Questions and Answers
Which of the following numbers is irrational?
The decimal expansion of an irrational number is non-terminating and non-repeating.
True
Is $0.25$ a rational or irrational number? Explain why.
Rational, because it can be expressed as the fraction $\frac{1}{4}$.
A number that can be expressed as a fraction of two integers is called a __________ number.
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Match the following numbers with their classification (Rational or Irrational):
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Which number is represented as a repeating decimal and is therefore rational?
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Study Notes
Rational and Irrational Numbers
- Rational numbers can be expressed as a fraction of two integers (e.g., ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b \neq 0 )).
- Irrational numbers cannot be represented as fractions and have non-terminating, non-repeating decimal expansions.
- Examples of rational numbers include ( \frac{5}{8} ), ( \frac{12}{7} ), and ( 0.25 ).
- Examples of irrational numbers include ( \sqrt{2} ) and ( \pi ).
Classification of Numbers
- A number like ( \sqrt{2} ) is classified as irrational because it cannot be expressed as a fraction and its decimal form continues without repeating.
- The number ( 4.5 ) is rational, being expressible as ( \frac{9}{2} ).
- ( 0.6666\ldots ) is a repeating decimal, thus classified as a rational number.
True or False Assertions
- ( \frac{12}{7} ) is a rational number (false for it being irrational).
- Irrational numbers have decimal expansions that are non-terminating and non-repeating (true).
- Every fraction is indeed a rational number (true).
Key Examples and Explanations
- ( 0.25 ) is a rational number because it can be expressed as ( \frac{1}{4} ).
- ( \frac{8}{3} ) is rational as it is a fraction of two integers (8 and 3).
- ( \sqrt{11} ) is irrational due to it not being expressible as a fraction of integers and having a non-repeating, non-terminating decimal expansion.
Fill in the Blanks
- A number that can be expressed as a fraction of two integers is called a rational number.
- The number ( \sqrt{9} ) is an example of a rational number (specifically equal to 3).
- The decimal representation of an irrational number does not terminate or repeat.
Matching Classifications
- ( \frac{5}{2} ): Rational
- ( \sqrt{8} ): Irrational
- ( 0.2 ): Rational
- ( \pi ): Irrational
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Description
Test your understanding of rational and irrational numbers with this quiz designed for 10th grade students. It includes multiple-choice questions that challenge your comprehension of various number types. Review your answers to strengthen your math skills!
