Rational numbers and whole numbers are two different sets of numbers, and they have a specific relationship based on their definitions.
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Rational Numbers: A rational number is any number that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \). This includes all integers, fractions, and finite or repeating decimals. Examples of rational numbers are \( \frac{1}{2} \), \( -3 \), \( 0 \), and \( 0.75 \) (which can be written as \( \frac{3}{4} \)).
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Whole Numbers: Whole numbers are the set of non-negative integers, which include \( 0, 1, 2, 3, \ldots \). They do not include negative numbers or fractions. Examples of whole numbers are \( 0, 1, 2, 3, 4, \) and so forth.
Relationship Between Rational Numbers and Whole Numbers
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Are all rational numbers also whole numbers? No, not all rational numbers are whole numbers. While every whole number can be expressed as a fraction (for example, \( 5 \) can be written as \( \frac{5}{1} \)), there are many rational numbers that are not whole numbers, such as \( \frac{1}{2} \) and \( -3 \).
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Are all whole numbers also rational numbers? Yes, all whole numbers are rational numbers. Whole numbers are a subset of rational numbers because any whole number \( n \) can be expressed as \( \frac{n}{1} \), which fits the definition of a rational number.
Examples
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Whole numbers that are also rational numbers: \( 0 \) (which can be written as \( \frac{0}{1} \)), \( 3 \) (which can be written as \( \frac{3}{1} \)), etc.
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Rational numbers that are not whole numbers: \( \frac{1}{3} \), \( -2.5 \), and \( \sqrt{2} \) (approximately \( 1.414 \), which cannot be expressed as a fraction of integers).
In summary, while all whole numbers are rational numbers, not all rational numbers fit within the category of whole numbers.