A rational number is a type of number that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. In other words, a rational number can be represented in the form \( \frac{a}{b} \), where \( a \) is an integer (which can be positive, negative, or zero) and \( b \) is a non-zero integer.
Some key characteristics of rational numbers include:
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Examples: Rational numbers include whole numbers (like 1, 2, and -3), fractions (such as \( \frac{1}{2} \) and \( \frac{-4}{3} \)), and terminating or repeating decimal numbers (like 0.75, which is \( \frac{3}{4} \), or 0.333..., which is \( \frac{1}{3} \)).
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Density: Between any two rational numbers, there are infinitely many other rational numbers. For example, between \( \frac{1}{2} \) and \( \frac{1}{3} \), you can find numerous rational numbers such as \( \frac{5}{12} \).
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Comparison to Irrational Numbers: Rational numbers are distinct from irrational numbers, which cannot be expressed as a fraction of two integers. Examples of irrational numbers include \(\sqrt{2}\), \(\pi\), and \(e\).
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Closure: The set of rational numbers is closed under addition, subtraction, multiplication, and division (except by zero).
Overall, rational numbers play a critical role in mathematics and are foundational in various applications across different fields.
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