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Rational numbers include all integers, fractions, and terminating or repeating decimals — any number that can be expressed as a ratio of two integers. This topic extends integer operations to fractions and decimals with negative signs. For example, −3/4 + 1/2 requires finding a common denominator and applying integer addition rules to the numerators. Similarly, −2.5 × 1.4 = −3.5 using decimal multiplication with sign rules. Fluency with rational number operations is essential because most real-world quantities are not whole numbers, and algebra constantly requires manipulating fractions and decimals.
Review fraction operations (common denominators for addition/subtraction, multiply straight across, flip and multiply for division) and layer on the integer sign rules. Practice mixed problems that combine fractions and decimals. Use number line placement to verify reasonableness of answers. Include word problems with real-world measurements.
A rational number is any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0. This family is larger than it might first appear: every integer is rational (3 = 3/1), every terminating decimal is rational (0.75 = 3/4), and every repeating decimal is rational (0.333… = 1/3). The word "rational" comes from "ratio" — it simply means expressible as a ratio of two integers.
You already know how to add fractions with unlike denominators (find the LCD, rewrite each fraction, add numerators) and how to apply sign rules to integers (same signs → positive, opposite signs → negative). Rational number operations combine both skills. For addition and subtraction, the denominator process is unchanged — you still need a common denominator — but now numerators can be negative. For example, −3/4 + 1/2: the LCD is 4, so rewrite as −3/4 + 2/4 = −1/4. The integer rule kicks in at the numerator level: −3 + 2 = −1, applying the same rule you use for negative integers on a number line.
Multiplication is simpler: multiply numerators together, multiply denominators together, then apply the sign. (−2/3) × (5/7) = −10/21 — negative times positive gives negative. Division means "multiply by the reciprocal": (−3/4) ÷ (1/2) = (−3/4) × (2/1) = −6/4 = −3/2. The key insight is that fractions obey the same sign rules as integers because a negative fraction like −3/4 is just a negative number that happens to sit between −1 and 0. There is no separate rule to learn; the sign lives with the numerator.
Decimals follow the same logic. −2.5 × 1.4: ignore signs to get 2.5 × 1.4 = 3.5, then apply the sign rule (negative × positive = negative), giving −3.5. For mixed expressions combining fractions and decimals, convert to a common form first — usually fractions, since decimals can introduce rounding. Fluency with these operations is the foundation for everything in algebra: nearly every equation you will encounter has rational coefficients, and manipulating those coefficients correctly requires exactly these skills.