Real Numbers

12 Key excerpts on "Real Numbers"

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  Quantity & Number Mathematics

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  Real Numbers are commonly opposed both to integers, such as 5 (whole numbers that express discrete rather than continuous quantities) and complex numbers (mathematical constructs that include Real Numbers as a special case). Real Numbers can be divided into rational numbers , such as 42 and −23/129, and irrational numbers, such as pi and the square root of two. A real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue indefinitely. The Real Numbers are sometimes thought of as points on an infinitely long number line. These descriptions of the Real Numbers, while intuitively accessible, are not sufficiently rigorous for the purposes of pure mathematics. The discovery of a suitably rigorous definition of the Real Numbers—indeed, the realization that a better definition was needed—was one of the most important developments of 19th century mathematics. Popular definitions in use today include equivalence classes of Cauchy sequences of rational numbers; Dedekind cuts; a more sophisticated version of decimal repre-sentation; and an axiomatic definition of the Real Numbers as the unique complete Archimedean ordered field. These definitions are all described in detail below. Real Numbers can be thought of as points on an infinitely long number line. ________________________ WORLD TECHNOLOGIES ________________________ Basic properties A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real Numbers are used to measure continuous qua-ntities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823122147… The ellipsis (three dots) indicate that there would still be more digits to come. More formally, Real Numbers have the two basic properties of being an ordered field, and having the least upper bound property.

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  Story Of Numbers, The

  • Asok Kumar Mallik(Author)
  • 2017(Publication Date)
  • World Scientific Publishing Company

    (Publisher)

  Chapter 3

  Real Numbers

  In Figure 2.1 we have seen that integers indicated by discrete points on a line, called the number line, have gaps between them. The concept of numbers is extended to include all the points on the number line. All such numbers to be discussed in this chapter are called Real Numbers and the number line will be referred to as the real number line. The Real Numbers have two basic classifications, viz., rational and irrational. Just like reference [7] for prime numbers, a real number dictionary and online internet-based reference for special Real Numbers is also available.f

  3.1.Rational Numbers

  Arithmetical operations like addition and multiplication on positive integers result in positive integers. It has already been mentioned that the process of subtraction of a larger integer from a smaller one necessitated the creation of negative integers. The first non-integers which were included to extend the concept of numbers are rational numbers. A rational number, r, is created by the process of division as

  where p and q are integers.

  If q > p, then we call the rational number, f, a rational fractiong

  For p > q, the rational number r = I + f, where I is the integer part and f is the fractional part. Integers are also rational numbers with q = 1 and f = 0. If we consider the segment of the real number line between 0 and 1, then all the rational fractions, f, fail to cover all the points. Similarly, the segment between any two integers cannot be filled up using only rational numbers [9]. To fill up the real number line in a continuous manner, we need the irrational numbers also.

  3.2.Irrational Numbers

  Euclid’s Elements contains a simple proof that is an irrational number. The number (if we agree to call it a number) cannot be expressed as a rational number. This can be proved by contradiction. Assuming this number to be rational, let it be expressed in its lowest form as , where p and q

