Real Numbers and Their Properties
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Questions and Answers

What is a real number?

  • A number that can be expressed only as a whole number.
  • A number that only has a finite decimal representation.
  • A number that can be represented as a point on the real line. (correct)
  • A number that can be expressed as a fraction.

    • What is the completeness property of real numbers primarily concerned with?

  • The representation of numbers as decimals.
  • The lack of gaps in the real number line. (correct)
  • The ability to divide real numbers by zero.
  • The ability to combine real numbers using various operations.

    • Which of the following describes a rational number?

  • A number that is always whole and positive.
  • A number that cannot be expressed as a fraction.
  • A number with a finite decimal expansion only.
  • A number that can be expressed as a fraction m/n where n ≠ 0. (correct)

    • When multiplying an inequality by a negative number, what happens to the direction of the inequality?

    <p>It is reversed.</p> Signup and view all the answers

    Which of the following is NOT included in the natural numbers?

    <p>0</p> Signup and view all the answers

    What is the result of dividing by zero in the context of real numbers?

    <p>It results in undefined behavior.</p> Signup and view all the answers

    Which of these can be expressed in the form of m/n where n ≠ 0?

    <p>−4/5</p> Signup and view all the answers

    What does the symbol R represent in mathematics?

    <p>The real number system or the real line.</p> Signup and view all the answers

    If a < b and c > 0, what conclusion can be drawn?

    <p>ac &lt; bc</p> Signup and view all the answers

    Which of the following statements regarding infinite decimal expansions is true?

    <p>Some numbers have two representations.</p> Signup and view all the answers

    Study Notes

    Real Numbers

    • Real numbers can be expressed in decimal form and include values such as -3/4, 1/3, and √2.
    • Infinite decimal expansions, such as 0.999..., represent the same real number as 1.000... .
    • Real numbers can be represented as points on a number line, known as the real line, denoted by the symbol R.
    • Properties of real numbers are categorized into algebraic properties, order properties, and completeness.

    Algebraic Properties

    • Real numbers can be added, subtracted, multiplied, and divided (except by 0).
    • Division by zero is undefined.

    Order Properties

    • If a, b, and c are real numbers, certain rules for inequalities apply:
      • a < b implies a + c < b + c.
      • a < b implies a - c < b - c.
      • a < b and c > 0 implies ac < bc.
      • a < b and c < 0 implies bc < ac, with the special case that a < b implies -b < -a.
      • If a > 0, then 1/a > 0.
      • For both positive or negative a and b, a < b implies 1/a < 1/b.
    • Multiplying by a positive number preserves an inequality, while multiplying by a negative number reverses it.

    Completeness Property

    • The completeness property ensures there are no "holes" in the real numbers, which is crucial for the concept of limits.
    • This property is foundational for many calculus theorems.

    Special Subsets of Real Numbers

    • Natural numbers include 1, 2, 3, 4, and so on.
    • Integers encompass 0, positive, and negative whole numbers (±1, ±2, ±3, ...).
    • Rational numbers are numbers expressible as a fraction m/n, where m and n are integers and n ≠ 0. Examples include 1/3, -9, and 4/200.

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    Quiz Team

    Description

    Explore the fascinating world of real numbers through this quiz. Understand their algebraic and order properties, as well as how they can be represented on a number line. Test your knowledge on concepts like infinite decimal expansions and the significance of division by zero.

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