Let A = {-6, -12⁄4, -5⁄8, -√3, 0, ¼, 1, 2π, 3, √12}. List all the elements of A that belong to each set. Rational numbers

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Recall that rational numbers are numbers that can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). This includes integers, fractions, and terminating or repeating decimals.

Examine each element of the set \( A = \{-6, -\frac{12}{4}, -\frac{5}{8}, -\sqrt{3}, 0, \frac{1}{4}, 1, 2\pi, 3, \sqrt{12} \} \) to determine if it can be written as a fraction of integers.

Identify \( -6 \) as rational because it is an integer, which can be written as \( \frac{-6}{1} \).

Simplify \( -\frac{12}{4} \) to \( -3 \), which is also an integer and therefore rational.

Recognize \( -\frac{5}{8} \), \( 0 \), \( \frac{1}{4} \), \( 1 \), and \( 3 \) as rational numbers since they are either fractions or integers.

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Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. This includes integers, fractions, and finite or repeating decimals. For example, 1/2, -3, and 0.75 are rational, while numbers like √3 or π are not.

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Rationalizing Denominators

Set Membership and Classification

Understanding set membership involves determining whether an element belongs to a particular set based on its properties. Classifying numbers into sets like rational or irrational requires analyzing their form and characteristics, such as whether they can be expressed as a fraction or not.

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Determining Different Coordinates for the Same Point Example 2

Simplification of Expressions

Simplifying expressions, such as fractions or radicals, helps identify the nature of numbers. For example, simplifying -12/4 to -3 or √12 to 2√3 clarifies whether the number is rational or irrational, aiding in accurate classification within sets.

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