Terminating and Repeating Decimal Expansions

• Classify decimals as terminating, non-terminating repeating, or non-terminating non-repeating.
• Apply the NCERT rule: p/q (lowest terms) terminates iff q = 2^m × 5^n.
• Convert repeating decimals to fractions using algebraic method.
• Link non-repeating non-terminating decimals to irrational numbers.
• Work with mixed recurring decimals like 0.4̄5.

1. Rational decimals — Every rational number has a decimal that either terminates or repeats a block forever.
2. Terminating rule — After simplifying p/q, if denominator has only prime factors 2 and 5, decimal terminates.
3. Examples — 7/8 = 0.875 (terminates); 1/3 = 0.3̄ (repeats); 1/7 = 0.142857… (period 6).
4. Irrational decimals — √2, π have non-terminating, non-repeating expansions.
5. Pure recurring — 0.abāb… = ab/99 (two-digit repeat block).
6. Conversion technique — Let x = decimal; multiply by 10^k to shift; subtract to eliminate repeating part.
7. 0.9̄ = 1 — Important identity showing two different decimal forms can equal same rational.
8. Mixed recurring — e.g. 0.23̄5: some digits fixed, then a block repeats.
9. NCERT Ex. 1.3 — Write 0.4̄7 in p/q form; identify terminating decimals without long division.
10. Comparison — To compare decimals, align place values or convert to fractions.

NCERT: Express 0.4̄7 as a rational number p/q

Let x = 0.4777…
Step 1 — One non-repeating digit (4), repeating digit (7): multiply x by 10 → 10x = 4.777…
Step 2 — Multiply by 100 → 100x = 47.777…
Step 3 — Subtract: 100x − 10x = 43 → 90x = 43
Step 4 — x = 43/90
Verify: 43 ÷ 90 = 0.4777… ✓
Note: 43 and 90 share no common factor → lowest terms

• Checking only numerator for terminating rule (must examine denominator in lowest terms).
• Thinking 0.999… < 1 (it equals 1 exactly).
• Using 10x − x for mixed recurring without correct power of 10.
• Confusing repeating with terminating when denominator has factor 3, 7, etc.
• Believing longer decimal means irrational (1/7 has long repeat but is rational).

• Does 13/125 terminate or repeat? Why?
• Write 0.6̄ as a fraction in lowest terms.
• Which has a repeating decimal: 3/16 or 5/12?
• Classify the decimal expansion of √5.
• Convert 0.2̄7 to p/q form.

Open the Practice tab for graded questions on Terminating and Repeating Decimal Expansions.