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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.

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Student-friendly explanation

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

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    Calling 0.666... irrational is wrong because the digit 6 repeats forever, so it is rational.

Definition

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.

Example

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

Rule to remember

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

Memory hook

Repeating means rational; no repeat means irrational.

Examples and method

Worked example

Classify 0.101001000100001... . The gaps between 1s keep changing, so no fixed block repeats. It is non-terminating non-repeating and therefore irrational.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

Calling 0.666... irrational is wrong because the digit 6 repeats forever, so it is rational.

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.

Quick check

Is 0.272727... rational or irrational?

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

Answer writing and exam use

1-mark answer

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-mark answer

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. Decimal test: terminating or repeating decimals are rational; non-terminating non-repeating decimals are irrational. √2 = 1.4142135... and π = 3.14159... are non-terminating non-repeating decimals, so they are irrational.

3-mark answer

Such decimals cannot be written exactly as p/q. This is the key decimal test used to distinguish irrational numbers from repeating rational decimals. Decimal test: terminating or repeating decimals are rational; non-terminating non-repeating decimals are irrational. Classify 0.101001000100001... . The gaps between 1s keep changing, so no fixed block repeats. It is non-terminating non-repeating and therefore irrational. Students may be asked to identify whether a given decimal is rational or irrational and give the reason. Calling 0.666... irrational is wrong because the digit 6 repeats forever, so it is rational.
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