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. Irrational numbers

검증된 단계별 안내

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 \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\).

Go through each element of the set \(A = \{-6, -\frac{12}{4}, -\frac{5}{8}, -\sqrt{3}, 0, \frac{1}{4}, 1, 2\pi, 3, \sqrt{12}\}\) and determine if it can be expressed as a fraction of integers or not.

Identify \(-6\), \(-\frac{12}{4}\), \(-\frac{5}{8}\), \(0\), \(\frac{1}{4}\), \(1\), and \(3\) as rational numbers because they can be written as fractions of integers.

Recognize that \(-\sqrt{3}\) and \(2\pi\) are irrational because \(\sqrt{3}\) and \(\pi\) are well-known irrational numbers, and multiplying by \(-1\) or \(2\) does not change their irrationality.

Check \(\sqrt{12}\) by simplifying it: \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\). Since \(\sqrt{3}\) is irrational, \(2\sqrt{3}\) is also irrational.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Irrational Numbers

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

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, irrational, integers, or real numbers helps organize and analyze them effectively.

Simplification of Radicals and Fractions

Simplifying radicals and fractions helps identify the nature of numbers. For example, √12 can be simplified to 2√3, which is irrational. Simplifying fractions like -12/4 to -3 helps determine if the number is rational or integer.