 number system is the common number system. 612 = 2 · 100 + 1 · 101 + 6 · 102 2 + 10 + 600 = 612 4365 = 5 · 100 + 6 · 101 + 3 · 102 + 4 · 103 5 + 60 + 300 + 4000 = 4365 Binary 101010101 = 1 · 20 + 0 · 21 + 1 · 22 + 0 · 23 + 1 · 24 + 0 · 25 + 1 · 26 + 0 · 27 + 1 · 28 = 341 1 + 0 + 4 + 0 + 16 + 0 + 64 + 0 + 256 = 341 The binary number system employs only two digits{0 and 1{and each digit is called a bit. A number can be represented in binary by a string of bits{each bit either a 0 or a 1 multiplied by a power of two. For example, to represent the decimal number 15 in binary you would write 1111. That string stands for 1 · 20 + 1 · 21 + 1 · 22 + 1 · 23 = 15 (1 + 2 + 4 + 8 = 15) History Gottfried Leibniz developed the modern binary system in 1679 and drew his inspiration from the ancient Chinese. Leibniz found a technique in the I Ching{an ancient chinese text{that was similar to the modern binary system. The system formed numbers by arranging a set of hexagrams in a specific order in a manner similar to modern binary. The systems base two nature was derived from the concepts of yin and yang. This system is at least as old as the Zhou Dynasty{it predates 700 BC. While Leibniz based his system of the work of the Chinese, other cultures had their own forms of base two number systems.
    [Show full text]
  • Realistic Approach of Strange Number System from Unodecimal to Vigesimal
    International Journal of Computer Science and Telecommunications [Volume 3, Issue 1, January 2012] 11 Realistic Approach of Strange Number System from Unodecimal to Vigesimal ISSN 2047-3338 Debasis Das and Dr. U. A. Lanjewar Abstract — Presently in computer science and technology, 11 digits: numeral 0 through 9 and alphabet A, a base 12 number system is based on some traditional number system viz. (duodecimal) numbers contain 12 digits numeral 0 through 9 decimal (base 10), binary (base-2), octal (base-8) and hexadecimal and alphabets A and B, a base 13 (tridecimal) number contains (base 16). The numbers in strange number system (SNS) are those numbers which are other than the numbers of traditional number 13 digits: numeral 0 through 9 and alphabet A, B and C and so system. Some of the strange numbers are unary, ternary, …, fourth. These are an alphanumeric number system because its Nonary, ..., unodecimal, …, vigesimal, …, sexagesimal, etc. The uses both alphabets and numerical to represent a particular strange numbers are not widely known or widely used as number. traditional numbers in computer science, but they have charms all It is well known that the design of computers begins with their own. Today, the complexity of traditional number system is the choice of number system, which determines many technical steadily increasing in computing. Due to this fact, strange number system is investigated for efficiently describing and implementing characteristics of computers. But in the modern computer, the in digital systems. In computing the study of strange number traditional number system is only used, especially binary system (SNS) will useful to all researchers.
    [Show full text]
  • Prelims 3..16
    Contents Preface to the First Edition xvii Preface to the Second Edition xix Preface to the Third Edition xx Preface to the Fourth Edition xxi Preface to the Fifth Edition xxii Hints on using the book xxiv Useful background information xxvi Part I: Foundation topics 1 Programme F.1 Arithmetic 3 Learning outcomes 3 Quiz F.1 4 Types of numbers ± The natural numbers, Numerals and place value, 5 Points on a line and order, The integers, Brackets, Addition and subtraction, Multiplication and division, Brackets and precedence rules, Basic laws ofarithmetic Revision summary 12 Revision exercise 13 Factors and prime numbers ± Factors, Prime numbers, Prime 14 factorization, Highest common factor (HCF), Lowest common multiple (LCM), Estimating, Rounding Revision summary 18 Revision exercise 18 Fractions ± Division of integers, Multiplying fractions, Of, Equivalent 20 fractions, Dividing fractions, Adding and subtracting fractions, Fractions on a calculator, Ratios, Percentages Revision summary 28 Revision exercise 28 Decimal numbers ± Division of integers, Rounding, Significant figures, 30 Decimal places, Trailing zeros, Fractions as decimals, Decimals as fractions, Unending decimals, Unending decimals as fractions, Rational, irrational and real numbers Revision summary 35 Revision exercise 35 Powers ± Raising a number to a power, The laws ofpowers, Powers on a 36 calculator, Fractional powers and roots, Surds, Multiplication and division by integer powers of10, Precedence rules, Standard form, Working in standard form, Using a calculator, Preferred
    [Show full text]
  • Number Systems
    E Number Systems Hereareonly numbers ratified. —WilliamShakespeare Naturehas some sort of arithmetic-geometrical OBJECTIVES coordinate system,because In this appendix you will learn: naturehas all kinds of models.What weexperience I To understand basicnumber systems concepts, suchas of natureis in models,and base,positional valueand symbol value. all of nature’s models are so I To understand how to work withnumbers represented in beautiful. It struckme that nature’s the binary,octaland hexadecimalnumber systems. system must bea realbeauty, I Toabbreviatebinary numbers as octalnumbers or becauseinchemistry wefind hexadecimalnumbers. that the associations are always in beautiful whole I Toconvert octalnumbers and hexadecimalnumbers to numbers—thereareno binary numbers. fractions. I Tocovnert backand forthbetween decimalnumbers and —RichardBuckminster Fuller their binary,octaland hexadecimalequivalents. I To understand binary arithmeticand how negative binary numbers are represented using two’s complement notation. 1404 Appendix ENumber Systems E.1 Introduction E.2 Abbreviating Binary Numbers as Octaland HexadecimalNumbers line E.3 Converting Octaland HexadecimalNumbers toBinary Numbers ut O E.4 Converting from Binary,Octalor Hexadecimal toDecimal E.5 Converting from Decimal toBinary,Octalor Hexadecimal E.6 NegativeBinary Numbers:Two’s Complement Notation Summary |Terminology |Self-Review Exercises |Answers toSelf-Review Exercises |Exercises E.1 Introduction In this appendix, weintroduce the key number systems that Javaprogrammers use,espe- cially when they are working on softwareprojects that requirecloseinteraction withma- chine-level hardware. Projects like this include operating systems,computer networking software,compilers,database systems and applications requiring high performance. When we writeaninteger suchas 227 or –63 in aJavaprogram, the number is assumed tobein the decimal(base10)number system. The digits in the decimalnumber system are 0,1, 2, 3,4,5, 6, 7,8and 9.
