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

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1. Recall that irrational numbers are numbers that cannot be expressed as a ratio of two integers, meaning they cannot be written as a simple fraction and their decimal expansions are non-repeating and non-terminating.
2. 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 is irrational.
3. Identify which elements are clearly rational: \(-6\) (an integer), \(- \frac{12}{4}\) (which simplifies to \(-3\), an integer), \(- \frac{5}{8}\) (a fraction), \$0\(, \)\frac{1}{4}\( (a fraction), \)1\(, and \)3$ are all rational numbers.
4. Focus on the elements involving square roots and \(\pi\): \(- \sqrt{3}\), \$2\pi\(, and \)\sqrt{12}\(. Recall that \)\sqrt{3}\( is irrational, \)\pi\( is irrational, and \)\sqrt{12}\( can be simplified to \)2\sqrt{3}$, which is also irrational.
5. Conclude that the irrational numbers in the set \(A\) are \(- \sqrt{3}\), \$2\pi\(, and \)\sqrt{12}$.

Key Concepts

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating. Examples include numbers like √3 and π, which cannot be written as simple fractions.

Set Membership and Classification

Set membership involves determining whether an element belongs to a particular set based on defined properties. Classifying numbers into sets like rational or irrational requires understanding their characteristics and applying these definitions to each element.

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.