The accurate hierarchy of number sets in the real number system from largest to smallest is as follows:
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Real Numbers (includes all rational and irrational numbers)
- This set encompasses all possible numbers that can be found on the number line, including both rational numbers (like fractions and whole numbers) and irrational numbers (like the square root of 2, pi, e, etc.).
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Rational Numbers (includes integers and fractions)
- These are numbers that can be expressed as the quotient of two integers (e.g., 1/2, 3.75). This set includes integers and whole numbers.
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Integers (includes whole numbers and their negatives)
- This set consists of all whole numbers and their negative counterparts (e.g., ... -3, -2, -1, 0, 1, 2, 3 ...).
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Whole Numbers (includes all non-negative integers)
- This set consists of all non-negative integers (e.g., 0, 1, 2, 3...).
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Natural Numbers (includes all positive integers)
- This set consists of all positive integers (e.g., 1, 2, 3, ...). Some definitions include 0 in this set, but traditionally, it refers only to positive integers.
If you're comparing the options you've mentioned, the correct order based on established mathematical conventions is:
- Real Numbers
- Rational Numbers
- Integers
- Whole Numbers
- Natural Numbers
Other enumerations you mentioned (like "national numbers," "phone numbers," etc.) do not fit into this mathematical hierarchy and are not considered standard number sets in mathematics. Additionally, "irrational numbers" are a subset of real numbers, showing up in more detail (e.g., square roots that are not perfect squares, π, e, etc.), but these numbers do not form a separate larger set like whole numbers.
In conclusion, the hierarchy is:
- Real Numbers
- Rational Numbers
- Integers
- Whole Numbers
- Natural Numbers
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