Number Line

5 Key excerpts on "Number Line"

• 

  eBook - ePub

  Teaching Mathematics Using Interactive Mapping

  • Sandra L. Arlinghaus, Joseph J. Kerski, William C. Arlinghaus(Authors)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  https://pro.arcgis.com/en/pro-app/latest/help/layouts/scale-bars.htm .

  Different Classes of Numbers Positioned on the Number Line

  We note these geometric characteristics of “lines” so that when we use lines to visualize number classes, we match their fundamental geometric characteristics to their fundamental numerical characteristics. Thus:

  • Use a ray to represent the natural numbers. Label the starting point of the ray with the numeral 1 and continue to move away from the starting point spacing successive numerals evenly along the visible extent of the ray, which extends infinitely in one direction.
  • Use a ray to represent the whole numbers. Label the starting point of the ray with the numeral 0 and continue to move away from the starting point spacing successive numerals evenly along the visible extent of the ray.
  • Use a line to represent the integers. Label a central point as 0 and move away from zero to the left, in the negative direction, spacing successive numerals evenly along the visible extent of the line, and to the right, in the positive direction, spacing successive numerals evenly along the visible extent of the line.
  • Rational numbers can be represented along a line. But, perhaps of particular interest is to represent a part of the set using a line segment. Choose any two rational numbers and connect them with a line segment and indicate selected intermediate values, using even or uneven spacing as desired. Remember: between any two rational numbers there is always another one.
  • Irrational numbers. These numbers are not rational. They can be placed on the Number Line for the rational numbers only in approximate form. For example, pi is an irrational number; one approximation to its numerical position is 22/7 which might also be used as an approximation to its geometric position on the Number Line. The value of 3.1416 is another commonly used value to estimate pi. What is critical to note, for both the numerical and geometric form of irrational numbers is that they are NOT rational; thus, any rational characterization of them is, necessarily, only an approximation. There is no “best” rational approximation for any irrational number; if there were, it would be the rational number itself (and it is not)

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  A Mathematical Bridge

  An Intuitive Journey in Higher Mathematics

  • Stephen Hewson(Author)
  • 2009(Publication Date)
  • WSPC

    (Publisher)

  These two ideas are so strongly interwoven that we adopt the hypothesis as a definition: an infinite mathematical line is simply a representation of all of the points of R , and only those points. This is a very useful visualisation, but is of lit-tle use in questions concerning the explicit construction of the various real numbers corresponding to different points on the line. Some real numbers can be constructed rather simply: the quadratic equation x 2 = 2 provides us with the real number which we call √ 2, and any rational number has a very simple representation n/m , which is the solution x to the linear equa-tion mx = n . How far can we take these ideas? How many real numbers can we construct starting from the integers and using only basic algebraic operations? We make a definition: • An algebraic number is any real number which is the solution to a polynomial equation with integer coefficients. Clearly all rational numbers, and all numbers √ n , where n is a natural number, are algebraic numbers. Furthermore, it is also possible to show, using a little theory of polynomials, that adding, subtracting, multiplying and dividing pairs of non-zero algebraic numbers yields numbers which are also algebraic. Therefore the set of all algebraic numbers forms a field and may thus be considered as a well-defined number system. Although it is a pleasant, mathematically clean notion that all of the real numbers could be generated in this way from the integers, it turns out that the real number Numbers 83 system is far more complex than simply the set of algebraic numbers. There exist real numbers which are not algebraic. Since such numbers transcend the algebraic construction, they are called transcendental : • A transcendental number is any real number which is not algebraic. We have actually met one of these transcendental numbers already: π .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  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).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Intermediate Algebra

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

    (Publisher)

  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. Unless otherwise noted, all content on this page is © Cengage Learning. Chapter 1 • Basic Concepts and Properties 22 Before we review the four basic operations with real numbers, let’s briefly discuss some concepts and terminology we commonly use with this material. It is often helpful to have a geometric representation of the set of real numbers as indicated in Figure 1.4. Such a repre-sentation, called the real Number Line , indicates a one-to-one correspondence between the set of real numbers and the points on a line. In other words, to each real number there cor-responds one and only one point on the line, and to each point on the line there corresponds one and only one real number. The number associated with each point on the line is called the coordinate of the point. -1 -2 -3 -4 -5 0 1 2 3 4 5 -π --π 1 2 1 2 2 2 Figure 1.4 Many operations, relations, properties, and concepts pertaining to real numbers can be given a geometric interpretation on the real Number Line. For example, the addition problem ( 2 1) 1 ( 2 2) can be depicted on the Number Line as in Figure 1.5. -1 -2 -3 -1 -4 -5 0 1 2 3 4 5 ( -1) + ( -2) = -3 -2 Figure 1.5 The inequality relations also have a geometric interpretation. The statement a . b (which is read “ a is greater than b ”) means that a is to the right of b , and the statement c , d (which is read “ c is less than d ”) means that c is to the left of d as shown in Figure 1.6. The symbol # means is less than or equal to , and the symbol $ means is greater than or equal to . The property 2 ( 2 x ) 5 x can be represented on the Number Line by following the se-quence of steps shown in Figure 1.7.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  With regards to the irrational numbers, we have to go beyond these operations.

  Shortly, we will learn that between any two rational numbers, there is always another rational number. A similar statement is true in the case of irrational numbers. (This is known as the property of denseness , which is studied later in this chapter.) Thus, if we plot only rational numbers on the Number Line, then there will be infinite number of holes throughout the line. These unoccupied positions must represent irrational numbers. Geometric constructions can be used to find points corresponding to certain irrational numbers, such as and so on. Points corresponding to other irrational numbers can be found by using decimal approximations.

  Every irrational number can be associated with a unique point on the x -axis, and every point that does not correspond to a rational number can be associated with an irrational number. This fact is guaranteed by the axiom of completeness and is discussed later at the end of this chapter.

  Note: We are familiar with the simpler properties of real numbers. It is now proposed to discuss some other properties of real numbers, which are not obvious. (This study will be found useful for building up necessary terminology, required for defining the “concept of limit ” in Chapters 7a and 7b.)

  The beauty and power of mathematics can be appreciated only if the properties of real numbers are properly understood. We give below the necessary material to make the study systematic and interesting. This material should be sufficient to meet the study requirements of this book and also serve as a good background for studying these concepts at higher levels.

  Sign up to read

  Learn more about book