Classifying Real Numbers Flashcards

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.