    [Show full text]
  • Binary Decimal Octal and Hexadecimal Number Systems
    Binary Decimal Octal and Hexadecimal number systems Conversion of binary to decimal ( base 2 to base 10) Example: convert (1000100)2 to decimal = 64 + 0 + 0+ 0 + 4 + 0 + 0 = (68)10 Conversion of decimal to binary ( base 10 to base 2) Example: convert (68)10 to binary 68/¸2 = 34 remainder is 0 34/ 2 = 17 remainder is 0 17 / 2 = 8 remainder is 1 8 / 2 = 4 remainder is 0 4 / 2 = 2 remainder is 0 2 / 2 = 1 remainder is 0 1 / 2 = 0 remainder is 1 Answer = 1 0 0 0 1 0 0 Note: the answer is read from bottom (MSB) to top (LSB) as 10001002 Conversion of decimal fraction to binary fraction •Instead of division , multiplication by 2 is carried out and the integer part of the result is saved and placed after the decimal point. The fractional part is again multiplied by 2 and the process repeated. Example: convert ( 0.68)10 to binary fraction. 0.68 * 2 = 1.36 integer part is 1 0.36 * 2 = 0.72 integer part is 0 0.72 * 2 = 1.44 integer part is 1 0.44 * 2 = 0.88 integer part is 0 Answer = 0. 1 0 1 0….. Example: convert ( 68.68)10 to binary equivalent. Answer = 1 0 0 0 1 0 0 . 1 0 1 0…. Octal Number System •Base or radix 8 number system. •1 octal digit is equivalent to 3 bits. •Octal numbers are 0 to7. (see the chart down below) •Numbers are expressed as powers of 8. Conversion of octal to decimal ( base 8 to base 10) Example: convert (632)8 to decimal = (6 x 82) + (3 x 81) + (2 x 80) = (6 x 64) + (3 x 8) + (2 x 1) = 384 + 24 + 2 = (410)10 Conversion of decimal to octal ( base 10 to base 8) Example: convert (177)10 to octal 177 / 8 = 22 remainder is 1 22 / 8 = 2 remainder is 6 2 / 8 = 0 remainder is 2 Answer = 2 6 1 Note: the answer is read from bottom to top as (261)8, the same as with the binary case.
    [Show full text]
  • A Multiples Method for Number System Conversions
    International Journal of Scientific Research in ______________________________ Research Paper . Computer Science and Engineering Vol.7, Issue.4, pp.14-17, August (2019) E-ISSN: 2320-7639 DOI: https://doi.org/10.26438/ijsrcse/v7i4.1417 A Multiples Method for Number System Conversions Jitin Kumar Department Of Computer Science, David Model Senior Secondary School, Delhi, India Corresponding Author: [email protected] Available online at: www.isroset.org Received: 20/Aug/2019, Accepted: 28/Aug/2019, Online: 31/Aug/2019 Abstract—Number systems are the technique, to represent numbers in the computer system architecture, every value that you save into computer memory has a defined number system. To understand computers, knowledge of number systems and knowledge of their inter conversion is essential. Computer architecture supports Binary number system, octal number system, Decimal number system and Hexadecimal number system. In this paper, the conversion of number systems is given as multiples. By using multiples method, we can easily convert one number system into another number system without taking too much time and in the multiples method you have not to consider integer and fractional as two parts, all will be converted in a single method. It is easy to understand as well as memorize. Keywords—Number System, binary, octal, decimal, hexadecimal, inter conversions, base, radix. I. INTRODUCTION technique, which is a multiples. And the rules are same for, integer and fractional part of number system. Any system that is used for representing numbers is a number system, also known as numeral system. The modern This paper is designed in such a way that it consists of three civilization is familiar with decimal number system using ten sections.
    [Show full text]