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Terminating and Recurring
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).
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Terminating and Recurring
What is it?
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).
In simple words
- 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.
Important terms
terminating: A terminating decimal stops after finitely many digits, such as 78 = 0.
78 = 0.875: A key idea used when studying Terminating And Recurring.
recurring: A recurring decimal goes on forever with a repeating block, such as 13 = 0.
13 = 0.333...: A key idea used when studying Terminating And Recurring.
sqrt(2): An irrational like sqrt(2) is non-terminating and non-recurring.
What you should learn
- 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.
Recommended for Class 9
Full school-level explanation
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).
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 = 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 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.
Remember
- terminating
- 78 = 0.875
- recurring
- 13 = 0.333...
- sqrt(2)
- non-terminating
- non-recurring
- lowest terms
Deeper understanding
Go beyond memorising Terminating and Recurring. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
WHAT
Key Points
HOW
Examples
Exam focus
Quick revision
- terminating
- 78 = 0.875
- recurring
- 13 = 0.333...
- terminating: A terminating decimal stops after finitely many digits, such as 78 = 0.
- 78 = 0.875: A key idea used when studying Terminating And Recurring.
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.
For long answers on Terminating and Recurring: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- terminating
- 78 = 0.875
- recurring
- 13 = 0.333...
Important formulas
- terminating
- 78 = 0.875
- recurring
Common exam traps
- Always cancel to lowest terms first.
- 615 = 25 terminates even though 15 has a 3, because after cancelling the denominator is 5.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Always cancel to lowest terms first.
- Mistake
615 = 25 terminates even though 15 has a 3, because after cancelling the denominator is 5.
- Mistake
Correction: Do not call sqrt(2) recurring; recurring decimals are still rational.
- Mistake
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;
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
terminating
78 = 0.875
recurring
13 = 0.333...
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...
78 = 0.875
- Definition
- 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...
- Formula
- 78 = 0.875
- Critical warning
- Always cancel to lowest terms first.
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Exam Practice
Define
Q1. Define Terminating and Recurring.
Explain
Q2. Explain: WHAT — Core idea.
Example
Q3. Give one clear example of Terminating and Recurring and say why it fits.
Mistake
Q4. State one common mistake students make about Terminating and Recurring.
Exam note
Q5. Write a short exam note on Terminating and Recurring: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.