Mathematics
Number Systems
Number systems are the different ways of representing numbers, including decimal, binary, octal, and hexadecimal systems. Each system has its own base and set of symbols for representing numbers. The decimal system, for example, is base 10 and uses the digits 0-9, while the binary system is base 2 and uses only 0 and 1.
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eBook - PDFComputer Programming
A Mixed Language Approach
- Marvin L. Stein, William D. Munro(Authors)
- 2014(Publication Date)
- Academic Press(Publisher)
CHAPTER 1 Number Systems 1 -0. Introduction and General Theory As implied by the name, the modern high speed digital computer does its arithmetic operations on digits. For reasons of design, however, most modern machines do not use the familiar decimal digits for their internal arithmetic. The most frequently used system is the binary and, in coding, the related octal system. For this reason the coder must become familiar with these systems and others which are occasionally used. In this chapter we will develop the theory of Number Systems to any valid base, thus laying the groundwork for the specific systems which will be employed later. In its simplest form a number system is a scheme for representing positive integers, and the system which human beings have used most is based on the integer ten, presumably because people have ten fingers (digits). By means of the number ten and the positional system of repre-sentation, we have a convenient and easily handled way of describing integers. For example, in the integer 13402, each digit has a position signifying a power of ten, and in writing 1, 3, 4, 0, 2 in order we really mean 1(10 4 ) + 3(10 3 ) + 4(10 2 ) + 0(H) 1 ) + 2(10°) In general, any decimal whole number with digits in order, d n d n -id n -2 · · · cWi^o is an abbreviation for 4 ( 1 0 ) + d^xUo*-1 ) + . . . + diCIO 1 ) + do (10°) 10 1-0. INTRODUCTION AND GENERAL THEORY 11 Looking at it this way, we see that our ordinary decimal representation of an integer is a shorthand way of writing a polynomial in powers of ten. Ten is called the base or radix of the decimal number system. We note that the digits or coefficients in the terms are restricted to 0 < d» < 9 and that we cannot omit terms with zero coefficients as we might do with poly-nomials written out (1342 is not the same as 13402). As we shall see, any integer r > 1, together with an appropriate set of digits, could be used in place of ten, and positive integers expressed as polynomials in r.
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
37 2 In this chapter we will explore the number system to help us see the similarities between the sets of numbers. The base ten numeration system, which most people use without thinking about it, is one of the greatest inventions in human history. A deeper understanding of our numeration system increases a person’s ability to make estimates and to solve more complex problems, to understand connections between whole numbers and decimals, to understand why the computation procedures work, and much more. For many thousands of years, whole numbers were adequate for most people’s needs. However, the limitations of whole numbers became more and more problematic as time went on, which created the need for decimals, fractions, integers, and irrational numbers. In this chapter, we will examine the meaning of each of these sets of numbers. As before, knowing why as well as how increases a person’s mathematical power. SECTION 2.1 Whole Numbers SECTION 2.2 Fractions SECTION 2.3 Decimals, Integers, and Real Numbers The Number System What do you think? l Why and when did humans invent numbers? l Why do many mathematicians regard the invention of zero as one of the most important developments in the entire history of mathematics? Explorations Manual 2.1 and 2.2 Whole Numbers SECTION 2.1 Did you know that the base ten numeration system you use every day is called the Hindu-Arabic numeration system because it was invented by the Hindus and transmitted to the West by the Arabs? Did you know that this system has been in widespread use in the West for only 500 years? INVESTIGATION 2.1a What Does “Place Value” Mean? A. Use the numeral 323 to explain what “place value” means and what it means to say we have a “base ten” place value system. B. What if we had a base five system instead of a base ten system? What would the numeral 323 mean then? 38 CHAPTER 2 The Number System DISCUSSION A. Our system is a base ten system because we group in tens.
