The Set of Rational Numbers Q: Fraction Form a/b, Equivalent Forms, and Comparing
In this lesson, we formalize the definition of 'Rational Numbers' (symbolized by the set Q). A rational number is any number that can be expressed as the ratio of two integers in the form a/b, where a is the numerator, b is the denominator, and crucially, b ≠ 0. Division by zero is mathematically undefined because no unique number multiplied by zero can recover a non-zero dividend. The set Q encompasses all natural numbers, whole numbers, negative integers, terminating decimals, repeating decimals, and percentages. We learn how to plot rational numbers precisely on the number line between integers, find equivalent fractions by scaling, and use the least common denominator (LCD) or cross-multiplication to compare and order rational quantities with rigorous accuracy.
1. Varieties of Rational Numbers
Equivalent representations in set Q:
- Integers as Rationals: Any integer n can be written as n/1 (e.g., -5 = -5/1).
- Terminating Decimals: 0.35 = 35/100 = 7/20.
- Repeating Decimals: 0.666... = 2/3, 0.333... = 1/3.
- Denominator Restriction: For 8/(x - 3) to be rational, x ≠ 3.
2. Comparison Strategies in Q
Techniques for comparing rational numbers:
- Common Denominators: Convert fractions to equivalent forms using LCD, then compare numerators.
- Cross-Multiplication: Compare a/b with c/d by comparing the products (a × d) and (b × c).
- Sign Rule: Any positive rational > 0 > any negative rational.
Problem Solving Workshop:
Cross-multiplication: (-2 × 5) = -10; (3 × -3) = -9.
Since -10 < -9, therefore -2/3 < -3/5.
A fraction is undefined when the denominator is zero: 3m - 12 = 0 -> 3m = 12 -> m = 4.
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