Rational numbers and whole numbers are both subsets of the real number system, but they have distinct characteristics and relationships.
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Definition of Rational Numbers: Rational numbers are any numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) is an integer and \( b \) is a non-zero integer. This includes fractions, integers, and whole numbers.
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Definition of Whole Numbers: Whole numbers are the set of non-negative integers, which includes 0, 1, 2, 3, and so on. Whole numbers do not include negative numbers or fractions.
Relationship Between Rational and Whole Numbers:
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Are all rational numbers also whole numbers?
- No: Not all rational numbers are whole numbers. For example, \( \frac{1}{2} \) is a rational number (because it can be expressed as a fraction), but it is not a whole number since it is not a non-negative integer.
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Are all whole numbers also rational numbers?
- Yes: All whole numbers are rational numbers. Each whole number can be expressed as a fraction by having it over 1. For example, the whole number 3 can be expressed as \( \frac{3}{1} \), which fits the definition of a rational number.
Examples:
- Rational but not Whole: \( \frac{3}{4} \), \(-2\), and \(1.5\) are all rational numbers, but none are whole numbers.
- Whole but also Rational: 0, 1, 2, and 3 are whole numbers, and they can all be expressed as rational numbers (\( \frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \frac{3}{1} \) respectively).
In summary, while all whole numbers are rational, not all rational numbers qualify as whole numbers.