eBook - ePubComputer Architecture
Software Aspects, Coding, and Hardware
- John Y. Hsu(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
C HAPTER 2 Number Systems 2.1 BASIC MATHEMATICS As all digital information is represented by binary digits, the mathematical backbone of designing a digital computer is Boolean algebra. Before learning instructions and data, it is necessary to know the different Number Systems, number conversions, and arithmetic operations performed by a computer. To accomplish this goal, we must review some basic mathematics and definitions. 2.1.1 Integer Part vs. Fraction Part Definition 1: If X is a real number, positive or negative, [X] is defined to be the greatest integer ≤ X, where ≤ stands for less than or equal to. For a positive integer X, after truncating the fraction part of the number, we obtain its integer part as the answer. A positive number in square brackets represents an integer but, for a negative number X we need to subtract one from the negative integer to get the correct result. The square bracket notation represents the floor function and some examples are given below: 1. [ 7.45 ] = 7 2. [ 3.14159 ] = 3 3. [ 3.00 ] = 3 4. [ - 5.6 ] = -6 5. [ -6.00 ] = -6 Definition 2: If X is a positive number, {X} is defined to be the fraction part of X. Therefore, we obtain, { X } = X- [ X ] Therefore, a positive number in curly brackets denotes a fraction, and some examples are listed below: 1. { 7.45 } = 7.45 – [ 7.45 ] = 7.45 – 7 =.45 2. { 8.00 } =.0 2.1.2 Modulus Concept Definition 3: Assuming that / is the arithmetic divide operator, and + is the add operator which has lower precedence than /, the Euclidean division algorithm for positive integers is shown below: A / B = Q + R / B where 0 R < B Q = [ A / B ] R / B = { A / B } Q is the quotient, and R is the. remainder. Definition 4: Integers A and B are said to be congruent of modulus N where N is an integer, if there exists an integer k, such that A - B = k * N where the - sign denotes the subtract operator, and * denotes the multiply operator
eBook - PDF- Camilla Gilmore, Silke M. Göbel, Matthew Inglis(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
7 Number Systems Which of these fractions is the numerically larger? 3 5 2 3 7.1 Going beyond the natural numbers In previous chapters we have looked at how children develop their understanding of numbers. But we have focused on a particular kind of number, typically referred to as whole numbers, counting numbers, or (the term mathematicians use) natural numbers: 1, 2, 3, 4, and so on. But there are many other types of numbers that arise in day-to-day life that children also need to understand: zero, negative numbers, fractions, and irrational numbers, for instance. And mathematicians have developed even more complicated systems of numbers that go beyond those needed in daily life. Students who go on to a scientific or engineering career may need to understand imaginary and complex numbers, and perhaps even infinite numbers. The cognitive processes involved in expanding the range of numbers a child can deal with are less well understood than those involved in developing understanding of the natural numbers, but we do have some ideas of how these shifts may occur. Before reviewing what is known about these issues, we briefly present a mathematical introduction to numbers beyond the naturals. Suppose we have developed a good understanding of the numbers 1,2,3,4, and so on. Mathematicians refer to these numbers as the naturals, and denote them 7.1 Going beyond the natural numbers 121 with the symbol N. Further suppose that we have learned to add, multiply, subtract and divide them. It soon becomes clear that there are some questions we simply cannot answer. For instance, what is 5 − 5? Or, to put it another way, which natural number n solves the equation 5 − 5 = n? Unfortunately there isn’t one. If we want to answer questions of this type, we need to expand our number system from the naturals to also include a number that represents zero. If we want to solve the equation 5 − 7 = n, then we will need negative numbers.
