Question

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

Accepted Answer

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. 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. 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. 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. Conclude that the irrational numbers in the set $$A$$ are $$- \sqrt{3}$$, $$2\pi$$, and $$\sqrt{12}.$$

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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Key Concepts

Irrational Numbers

Set Membership and Classification

Simplification of Expressions