Rational numbers and whole numbers are both sets of numbers that belong to the broader category of numbers, but they have distinct definitions.
Definitions:
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Rational Numbers: A rational number is any number 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 means that rational numbers include integers, fractions, and finite or repeating decimals. Examples of rational numbers are \( \frac{1}{2} \), \( -3 \) (which can be expressed as \( \frac{-3}{1} \)), and \( 0.75 \) (which can be expressed as \( \frac{3}{4} \)).
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Whole Numbers: Whole numbers are the set of non-negative integers, which include zero and all positive integers. The set of whole numbers is: \( 0, 1, 2, 3, 4, \ldots \)
Relationships:
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Are all rational numbers also whole numbers?
- No. Not all rational numbers are whole numbers. For example, the rational number \( \frac{1}{2} \) is a fraction and cannot be classified as a whole number because it is not a non-negative integer. Furthermore, negative rational numbers like \( -3 \) or \( -\frac{5}{4} \) also do not qualify as whole numbers.
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Are all whole numbers also rational numbers?
- Yes. Every whole number is a rational number. This is because any whole number \( n \) can be expressed as the fraction \( \frac{n}{1} \). For example, the whole number \( 4 \) can be expressed as \( \frac{4}{1} \), which is a valid rational number. Similarly, \( 0 \) can be expressed as \( \frac{0}{1} \).
Summary:
- Rational Numbers: Include integers, fractions, and can represent whole numbers.
- Whole Numbers: Only non-negative integers, starting from zero.
Examples:
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Rational Numbers that are not Whole Numbers:
- \( \frac{1}{3} \) (a fraction)
- \( -2 \) (a negative integer)
- \( 0.25 \) (a terminating decimal)
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Whole Numbers that are Rational Numbers:
- \( 0 \) (can be expressed as \( \frac{0}{1} \))
- \( 7 \) (can be expressed as \( \frac{7}{1} \))
- \( 100 \) (can be expressed as \( \frac{100}{1} \))
In conclusion, while all whole numbers fall under the category of rational numbers, not all rational numbers are whole numbers.