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  Intermediate Algebra

  • Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  2 5 1.414213562373095 . . . 3 5 1.73205080756887 . . . p 5 3.14159265358979 . . . The set of Real Numbers is composed of the rational numbers along with the irrational numbers. Every real number is either a rational number or an irrational number. The following tree diagram summarizes the various classifications of the real number system. Real Numbers Rational numbers Irrational numbers Integers Nonintegers 0 We can trace any real number down through the diagram as follows: 7 is real, rational, an integer, and positive 2 2 3 is real, rational, noninteger, and negative 7 is real, irrational, and positive 0.38 is real, rational, noninteger, and positive Remark: We usually refer to the set of nonnegative integers, 5 0, 1, 2, 3, . . . 6 , as the set of whole numbers , and we refer to the set of positive integers, 5 1, 2, 3, . . . 6 , as the set of natural numbers . The set of whole numbers differs from the set of natural numbers by the inclusion of the number zero. The concept of subset is convenient to discuss at this time. A set A is a subset of a set B if and only if every element of A is also an element of B . This is written as A # B and read as “ A is a subset of B .” For example, if A 5 5 1, 2, 3 6 and B 5 5 1, 2, 3, 5, 9 6 , then A # B because every element of A is also an element of B . The slash mark denotes nega-tion, so if A 5 5 1, 2, 5 6 and B 5 5 2, 4, 7 6 , we can say that A is not a subset of B by writ-ing A s B . Figure 1.3 represents the subset relationships for the set of Real Numbers. Refer to Figure 1.3 as you study the following statements, which use subset vocabulary and sub-set symbolism. 1. The set of whole numbers is a subset of the set of integers. 5 0, 1, 2, 3, . . . 6 # 5 . . . , 2 2, 2 1, 0, 1, 2, . . . 6 2. The set of integers is a subset of the set of rational numbers. 5 . . . , 2 2, 2 1, 0, 1, 2, . . . 6 # 5 x 0 x is a rational number 6 3. The set of rational numbers is a subset of the set of Real Numbers.

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  Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Real Numbers include whole numbers, fractions, decimals, and other numbers that are not as familiar to us as these. Equivalent Fractions on the Number Line As we proceed through the chapter, from time to time we will review some of the major concepts associated with fractions. To begin, here is the formal definition of a fraction: The number line can be used to visualize fractions. Recall that for the fraction a _ b , a is called the numerator and b is called the denominator. The denominator indicates the number of equal parts in the interval from 0 to 1 on the number line. The numerator indicates how many of those parts we have. If we take that part of the number line from 0 to 1 and divide it into three equal parts , we say that we have EXAMPLE 1 VIDEO EXAMPLES SECTION 1.1 Note: There are other numbers on the number line that you may not be as familiar with. They are irrational numbers such as π , √ — 2, √ — 3. You will see these later in the book. -4 -3 -2 -1 0 1 2 3 4 -3.5 -1 1 4 1 2 3 4 2.5 The number associated with a point on the real number line is called the coordinate of that point. coordinate DEFINITION The numbers that can be represented with points on the real number line are called Real Numbers . Real Numbers DEFINITION If a and b are Real Numbers with b ≠ 0, then the expression a __ b is called a fraction . The top number a is called the numerator , and the bottom number b is called the denominator . The restriction b ≠ 0 keeps us from writing an expression that is undefined. (As you will see later in this chapter, division by zero is not allowed.) fraction DEFINITION 1.1 The Real Numbers 5 divided it into thirds (Figure 1). Each of the three segments is 1 _ 3 (one third) of the whole segment from 0 to 1. Two of these smaller segments together are 2 _ 3 (two thirds) of the whole segment. Three of them would be 3 _ 3 (three thirds), or the whole segment. Let’s do the same thing again with six equal divisions of the segment from 0 to 1 (Figure 2).

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  Mathematics

  A Second Start

  • S. Page, J Berry, H Hampson(Authors)
  • 2002(Publication Date)
  • Woodhead Publishing

    (Publisher)

  We therefore state them without proof. However, you will have an opportunity to verify some of them in the next chapter. 1. The Real Numbers consist of The natural numbers (positive integers) together with zero The negative integers Rational numbers (positive and negative) Irrational numbers (positive and negative) Together with four binary operations +, - and *, +. 2. The complete set of Real Numbers is closed under all these operations but subsets may not be closed. For example, a real number + a real number = a real number (always!). But the set of irrational numbers is not closed under multiplication, since, for example, V2 χ V2 = 2, which is rational! 3 Similarly, the set of integers is not closed under division since, for example, — 4 is not an integer. 3. The Real Numbers are ordered. For example: -3 < -2 < -/2 < -1 < -0.5 < 0 < 0.5 < 1< /3 < 2 < e < 3 < π < 4 and so on. 4. + and x are commutative on any subset - and + are not + and χ are associative on any subset - and -s-are not This means that we can add and multiply numbers in any order but changing the order of subtractions and divisions will change the answer! 5. χ is distributive over + and - (in both directions) So, if we want to multiply a list of numbers by a certain number, every member of the list is multiplied by that number. + is distributive over + and - (in one direction only) SUMMARY

