MathematicsChapter 1: Real Numbers
Terminating and Recurring
A terminating decimal stops after finitely many digits, such as 7/8 = 0.875.
Learning objectives
- Explain Terminating And Recurring in simple mathematical language.
- Read the comparison table and choose the correct rule or method.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
What this grid covers
A terminating decimal stops after finitely many digits, such as 78 = 0.875. A recurring decimal goes on forever with a repeating block, such as 13 = 0.333... . An irrational like sqrt(2) is non-terminating and non-recurring. A fraction in lowest terms terminates if and only if the denominator's primes are only 2 and/or 5. So 38 terminates (8 = 23) while 56 is recurring (6 = 2 x 3).
Where / symbols
- terminating
- A terminating decimal stops after finitely many digits, such as 7/8 = 0.
- 7/8 = 0.875
- A key idea used when studying Terminating And Recurring.
- recurring
- A recurring decimal goes on forever with a repeating block, such as 1/3 = 0.
- 1/3 = 0.333...
- A key idea used when studying Terminating And Recurring.
- sqrt(2)
- An irrational like sqrt(2) is non-terminating and non-recurring.
- non-terminating
- An irrational like sqrt(2) is non-terminating and non-recurring.
- non-recurring
- An irrational like sqrt(2) is non-terminating and non-recurring.
- lowest terms
- A fraction in lowest terms terminates if and only if the denominator's primes are only 2 and/or 5.
- 2 and/or 5
- A fraction in lowest terms terminates if and only if the denominator's primes are only 2 and/or 5.
- 3/8
- So 3/8 terminates (8 = 2^3) while 5/6 is recurring (6 = 2 x 3).
Relationship
- 78 = 0.875
- 13 = 0.333...
WHAT — Core idea
A terminating decimal stops after finitely many digits, such as 78 = 0.875. A recurring decimal goes on forever with a repeating block, such as 13 = 0.333... . An irrational like sqrt(2) is non-terminating and non-recurring. A fraction in lowest terms terminates if and only if the denominator's primes are only 2 and/or 5. So 38 terminates (8 = 23) while 56 is recurring (6 = 2 x 3).
WHY — Why it matters
This idea helps you recognise the right mathematical pattern, choose a safe method, and explain your working clearly.
HOW — Method and examples
78 = 0.875: Terminating — Denom 8 = 2^3, only prime 2 38 = 0.375: Terminating — Denom 8 = 23, only prime 2 12 = 0.5: Terminating — Denom 2 is allowed 15 = 0.2: Terminating — Denom 5 is allowed 13 = 0.333...: Recurring — Denom 3, prime 3 not only 2 or 5 56 = 0.8333...: Recurring — Denom 6 = 2 x 3, extra prime 3
HOW — Answer in an exam
Always cancel to lowest terms first. 615 = 25 terminates even though 15 has a 3, because after cancelling the denominator is 5. Do not call sqrt(2) recurring; recurring decimals are still rational.
Read the denominator in lowest terms, then name the decimal type:
| Number | Decimal type | Reason |
|---|---|---|
| 1. 7/8 = 0.875 | Terminating | Denom 8 = 2^3, only prime 2 |
| 2. 3/8 = 0.375 | Terminating | Denom 8 = 2^3, only prime 2 |
| 3. 1/2 = 0.5 | Terminating | Denom 2 is allowed |
| 4. 1/5 = 0.2 | Terminating | Denom 5 is allowed |
| 5. 1/3 = 0.333... | Recurring | Denom 3, prime 3 not only 2 or 5 |
| 6. 5/6 = 0.8333... | Recurring | Denom 6 = 2 x 3, extra prime 3 |
| 7. sqrt(2) | Non-terminating non-recurring | Irrational, not a ratio a/b |
In simple words
Always cancel to lowest terms first. 615 = 25 terminates even though 15 has a 3, because after cancelling the denominator is 5. Do not call sqrt(2) recurring; recurring decimals are still rational.
Key points
- terminating
- 78 = 0.875
- recurring
- 13 = 0.333...
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Terminating and Recurring
Sticky Notes · Quick Revision
WHAT
- A terminating decimal stops after finitely many digits, such as 78 = 0.
Key Points
- A terminating decimal stops after finitely many digits, such as 78 = 0.
- Read the denominator in lowest terms, then name the decimal type: Review each row and apply its rule carefully.
- Always cancel to lowest terms first.
HOW
- 78 = 0.875: Terminating — Denom 8 = 2^3, only prime 2
- 38 = 0.375: Terminating — Denom 8 = 2^3, only prime 2
- 12 = 0.5: Terminating — Denom 2 is allowed
- 15 = 0.2: Terminating — Denom 5 is allowed
- 13 = 0.333...: Recurring — Denom 3, prime 3 not only 2 or 5
Examples
- 78 = 0.875: Terminating
- 38 = 0.375: Terminating
- 12 = 0.5: Terminating
- 15 = 0.2: Terminating
- 13 = 0.333...: Recurring
Common Mistake
- Always cancel to lowest terms first.
- 615 = 25 terminates even though 15 has a 3, because after cancelling the denominator is 5.
Exam Tip
- Always cancel to lowest terms first. 615 = 25 terminates even though 15 has a 3, because after cancelling the denominator is 5. Do not call sqrt(2) recurring; recurring decimals are still rational.
60-second revision
- A terminating decimal stops after finitely many digits, such as 78 = 0.
- Read the denominator in lowest terms, then name the decimal type: Review each row and apply its rule carefully.
- Always cancel to lowest terms first.
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