Irrational numbers
An irrational number cannot be written in the form p/q, where p and q are integers and q ≠ 0.
In one line: Rational repeats or ends; irrational does neither.
Practice This ConceptUnderstand it
These numbers have decimal expansions that never end and never repeat. In Class 10, square roots like √2, √3, and √5 become important examples. They are not fractions, even if they look neat on the number line.
Example
Formula and rule
An irrational number cannot be expressed as p/q with integers p and q, where q ≠ 0.
See it worked out
Follow the example line by line, then use the same method on new questions.
Worked example
√5 is irrational because it cannot be written exactly as a fraction and its decimal continues without repetition.
Method you can reuse
- 1Try to write the number as a fraction.
- 2Check its decimal pattern.
- 3If it neither terminates nor repeats, classify it as irrational.
- 4Use proof questions when the number is a square root.
Visual tip
A number line can show irrational numbers lying between two nearby rational numbers.
Check yourself
Answer the question in your head before opening the answer.
Try it yourself
A decimal never ends and never repeats. What conclusion should you draw if it comes from √2?
Think first, then show answerHide answer
Common wrong answer
Thinking 0.333... is irrational.
Why it loses marks: It repeats and equals 1/3.
Don't mix this up
Exam tip
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
- 1
Enough for 1 mark
An irrational number cannot be written in the form p/q, where p and q are integers and q ≠ 0.
- 2
Add for 2 marks
Example: √2 and π are irrational numbers.
- 3
Add for 3 marks
These numbers have decimal expansions that never end and never repeat.
Practice this concept with focused MCQs
Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
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