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  Beginning and Intermediate Algebra

  A Guided Approach

  • Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Because the set of natural numbers and the set of whole numbers are included within the set of integers, these sets are called subsets of the set of integers. Integers cannot describe every real-life situation. For example, a student might study 3 1 2 hours, or a TV set might cost $217.37. To describe these situations, we need fractions, more formally called rational numbers. We cannot list the set of rational numbers as we have listed the previous sets in this section. Instead, we will use set-builder notation . This notation uses a variable (or vari-ables) to represent the elements in a set and a rule to determine the possible values of the variable. THE SET OF INTEGERS 5 . . . , 2 5, 2 4, 2 3, 2 2, 2 1, 0, 1, 2, 3, 4, 5, . . . 6 they are denoted with a 2 sign. For example, a debt of $1,500 can be denoted as 2 $1,500, Copyright 2015 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1.1 Real Numbers and Their Graphs 5 Unless otherwise noted, all content on this page is © Cengage Learning. p < 3.141592654 Using a scientific calculator, press p . Using a graphing calculator, press 2nd ^ ( p ) ENTER . Read < as “is approximately equal to.” 2 < 1.414213562 Using a scientific calculator, press 2 . Using a graphing calculator, press 2nd x 2 ( ! ) 2 ENTER . If we combine the rational and the irrational numbers, we have the set of Real Numbers . The previous notation is read as “the set of all numbers x such that x is either a rational number or an irrational number.” Figure 1-1 illustrates how the various sets of numbers are interrelated.

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  Mathematical Practices, Mathematics for Teachers

  Activities, Models, and Real-Life Examples

  • Ron Larson, Robyn Silbey(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 350 Chapter 9 Rational Numbers and Real Numbers Completeness of the Real Number Line Each of the number lines you have studied so far has holes or gaps. For instance, the rational number line has a gap at the point corresponding to √ — 2. -3 3 2 2 1 0 The rational number line -1 -2 In fact, the rational number line has infinitely many gaps because there are infinitely many irrational numbers. The real number line, on the other hand, has no holes or gaps. In this sense, the set of Real Numbers is called “complete” because there are no other numbers that can be “squeezed” onto the real number line. Completeness of the Real Number Line The real number line is called complete because it has no holes or gaps. This implies the following two statements. 1. Every decimal representation (terminating, repeating, or nonrepeating) corresponds to exactly one point on the real number line. 2. Every point on the real number line corresponds to exactly one decimal representation. EXAMPLE 2 Solving a Real-Life Problem Use a number line to estimate the length of the diagonal of the square tile. 1 ft SOLUTION Place the tile on a number line so that a 1-foot side lies exactly on the number line, as shown. Be sure to use the appropriate tick marks. Next, rotate the tile about zero on the number line so that the diagonal aligns with the number line. Then use the number line to estimate the length of the diagonal. 2 1 0 2 1 1.4 0 So, the length of the diagonal of the square tile is about 1.4 feet. Note that the exact length of the diagonal is on the number line, but it is not practical to accurately determine the length past one or two decimal places. A more straightforward approach is to use the Pythagorean Theorem that you studied in high school geometry. This states that a 2 + b 2 = c 2 for a right triangle shown below.

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  Emergence Of Number, The

  • John Newsome Crossley(Author)
  • 1987(Publication Date)
  • World Scientific

    (Publisher)

