Rational numbers and whole numbers are both types of numbers, but they occupy different subsets within the broader number system.
Definitions
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Rational Numbers: A rational number is any number that can be expressed as the quotient of two integers \(a\) and \(b\) (where \(b \neq 0\)). This means that rational numbers can take forms such as \( \frac{1}{2} \), \(-3\), \(0\), or \(2.5\). They can be whole numbers, fractions, or terminating/recurring decimals.
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Whole Numbers: Whole numbers are the set of non-negative integers, which include \(0, 1, 2, 3, \ldots\). Whole numbers do not include negative numbers or fractions.
Relationships
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Are all rational numbers also whole numbers?
- No, not all rational numbers are whole numbers. While some rational numbers can be whole numbers (such as \(0\), \(1\), or \(2\)), there are many rational numbers that are not whole numbers, such as \(-1\), \(\frac{1}{2}\), and \(3.5\).
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Are all whole numbers also rational numbers?
- Yes, all whole numbers are rational numbers. This is because any whole number \(n\) can be expressed in the form \( \frac{n}{1} \), which fits the definition of a rational number (where both \(n\) and \(1\) are integers, and \(1 \neq 0\)). For example, the whole number \(3\) can be written as \( \frac{3}{1} \).
Examples
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Examples of rational numbers that are not whole numbers:
- \(-2\) (a negative integer)
- \(\frac{3}{4}\) (a proper fraction)
- \(2.5\) (a terminating decimal)
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Examples of whole numbers that are also rational numbers:
- \(0\), which can be written as \( \frac{0}{1} \)
- \(1\), which can be expressed as \( \frac{1}{1} \)
- \(5\), which is \( \frac{5}{1} \)
In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers. The subsets overlap, but they are not identical.
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