Classifying Real Numbers Flashcards
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Questions and Answers

Which of the following numbers are rational? (Select all that apply)

  • √49 (correct)
  • √144 (correct)
  • √3,100
  • 28.788712...
  • 6.87441 (correct)

    • Which of the following numbers are irrational? (Select all that apply)

  • √36
  • √81
  • √17 (correct)
  • √44 (correct)

    • Which of the following are rational numbers? (Select all that apply)

  • 0.816¯¯¯ (repeat bar on 6) (correct)
  • 0.418302...
  • 2.63¯¯¯ (repeat bar on 3) (correct)
  • 2.515115111...
  • 0.125 (correct)
  • 0.29 (correct)

    • Which of the following sets does the number √49 belong to? (Select all that apply)

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

    Which of the following sets does the number 2 belong to? (Select all that apply)

    <p>Rational numbers</p> Signup and view all the answers

    Which of the following are real numbers? (Select all that apply)

    <p>−√81</p> Signup and view all the answers

    Which of the following is a whole number?

    <p>11</p> Signup and view all the answers

    Which of the following is an integer? (Select all that apply)

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

    Which of the numbers given are rational? (Select all that apply)

    <p>√16</p> Signup and view all the answers

    Which of the following numbers are irrational numbers? (Select all that apply)

    <p>−9.72827...</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>√49</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>32.74178¯¯¯ (repeat bar on 78)</p> Signup and view all the answers

    Which of the following sets does the number −√3 belong to? (Select all that apply)

    <p>Irrational numbers</p> Signup and view all the answers

    Which of the following is an integer?

    <p>9</p> Signup and view all the answers

    Which of the following is a whole number? (Select all that apply)

    <p>11</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>27.20058</p> Signup and view all the answers

    Write 9.279 as a ratio of two integers.

    <p>9279/1000</p> Signup and view all the answers

    What defines a rational number?

    <p>A number of the form p/q where p and q are integers and q≠0.</p> Signup and view all the answers

    What is a real number?

    <p>A number that is either rational or irrational.</p> Signup and view all the answers

    Can we simplify √-25?

    <p>No, there is no real number whose square is −25.</p> Signup and view all the answers

    Identify whether the following numbers are real or not real: √-169, √-64.

    <p>√-64 is a real number</p> Signup and view all the answers

    Which of the following are real numbers? (Select all that apply)

    <p>−√49</p> Signup and view all the answers

    Which of the following is a whole number? (Select all that apply)

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

    Which of the following sets does the number 2 belong to? (Select all that apply)

    <p>whole numbers</p> Signup and view all the answers

    Write 3.19 as a ratio of two integers.

    <p>319/100</p> Signup and view all the answers

    Write 4.399 as a ratio of two integers.

    <p>4399/1000</p> Signup and view all the answers

    Write −12 as a ratio of two integers.

    <p>-12/1</p> Signup and view all the answers

    Write as the ratio of two integers: 8.41.

    <p>841/100</p> Signup and view all the answers

    Given the numbers 0.29, 0.816 (repeat bar over 6), 2.515115111..., 2.63 (repeat bar over 3), 0.125, and 0.418302..., select all of the rational numbers.

    <p>0.816 (repeat bar over 6)</p> Signup and view all the answers

    Given the numbers √36, √44, √81, and √17, select all of the irrational numbers.

    <p>√44</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>8.21677</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>33.2406 (repeat bar over 6)</p> Signup and view all the answers

    Which of the following is an integer? (Select all that apply)

    <p>−10</p> Signup and view all the answers

    Which of the following is a whole number? (Select all that apply)

    <p>7</p> Signup and view all the answers

    Which of the following sets does the number 0.1453 (repeat bar on 53) belong to? (Select all that apply)

    <p>rational numbers</p> Signup and view all the answers

    Which of the following sets does the number 2 and 1/3 belong to? (Select all that apply)

    <p>rational numbers</p> Signup and view all the answers

    Which of the following sets does the number -√3 belong to? (Select all that apply)

    <p>irrational numbers</p> Signup and view all the answers

    Which of the following sets does the number 12/5 belong to? (Select all that apply)

    <p>rational numbers</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>√16</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>5.50233 (repeat bar on 233)</p> Signup and view all the answers

    Which of the following sets does the number 265 belong to? (Select all that apply)

    <p>real numbers</p> Signup and view all the answers

    Which of the numbers are irrational? (Select all that apply)

    <p>21.10675...</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>29.46660</p> Signup and view all the answers

    Which of the numbers given are rational? (Select all that apply)

    <p>18.12760 (repeat bar on 60)</p> Signup and view all the answers

    Which of the following is a whole number? (Select all that apply)

    <p>54</p> Signup and view all the answers

    Which of the following sets does the number −8 belong to? (Select all that apply)

    <p>rational numbers</p> Signup and view all the answers

    Which of the following are real numbers? (Select all that apply)

    <p>−√81</p> Signup and view all the answers

    Which of the following sets does the number −12.12532... belong to? (Select all that apply)

    <p>irrational numbers</p> Signup and view all the answers

    Study Notes

    Real Numbers Classification

    • Real numbers include both rational and irrational numbers.
    • Rational numbers can be expressed as a fraction ( p/q ) where ( p ) and ( q ) are integers, and ( q \neq 0 ).
    • Irrational numbers cannot be expressed as a fraction and have non-terminating, non-repeating decimal expansions.