  . . 1 a < a'', a < b'' , a < c'' ,. . . J The fact that all such Real Numbers could then be compared would, however, have required a considerable amount of technical argument and development to show, for example, that the Real Numbers can be ordered and 143 Real Numbers that all are comparable (under the obvious definition, or should we say extension, of < ). Because Hamilton insisted on a law for defining his sets of ratios which would determine a real number, it is not clear that Hamilton in fact intended or believed that he was treating all Real Numbers. (Kline describes Hamilton's work as unfinished.) The problem of treating continuously variable quantities had been attacked by Descartes. His use of letters in what we now call Cartesian or co-ordinate geometry allowed him to talk of operations on varying lengths. For example, a/b is determined by the geometric solution of the proportionality a : b = x : 1. a is to b as a/b is to 1 (Descartes [1954], p.3). This construction is in Euclid. It is clear that this works for all lengths a and b. However, it was not until the second half of the nineteenth century that a satisfactory theory of real (including irrational) numbers developed. The Greeks had deftly sidestepped the problem of incommensurables by substituting magnitude for number (wilder [19731, p.lll). The nineteenth century growth of analysis and the study of differential equations demanded a solution. Ch. Meray was the first to fina a purely arithmetic meaning for the expression irrational number (Hoik [1904], p.148)' 1 * Meray defines Real Numbers as equivalence classes of Cauchy sequences; at least that is the modern descriptions. He uses sequences of the form v , permitting several indices unlike our present practice of using only one index. He says: ...the necessary and sufficient condition for a given variable v to tend to a commensurable or incommensurable limit is that it be convergent, that is to say, that the difference v - v ■,n,...

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  The How and Why of One Variable Calculus

  • Amol Sasane(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Real Numbers What about the construction of the real number system R ? In this book, we treat the real number system R as a given. But one might wonder if we can take the existence of Real Numbers on faith alone. It turns out that a mathematical proof of its existence can be given. There are several ways of doing this. One is by a method called ‘completion of Q ’, where one considers ‘Cauchy sequences’ in Q , and defines R to be ‘equivalence classes of Cauchy sequences under a certain equivalence relation’. We refer the interested student to [ S2 , Prob-lem 1, p. 588] or [ R , Exercises 24, 25, p. 82] for details about this. Another way, which is more intuitive, is via ‘(Dedekind) Cuts’, where we identify each real number by means of two sets A and B associated with it: A is the set of rationals less than the real number we are defining, and B is set of rational numbers at least as big as the real number we are trying to identify. In other words, if we view the rational numbers lying on the number line, and think of the sets A and B (described above) corresponding to a real number, then this real number is the place along this rational number line where it can be cut, with A lying on the left side of this cut, and B lying on the right side of this cut. See Figure 1.5. More precisely, a cut ( A , B ) in Q is a pair of subsets A , B of Q such that A B = Q , A = ∅ , B = ∅ , A B = ∅ , if a ∈ A and b ∈ B then a < b , and A contains no largest element. R is then taken as the set of all cuts ( A , B ) . Here are two examples of cuts: ( A , B ) = { r ∈ Q : r < 0 } , { r ∈ Q : r ≥ 0 } (giving the real number ‘0’) ( A , B ) = { r ∈ Q : r ≤ 0 or r 2 < 2 } , { r ∈ Q : r > 0 and r 2 ≥ 2 } (‘ √ 2’). A B Figure 1.5 Dedekind cut. 28 THE HOW AND WHY OF ONE VARIABLE CALCULUS It turns out that R is a field containing Q , and it can be shown to possess the Least Upper Bound Property.

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  Numbers, Sequences and Series

  • Keith Hirst(Author)
  • 1994(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  We now look at an example of an algebraic relationship involving rational and irrational numbers. rhe Real Numbers 81 Example 6 Suppose that a and b are rational numbers satisfying a + bJ2 = o. Prove that -a = b = O. If b -1= 0 then we can rearrange this equation to give J2 ::::: +alb. But -alb is a rational number. This is a contradication since J2 is irrational. So we must have b = 0, and therefore a = O. Let us now generalize this a little and consider the equation a + bJ2.+ cJ3 = 0, where a, b, c are rational numbers. Rearrange the equation by subtracting a from both sides. Squaring both sides of the new equation will give a relationship similar to that of Example 6 but involving Use this to show that we must have a = = C ::::: O. EXERCISES 5.4 1. Find a rational number between 71 2 and J2. 2. Find an irrational number between 2/3 and 3/4. 3. Decide whether each of the following is always true, sometimes true, or always false: (i) rational x rational = rational, (ii) irrational + irrational = irrational, (iii) rational + irrational = irrational, (iv) rational x irrational = irrational, (v) irrational x irrational = irrational. Summary The topics discussed in this chapter are among the most conceptually demanding in basic mathematics. The brief historical sketch in §5.2 indicates the difficulties involved in developing the ideas to the form we have them today. We mentioned that a well-developed theory of Real Numbers is necessary for a theory of limits, and we shall look at this in relation to sequences in Chapter 7. The ideas are also vital for a proper foundation for the theory of functions and calculus, and are discussed in a sequel to this book, on Analysis. EXERCISES ON CHAPTER 5 1. For each of the sets described below, say whether it is bounded above, and/or bounded below, and if so what its least upper bound and/or greatest lower

