The hierarchy of number sets within the real number system, from largest to smallest, typically includes the following:
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Real Numbers (\(\mathbb{R}\)): This set includes all rational and irrational numbers.
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Rational Numbers (\(\mathbb{Q}\)): This set includes all numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Rational numbers include both integers and fractions.
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Integers (\(\mathbb{Z}\)): This set includes all whole numbers, both positive and negative, as well as zero. Examples: \(-2, -1, 0, 1, 2\).
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Whole Numbers: This set includes all non-negative integers. Examples: \(0, 1, 2, 3, \ldots\).
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Natural Numbers (\(\mathbb{N}\)): This set typically includes all positive integers. Depending on context, it may or may not include zero. Examples: \(1, 2, 3, \ldots\) (or \(0, 1, 2, 3, \ldots\) if considered to include zero).
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Irrational Numbers: Although irrational numbers are part of the real number set, they do not form a separate tier in the hierarchy as they are not a subset of the sets listed above but rather complement the rational numbers to form the complete set of real numbers. Examples: \(\sqrt{2}, \pi, e\).
So, while we can consider the inclusion of irrational numbers within the real numbers, the hierarchy hierarchically arranges mostly the subsets of rational and integral values. If we were to include irrational numbers in a strict hierarchy based on their nature, they would remain part of \( \mathbb{R} \) rather than being distinct in this context.
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