Class 9 Maths: Irrational Numbers — Practice Questions with Answers

Explanation: 22/7 is a ratio of two integers, so it is rational, even though it is often used to approximate π. √25 = 5 and 3.272727... is a repeating decimal, so both are rational. The decimal 0.101001000100001... never terminates and never repeats because the number of zeros keeps increasing, so it cannot be written as p/q and is irrational.

Explanation: By definition, a rational number is any number that can be written as p/q where p and q are integers and q ≠ 0. The condition q ≠ 0 is needed because division by zero is not defined. Restricting to natural numbers would wrongly exclude negative rationals like −3/4, and demanding p ≠ 0 would wrongly exclude 0, which is rational since 0 = 0/1.

Explanation: A rational number always has a decimal expansion that either terminates or repeats a block of digits. An irrational number cannot be written as p/q, so its decimal expansion goes on forever without any repeating block. For example, √2 = 1.41421356... never terminates and never repeats.

Explanation: Look for perfect squares on either side of 50. Since 7² = 49 and 8² = 64, we have 49 < 50 < 64. Taking square roots gives 7 < √50 < 8, so √50 lies between 7 and 8.

Explanation: Use the identity (a + b)(a − b) = a² − b² with a = 2 and b = √3. This gives 2² − (√3)² = 4 − 3 = 1. Note that the product of these two irrational numbers is a rational number.

Explanation: In (3 + √5) + (3 − √5), the +√5 and −√5 cancel, leaving 3 + 3 = 6, which is rational. In every other option the irrational part does not cancel: √2 + √3 is irrational, (2 + √3) + (2 + √3) = 4 + 2√3 is irrational, and π + 2 is irrational.

Explanation: For square roots, √a × √b = √(a × b). So √8 × √2 = √(8 × 2) = √16 = 4. A common error is to add under the root and get √10, but roots multiply, they do not add.

Explanation: Write 32 as 16 × 2, so √32 = √16 × √2 = 4√2. Then 4√2 ≈ 4 × 1.414 = 5.656. Choosing 2.828 or 11.312 comes from wrongly simplifying √32 as 2√2 or 8√2.

Explanation: Suppose r is rational, s is irrational, and r + s were rational. Then s = (r + s) − r would be a difference of two rationals, hence rational, which is a contradiction; so r + s must be irrational. The other statements fail by counterexample: √2 + (−√2) = 0 is rational, √2 × √2 = 2 is rational, and √4 = 2 shows the square root of a positive integer can be rational.

Explanation: From √2 = p/q, squaring gives p² = 2q², so p² is even, which forces p to be even, say p = 2m. Substituting gives 4m² = 2q², so q² = 2m², which forces q to be even as well. Then p and q share the common factor 2, contradicting the assumption that they are coprime, so √2 cannot be written as p/q.

Explanation: Rationalise 1/x: multiply 1/(3 + 2√2) by (3 − 2√2)/(3 − 2√2) to get (3 − 2√2)/(9 − 8) = 3 − 2√2. So x + 1/x = (3 + 2√2) + (3 − 2√2) = 6. The irrational parts cancel exactly, leaving a rational answer.

Explanation: Expand each square: (√5 + √3)² = 5 + 2√15 + 3 = 8 + 2√15, and (√5 − √3)² = 5 − 2√15 + 3 = 8 − 2√15. Adding them, the 2√15 terms cancel: (8 + 2√15) + (8 − 2√15) = 16. A sign slip that adds the middle terms instead of cancelling them gives the wrong answer 16 + 4√15.