    Integers and Whole Numbers

    • Integers are whole numbers and their negatives: (..., −3, −2, −1, 0, 1, 2, 3, ...).
    • Whole numbers encompass all non-negative integers: (0, 1, 2, 3,...).

    Identifying Rational Numbers

    • Rational numbers include:
      • Perfect squares, e.g., ( \sqrt{16} = 4 ), ( \sqrt{49} = 7 ).
      • Terminating decimals, e.g., 15.567, 0.29.
      • Repeating decimals, e.g., 0.816 (with repeating 6).
    • Non-repeating, non-terminating decimals are considered irrational.

    Recognizing Irrational Numbers

    • Examples include square roots of non-perfect squares (e.g., ( \sqrt{17}, \sqrt{44} )).
    • Decimal representations that do not terminate or repeat (e.g., 21.10675...).

    Writing Numbers as Ratios

    • Decimal numbers can be converted to fractions:
      • 9.279 can be expressed as ( \frac{9279}{1000} ).
      • Any integer, e.g., -12 can be written as ( \frac{-12}{1} ).

    Special Cases and Non-existent Real Numbers

    • ( \sqrt{-25} ) and other square roots of negative numbers do not represent real numbers as no real number squared gives a negative result.
    • Roots of negative numbers or complex numbers are not part of the real number set.

    Specific Examples to Classify

    • Numbers like ( 265 ) belong to whole numbers, integers, rational numbers, and real numbers.
    • Mixed numbers, like ( 2 \frac{1}{3} ), are rational and real.

    Repeated and Non-repeated Decimals

    • Decimal numbers, such as ( 31.7408\overline{8} ) and ( 21.90062\overline{62} ), are classified as rational because they repeat.
    • Non-repeating decimals, e.g., ( 4.381293...), are irrational.

    Conclusion

    • A thorough understanding of the definitions and examples of integers, whole numbers, rational and irrational numbers helps in identifying and classifying numbers accurately in various mathematical contexts.### Irrational and Rational Numbers
    • Irrational numbers cannot be expressed as fractions; their decimals do not terminate or repeat.
    • Examples include √44 and √17; both do not simplify to whole numbers.

    Identifying Rational Numbers

    • Rational numbers can be identified through their decimal representations, which either terminate or repeat.
    • Examples include:
      • 19.30764¯¯¯¯¯ (repeating decimal)
      • 35.45449 (terminating decimal)
      • 1 (perfect square) simplifies from √1.

    Perfect Squares and Rationality

    • Perfect squares simplify to whole numbers, classifying them as rational. For instance, √36 = 6 and √81 = 9.

    Real Numbers

    • Real numbers consist of rational and irrational numbers.
    • -√81 and -√49 are real numbers; however, square roots of negative numbers (e.g., √-196, √-121) are not real.

    Whole Numbers and Integers

    • Whole numbers include non-negative integers (0, 1, 2, 3, ...).
    • Integers include whole numbers and their negatives, exemplified by numbers like -4 and 6.

    Sets of Numbers

    • √49 = 7 belongs to multiple sets: whole numbers, integers, rational numbers, and real numbers.
    • The number 2 also belongs to all these sets for the same reasons.

    Identifying Rational vs. Irrational Decimals

    • Decimals such as 15.567 and 27.20058 are rational (terminate).
    • 21.10675... and √526 are irrational (do not terminate or repeat).

    Analyzing Decimal Patterns

    • A decimal that repeats qualifies as rational, reflected in numbers like 18.12270¯¯¯¯¯ and 32.74178¯¯¯¯¯.
    • Conversely, decimals without repeating patterns or that don’t terminate, like 1.421490..., are classified as irrational.

    Additional Insights

    • 0.1453¯¯¯¯¯ is rational as it is a repeating decimal but is not classified as a whole number or integer.
    • The square root of a negative number (e.g., -√3) does not fall into rational or integer categories but is categorized as irrational and real.

    Summary of Sets

    • Categories for numbers based on their decimal forms or simplification include:
      • Whole Numbers: Non-negative integers.
      • Integers: Whole numbers and their negatives.
      • Rational Numbers: Terminating or repeating decimals.
      • Irrational Numbers: Non-repeating, non-terminating decimals.
      • Real Numbers: All rational and irrational numbers.

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    Description

    This quiz focuses on classifying real numbers, including integers and whole numbers. Use these flashcards to test your knowledge and understanding of the different categories of real numbers. Perfect for students looking to enhance their math skills.

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