Rational numbers and whole numbers are both subsets of the real number system, but they have distinct definitions and characteristics that outline their relationship.
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Definitions:
- Whole Numbers: Whole numbers are the set of non-negative integers, which includes 0, 1, 2, 3, and so on. In mathematical terms, the set can be represented as \( W = {0, 1, 2, 3, \ldots} \).
- Rational Numbers: Rational numbers include all numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In formal notation, a rational number can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \).
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Inclusion:
- Every whole number is a rational number. This is because any whole number \( n \) can be expressed as \( \frac{n}{1} \). Thus, the set of whole numbers (W) is a subset of the set of rational numbers (Q). In set notation, we can express this relationship as \( W \subseteq Q \).
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Examples:
- Examples of whole numbers: 0, 1, 2, 3, ...
- Examples of rational numbers: \( \frac{1}{2}, -\frac{3}{4}, 2, 0, \frac{0}{1} \) (where 2 and 0 are whole numbers).
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Decimal Representation:
- Rational numbers can also be expressed in decimal form. They may terminate (e.g., \( \frac{1}{4} = 0.25 \)) or repeat (e.g., \( \frac{1}{3} = 0.333...\)). Whole numbers, when expressed in decimal form, are simply represented by themselves (e.g., 0, 1, 2, ...).
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Conclusion:
- The relationship between rational numbers and whole numbers is that all whole numbers are rational, but not all rational numbers are whole numbers. Whole numbers are a specific, limited category within the broader category of rational numbers.