Tricks : What is a Recurring Decimal & Not Recurring Decimal ?

Recurring Decimals : A recurring decimal is a decimal number in which one or more digits repeat forever after the decimal point.


Major Points


1.      A recurring decimal repeats a fixed pattern infinitely.


2.      It is non-terminating but repeating.


3.      Every recurring decimal can be written as a fraction (rational number).


4.      The repeating part is shown with a bar ( ̅ ) above digits.


o    Ex: 0.(3) = 0.333…


o    Ex: 0.(76)= 0.767676…


Ex



  • 0.(3) → 0.33333…

  • 0.(12)→ 0.121212…

  • 5.(42) → 5.424242…


Types of Decimals


1. Terminating Decimal (Ends)


Ex: 0.25, 0.75, 2.4


2. Non-terminating Decimal (Does not end):



  • Repeating / Recurring → pattern repeats
    Ex: 0.181818…

  • Non-repeating → digits never repeat
    Ex: 3.1415926…


Not Recurring Decimal (no digit repeats) : A not recurring decimal is a decimal number in which no digit repeats in a fixed pattern.


Major Points



  • Digits do NOT repeat.

  • It can be:
    (A) Terminating → ends
    (B) Non-terminating but non-repeating → goes on forever but no pattern


Ex


1. Terminating (ends)



  • 0.5

  • 0.75

  • 3.125


2. Non-terminating but non-repeating



  • 0.101001000100001…

  • √2 = 1.414213562…


Easy Way to Remember : Types of Decimals


1. Recurring Decimal



  • Meaning: Digits repeat in a fixed pattern.

  • Ex 0.272727… (27 keeps repeating)


2. Not Recurring Decimal



  • Meaning: No digit repeats in a fixed pattern.

  • Ex: 0.47


3. Terminating Decimal



  • Meaning: The decimal stops after a limited number of digits.

  • Ex 1.25


4. Non-terminating Decimal



  • Meaning: The decimal does not stop and continues forever.

  • Ex: 3.14…


Tricks


1. To check if a decimal is recurring → look for repeating digits.


Ex: 0.727272… = recurring.


2. To convert recurring decimals to fractions:



  • 1-digit repeating → divide by 9 

  •  2-digit repeating → divide by 99 

  •  3-digit repeating → divide by 999


3. If there is a non-repeating part before repeating starts:



  • Multiply 9’s with 10’s

  • Rule:


Numerator = (all decimal digits) – (non-recurring digits)


Denominator = 9 for each repeating + 0 for each non-repeating


Important Rule (Exam Trick):


1. A fraction is recurring if the denominator has prime factors other than 2 or 5


Ex:


i. 1/3 (has 3 → recurring)


ii. 2/7 (has 7 → recurring) 


iii. 5/6 (has 3 → recurring)


2. A fraction is NOT recurring if its denominator has only 2 or 5


Ex:


i. 1/4 → terminating (only 2)


ii. 3/20 → terminating (2, 5)


iii. 7/8 → terminating (2)


Summary



  • Recurring → repeats forever → can be fractions like 1/3, 2/7

  • Not recurring → no repeating pattern → can be terminating or irrational

  • Terminating → stops

  • Non-terminating repeating → recurring

  • Non-terminating non-repeating → irrational (like √2)