📘 Question
Look at several examples of rational numbers \( \frac{p}{q} \) (where \(q \ne 0\), and \(p, q\) have no common factors except 1) that have terminating decimal representations. What property must \(q\) satisfy?
✏️ Explanation
Step 1: Observe examples
- \(\frac{1}{2} = 0.5\)
- \(\frac{3}{4} = 0.75\)
- \(\frac{7}{5} = 1.4\)
- \(\frac{9}{20} = 0.45\)
Step 2: Factor denominators
- \(2 = 2\)
- \(4 = 2^2\)
- \(5 = 5\)
- \(20 = 2^2 \times 5\)
Step 3: Identify pattern
All denominators contain only prime factors:
\[
2 \quad \text{and/or} \quad 5
\]
—
Step 4: General rule
A rational number \( \frac{p}{q} \) (in lowest form) has a terminating decimal if:
\[
q = 2^m \times 5^n
\]
where \(m, n\) are non-negative integers.
—✅ Final Answer
Denominator \(q\) must have only prime factors 2 and/or 5.
—
💡 Key Concept
- If denominator has only 2 and 5 → terminating decimal
- Otherwise → recurring decimal