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  Advanced Calculus

  Theory and Practice

  • John Petrovic(Author)
  • 2013(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  2 Real Numbers In the previous chapter we used some very powerful results about sequences, such as the Monotone Convergence Theorem (Theorem 1.4.7) or Cauchy’s Test (Theorem 1.6.6). In this chapter, our goal is to prove these theorems. When Cauchy did that in Cours d’analyse he took some properties of Real Numbers as self-evident. In the course of the 19th century it became clear that these needed to be proved as well, and for that it was necessary to make a precise definition of Real Numbers. This task was accomplished around 1872 by the independent efforts of Dedekind, Cantor, Heine, and M´ eray. 2.1 Axioms of the Set R In any area of mathematics, statements need to be proved, and this always involves the use of previously established results. This approach, the deductive method , has as its foundation a set of axioms , from which other assertions can be derived. This is a modern way of thinking. In the 19th century, mathematicians were more concerned with the construction of Real Numbers from the rationals than finding a system of axioms that would capture the essence of the set R . These were important efforts and we will return to them later. Right now, we will fast forward to the year 1900, and a paper [65] written by a German mathematician David Hilbert (1862–1943). In this paper he gave a list of axioms that characterize the Real Numbers. What were Hilbert’s axioms for R ? The first group of axioms took care of the usual operations on Real Numbers and their properties. These algebraic properties mean that the set of Real Numbers R , together with the operations of addition and multiplication, is a field. We list them in Table 2.1. While the first group of axioms deals with equalities, the second group is all about inequalities. The rules concerning the relation ≤ are listed in Table 2.2. Together, Field Axioms and Order Axioms make the set R an ordered field . However, they are not sufficient to describe R , and Hilbert was forced to include a third group.

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  From Numbers to Analysis

  • Inder K Rana(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  Chapter 6 PROPERTIES OF Real Numbers 6.1 Historical comments As stated in section 4.1, by the end of the 18th century mathe-maticians were worried about the tremendous obscurity in various branches of analysis. Efforts were made to give rigorous proofs of properties of functions on Real Numbers. The concept of func-tion at that time was understood as a relation between variables in terms of algebraic equations. The need to define a function precisely was felt during the second half of the eighteenth century when the works of Jean le d'Alembert (1717-83), Leonhard Euler (1707-83) and Daniell Bernoulli (1700-82) on the vibrating string problem started a debate about the concept of a function. By the end of the 18th century, the accepted concept of a function was as proposed by Euler in 1755: If some quantities depend upon the others in such a way as to undergo variations when the latter are varied, then the former are called functions of the latter. How-ever, it was still believed that the graph of every function could be traced from a 'free motion of the hand', i.e., functions were be-lieved to be 'piecewise smooth'. Interest in the notion of function 233 234 Ch. 6: Properties of Real Numbers was revived with the work of Joseph Fourier (1768-1830) on the 'Analytic Theory of Heat' in 1807. It was only in 1837 that Gustav Lejeune-Dirichlet (1805-59) gave the definition of a function, em-ployed most often now. He also gave the example of the function / : [0,1] —> IR given by f(x) := 1 if x is a rational, and f(x) := 0 if x is an irrational. This is an example of a function which is neither defined by a formula nor could its graph be drawn. This exam-ple not only crystallized the concept of a function, but also made mathematicians treat continuous and discontinuous functions with equal vigor, and later this motivated Bernhard Riemann (1826-66) to define the notion of integral.

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