Rational numbers are fractions in disguise — they are numbers that can be written in the form p/q where p and q are integers and q is not zero. You have already worked with fractions and integers. This chapter shows that both are part of the same big family: rational numbers.
Key Concepts and Definitions
A rational number is any number expressible as p/q, where p and q are integers and q ≠ 0. Examples: 3/4, -5/7, 0, -2, 11/1.
Equivalent rational numbers: Multiplying or dividing both numerator and denominator by the same non-zero integer gives an equivalent rational number. Example: 3/4 = 6/8 = 9/12.
Standard form: A rational number p/q is in standard form when q > 0 and GCD(p, q) = 1. Example: -6/10 in standard form is -3/5.
Comparison: To compare p/q and r/s, convert to a common denominator (LCM of q and s) and compare numerators.
- Operations on rational numbers:
- Addition: a/b + c/d = (ad + bc)/(bd)
- Subtraction: a/b - c/d = (ad - bc)/(bd)
- Multiplication: (a/b) × (c/d) = (ac)/(bd)
- Division: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/(bc)
Write -18/24 in standard form.
Step 1: GCD(18, 24) = 6.
Step 2: Divide both by 6 → -18/24 = -3/4.
Step 3: Denominator is positive, so standard form is -3/4.
Add 3/5 + (-7/10).
Step 1: LCM(5, 10) = 10.
Step 2: 3/5 = 6/10.
Step 3: 6/10 + (-7/10) = (6 - 7)/10 = -1/10.
Subtract (-2/3) - (5/6).
Step 1: LCM(3, 6) = 6.
Step 2: -2/3 = -4/6.
Step 3: -4/6 - 5/6 = -9/6 = -3/2.
Multiply (3/7) × (-14/9).
Step 1: Multiply numerators: 3 × (-14) = -42.
Step 2: Multiply denominators: 7 × 9 = 63.
Step 3: -42/63. GCD(42, 63) = 21. Result: -2/3.
Divide (-4/5) ÷ (8/15).
Step 1: Flip the divisor: 15/8.
Step 2: (-4/5) × (15/8) = -60/40 = -3/2.
Find x if x + 1/3 = 5/6.
Step 1: x = 5/6 - 1/3 = 5/6 - 2/6 = 3/6 = 1/2.
Arrange -2/3, 5/(-6), and 1/2 in ascending order.
Step 1: Rewrite: -2/3, -5/6, 1/2.
Step 2: Common denominator = 6 → -4/6, -5/6, 3/6.
Step 3: Ascending order: -5/6, -4/6, 3/6 → -5/6 < -2/3 < 1/2.
- Key formulas:
- Additive inverse of p/q is -p/q.
- Multiplicative inverse (reciprocal) of p/q is q/p (p ≠ 0).
- Between any two rational numbers there exist infinitely many rational numbers.
Common mistakes
- Forgetting to make the denominator positive when writing in standard form.
- Adding numerators without first finding a common denominator.
- Confusing the additive inverse (-p/q) with the multiplicative inverse (q/p).
Summary
Rational numbers extend our number system beyond integers. They are closed under all four operations (except division by zero), and every rational number has an additive inverse and a multiplicative inverse.