medium difficulty6 min

Non-Terminating Non-Repeating Decimals

A non-terminating non-repeating decimal is a decimal that goes on forever without a repeating block of digits, and it represents an irrational number.

In one line: Repeating means rational; no repeat means irrational.

Practice This Concept

Understand it

Such decimals cannot be written exactly as p/q. This is the key decimal test used to distinguish irrational numbers from repeating rational decimals.

Example

√2 = 1.4142135... and π = 3.14159... are non-terminating non-repeating decimals, so they are irrational.

Formula and rule

Decimal test: terminating or repeating decimals are rational; non-terminating non-repeating decimals are irrational.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Classify 0.101001000100001... .

1The gaps between 1s keep changing, so no fixed block repeats.
2It is non-terminating non-repeating and therefore irrational.

Method you can reuse

1Check if the decimal stops.
2If it stops, it is rational.
3If it continues, look for a repeating block.
4If a block repeats, it is rational; if no fixed block repeats, it is irrational.

Sketch it in your answer

Use a two-column comparison showing 0.333... as repeating rational and 1.414213... as non-repeating irrational.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Is 0.272727... rational or irrational?

Think first, then show answer

0.272727... is rational because the block 27 repeats continuously, so it is non-terminating repeating, not non-repeating.

Common wrong answer

Calling 0.666... irrational.

Why it loses marks: The digit 6 repeats forever, so it is rational.

Don't mix this up

Students may see a long decimal and call it irrational even when a repeating pattern is present.

Exam tip

Look for a fixed repeated block. If a block repeats forever, the decimal is rational; if no block repeats and it does not terminate, it is irrational.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to identify whether a given decimal is rational or irrational and give the reason.

1
Enough for 1 mark
A non-terminating non-repeating decimal is a decimal that goes on forever without a repeating block of digits, and it represents an irrational number.
2
Add for 2 marks
Example: √2 = 1.4142135... and π = 3.14159... are non-terminating non-repeating decimals, so they are irrational.
3
Add for 3 marks
Such decimals cannot be written exactly as p/q.

MCQ Quiz

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