eBook - PDF- Gideon Langholz, Abraham Kandel;Joe L Mott;;(Authors)
- 1998(Publication Date)
- WSPC(Publisher)
2 Number Systems, Arithmetic Operations and Codes 2.1 INTRODUCTION Digital systems are characterized by signals restricted to two possible values which are represented by an alphabet of only two characters, commonly denoted by 0 and 1. Binary data representations are therefore of fundamental importance to the analysis and design of digital systems. We begin this chapter by considering numerical data representations. Working with digital systems usually requires familiarity with several Number Systems: Decimal Binary Octal Hexadecimal Binary-coded decimal (BCD) The binary number system is the most natural to use in a digital system. Sometimes, however, it is more convenient to employ the other Number Systems, in particular the familiar decimal number system. In such cases, the numbers are manipulated within the number system in use but are, nevertheless, represented as binary numbers within the digital system. We will introduce arithmetic operations to provide a basis for understanding how digital systems handle numbers in arithmetic computations. We will discuss unsigned and signed binary number representations and consider the four basic arithmetic operations: 21 22 CHAPTER 2 Number Systems addition, subtraction, multiplication, and division. Binary coding is another important issue. Numbers can be represented in a variety of codes, some of which are more useful than others in particular situations, and can be operated upon. Nevertheless, since arithmetic (numerical) data are not the only data type used, we also discuss codes for representing nonnumerical data, such as letters of the alphabet. Finally, we provide a brief introduction to the fundamentals of protecting against various transmission errors when data are transferred over communication links, and the principles of coding control instructions in a digital system.
eBook - PDFModern Syllabus Algebra
The Commonwealth and International Library: Mathematical Topics
- D.G.H.B. Lloyd, C. Plumpton(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER 6 Number Systems 1. Peano's Axioms Elementary arithmetic could be defined as the manipulation of numer-ical quantities. It is therefore surprising that many arithmetic courses pay little attention to the nature of these quantities. 3, + ^ 3 , 34, ^ 3 and n are all regarded as numbers without recognising that each of these is, in fact, a different type of number. Using the language developed earlier in this book, each of these belongs to a different set and is combined with other elements by different operations. We shall study each type of num-ber separately and try to see how one set develops out of another. We deal first with numbers of the type 1, 2, 3, 4,..., the counting num-bers. These must have been our earliest mathematical experience both as individuals in our pre-school childhood and as the human race in the dawn of history. Doubtless long familiarity with these numbers enabled them to be accepted without much investigation. In the event it was not until the end of the nineteenth century that an Italian mathematician, Giuseppe Peano (1858-1932), studied the question What do we mean by a number system? and produced in answer a set of five axioms now known by his name. These may be stated as follows: A set TV is a natural number system if: P I There is a number 1 in it. P II Every number, a, in the system has a successor a* which is also in the system. P III The number 1 is not the successor of any number. P IV If two numbers have the same successor then the numbers them-selves are the same. 102 Number Systems 103 P V Any set which contains 1 and also the successor of each of its elements contains the whole natural number system. These axioms may be written more succinctly in symbols as P I 31eN PII VaeN 3a*eN P III 3 a e N; a* = 1 P IV a* = b*=>a = b P V 1 e M and VzeM 3 z* e M => M = N. The following should be noted concerning the natural number system defined in this way. First it is an infinite set; this follows from P II.
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Section 2.3 Number Lines and Numeral Systems 59 2.3 Number Lines and Numeral Systems Understand and use the properties of the number line for whole numbers. Understand the concept of base for a positional numeral system, and represent numbers using alternative numeral systems. The Number Line for Whole Numbers It is uncertain exactly when humans first learned the concept of numbers. Whenever it was, it was some time later when humans assigned names to different numbers. And it was even later when humans assigned symbols or numerals to those numbers. By the time the early civilizations developed, humans had established numeral systems. Within these systems, they could write, add, subtract, multiply, and divide numbers. They also knew that whole numbers could be ordered. In ancient Egypt, construction guidelines were labeled at regular intervals using whole numbers. The Number Line for Whole Numbers A number line is a line on which numbers are marked at intervals. The concept that numbers are ordered is independent of the symbols or numerals used to represent the numbers. Below are some properties of the number line that are important for elementary school students to know. Students can use a number line to compare and order whole numbers. 1. The dots or tic marks specify the whole numbers. They are placed at evenly-spaced intervals. 2. The numbers increase from left to right. 3. The dots or tic marks are labeled. 0 1 2 3 4 5 EXAMPLE 1 Comparing Whole Numbers Using a Number Line Compare 7 and 3. SOLUTION Graph 3 on a number line. Graph 7 on a number line. 3 is to the left of 7 on the number line. 0 1 2 3 4 5 6 7 So, 3 is less than 7, or 3 < 7. Standards Grades K–2 Number and Operations in Base Ten Students should understand place value. Grades 3–5 Number and Operations in Base Ten Students should generalize place-value understanding for multi-digit whole numbers. iStockphoto.com/susaro Copyright 2014 Cengage Learning.
