10th Class Math: Rational and Irrational Numbers

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