Rational numbers are numbers that can be expressed in the form of a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \) is not equal to zero. This includes integers (which can be written as fractions with a denominator of 1), as well as fractions and terminating or repeating decimals. Examples of rational numbers include \( \frac{1}{2} \), \( -3 \), \( 0.75 \) (which is \( \frac{3}{4} \)), and \( 2.333... \) (which is \( \frac{7}{3} \)).
To compare rational numbers, you can use several methods:
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Common Denominator: Convert both fractions to have a common denominator, making it easier to compare their numerators. For example, to compare \( \frac{1}{3} \) with \( \frac{2}{5} \), you could convert both to have a common denominator, say 15:
- \( \frac{1}{3} = \frac{5}{15} \)
- \( \frac{2}{5} = \frac{6}{15} \)
- Since \( \frac{5}{15} < \frac{6}{15} \), \( \frac{1}{3} < \frac{2}{5} \).
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Decimal Form: Convert both rational numbers to decimal form. For instance:
- \( \frac{1}{4} = 0.25 \)
- \( \frac{1}{3} \approx 0.333... \)
- By comparing their decimal representations, it is clear that \( 0.25 < 0.333... \).
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Cross-Multiplication: For two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), compare them by cross-multiplying:
- Compare \( a \cdot d \) to \( b \cdot c \).
- If \( a \cdot d > b \cdot c \), then \( \frac{a}{b} > \frac{c}{d} \).
- If \( a \cdot d < b \cdot c \), then \( \frac{a}{b} < \frac{c}{d} \).
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Number Line: If you're working with a small set of rational numbers, you can plot them on a number line to visually determine their order.
By using these methods, you can efficiently compare and order rational numbers.
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