To determine the smallest subset into which \(\sqrt{37}\) fits, we first need to recognize the type of number \(\sqrt{37}\) is.
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Natural Numbers (\(\mathbb{N}\)): These are the positive integers (1, 2, 3, ...). \(\sqrt{37}\) is not a natural number.
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Whole Numbers (\(\mathbb{W}\)): These include all natural numbers and zero (0, 1, 2, 3, ...). \(\sqrt{37}\) is not a whole number.
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Integers (\(\mathbb{Z}\)): This set includes positive integers, negative integers, and zero (..., -2, -1, 0, 1, 2, ...). \(\sqrt{37}\) is not an integer.
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Rational Numbers (\(\mathbb{Q}\)): A rational number is defined as the ratio of two integers (where the denominator is not zero). \(\sqrt{37}\) is an irrational number because it cannot be expressed as a fraction of two integers.
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Irrational Numbers: This subset includes numbers that cannot be expressed as a fraction. Since \(\sqrt{37}\) does not fit into any of the previous sets, it is an irrational number.
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Real Numbers (\(\mathbb{R}\)): This set includes all rational and irrational numbers. \(\sqrt{37}\) is a real number.
Thus, the smallest subset that \(\sqrt{37}\) fits into is the set of irrational numbers.