eBook - ePub- Steven G. Krantz(Author)
- 2022(Publication Date)
- Chapman and Hall/CRC(Publisher)
5 Number SystemsDOI: 10.1201/9781003214564-5In this chapter we treat all the basic Number Systems. This includes the natural numbers, the integers, the rational numbers, the real numbers, the complex numbers, and beyond. We do this rigorously—certainly more rigorously than anything you have ever seen before.You will see right away in Section 5.1 that we begin with an axiom system for the natural numbers. You will have to read carefully to get the most from this treatment.5.1 The Natural Number System
Giuseppe Peano's axioms for the natural numbers are as follows. In this discussion, we will follow tradition and use the notation ′ to denote the “successor” of a natural number. For instance, the successor of 2 is 2′ . Intuitively, the successor of n is the numbern + 1. However addition is something that comes later; so we formulate the basic properties of the natural numbers in terms of the successor function.PEANO's AXIOMS FOR THE NATURAL NUMBERS- P11 ∈ ℕ.
- P2 Ifn ∈ ℕ, then.n ′∈ ℕ
- P3 There is no natural number n such that.n ′= 1
- P4 If m and n are natural numbers and if, thenm ′=n ′m = n.
- P5 Let P be a property. If
- P ( 1 )is true;
- P ( j ) ⇒ P (for everyj ′)j ∈ ℕ
thenP ( n )is true for everyn ∈ ℕ.As Suppes says in [SUP, pp. 121 ff.] , these axioms for the natural numbers are almost universally accepted (although E. Nelson [NEL] , among others, has found it useful to explore how to develop arithmetic without Axiom P5
eBook - ePub- Christine Hopkins, Ann Pope, Sandy Pepperell(Authors)
- 2013(Publication Date)
- David Fulton Publishers(Publisher)
n where the numerator is always one) are known as ‘unit’ fractions and were used mainly for sharing out resources, grain, beer, etc. All fractional representation was in terms of unit fractions, so for example, 3/4 was written as two separate unit fractions 1/2, 1/4. It is possible to obtain many different kinds of unit fractions by judicious use of various ‘natural’ fractions.The introduction by the Dutchman Simon Stevin in The Art of Tenths 1585 of a ‘complete’ decimal system where he developed symbols for decimal fractions and showed how to use these for counting and calculating, still did not persuade many people of the advantages of a unified system. Even now, many calculations are still done by goldsmiths, apothecaries and others using the traditional fraction systems; and astronomical measurements still use sexagesimal fractions (sixtieths).Futher ReadingJoseph, G. 1991 The crest of the Peacock: non-European roots of Mathematics London: TaurisKatz V.J. 1998 A History of Mathematics: an introduction Harlow: Addison-Wesley1.2 DEVELOPING THE NUMBER SYSTEM
NATURAL OR COUNTING NUMBERS
A number system requires an ordered set of number names and a symbol for writing each name. Around the world a rich variety of systems for naming and writing numbers was developed. One way of reducing the number of symbols is shown in the Mayan system where the symbol for 7 is a combination of the symbol for 5 and the symbol for 2.In the Aztec system the number of symbols you had to draw was reduced by inventing symbols for large numbers.CARDINAL AND ORDINAL
To count some objects in any of these systems you recite the number names in order whilst pointing to each object in turn. The number of the final object is the total number of objects.
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