Multiplication and Division of Rational Numbers: Reciprocals, Cross-Canceling, and Distributive Property
Multiplication and division of rational numbers offer remarkable algorithmic clarity: unlike addition, multiplication does not require a common denominator! To multiply two rational numbers, multiply the numerators directly and multiply the denominators directly: (a/b) × (c/d) = (a × c) / (b × d). We apply standard sign rules: positive × positive = positive, negative × negative = positive, and positive × negative = negative. An essential concept in this lesson is the 'multiplicative inverse' or 'reciprocal': for any non-zero rational number a/b, its reciprocal is b/a, such that (a/b) × (b/a) = 1. The number 0 is the unique rational number that has no reciprocal because division by zero is undefined. For division, we convert the operation to multiplication by taking the reciprocal of the divisor ('Keep - Change - Flip'). We also master the distributive property of multiplication over addition: a × (b + c) = (a × b) + (a × c).
1. Multiplication & Reciprocals
Properties of rational multiplication:
- Direct Multiplication: (3/5) × (2/7) = (3 × 2) / (5 × 7) = 6/35.
- Multiplicative Inverse: Reciprocal of 4/7 is 7/4; reciprocal of -3 is -1/3.
- Multiplicative Identity: 1 is the identity element: a × 1 = a.
- Multiplication by Zero: a × 0 = 0.
2. Division & Distributive Law
Division algorithm and mental algebra:
- Division Algorithm: (a/b) ÷ (c/d) = (a/b) × (d/c) where c ≠ 0.
- Example: (2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12 = 5/6.
- Distributive Law: a(b + c) = ab + ac (simplifies factoring and arithmetic).
Applied Arithmetic Practice:
Divide 7 and 14 by 7 (gives 1 and 2); divide -6 and 9 by 3 (gives -2 and 3).
Product = (-2 × 2) / (1 × 3) = -4/3.
Factor out (4/9): (4/9) × (13 + 5) = (4/9) × 18 = 4 × 2 = 8.
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