Sec 1 Maths · Integers, rational and real numbers

Rational and irrational numbers questions with answers

Practice questions on rational and irrational numbers, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

State whether $\dfrac{13}{4}$ is a rational number or an irrational number.

Step 1 of 2

A rational number can be written as a fraction with an integer on top and a non-zero integer below

\dfrac{13}{4}

is a fraction of two integers.

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Questions

Question 1Rational or irrational?
State whether $\sqrt{75}$ is a rational number or an irrational number.
Show the worked solution
1. Step 1: A square root is rational only when the number under it is a perfect square $75$ is not a perfect square, so $\sqrt{75}$ is a non-terminating, non-recurring decimal.
2. Step 2: Irrational: it cannot be written as a fraction of two integersSo $\sqrt{75}$ is irrational.
Question 2Pick the odd one out
Which one of the following numbers is an irrational number?
Show the worked solution
1. Step 1: Decide for each number: can it be written as a fraction of two integers?A: $\sqrt{36}$ is rational.B: $\sqrt[3]{8}$ is rational.C: $\sqrt{20}$ is irrational.D: $\dfrac{22}{7}$ is rational.
2. Step 2: Choose the irrational numberThe answer is C.
Question 3Square roots on a number line
The points P, Q, R and S are marked on the number line. Which point represents $\sqrt{10}$ ?
Show the worked solution
1. Step 1: Find the perfect squares on either side $9 < 10 < 16$
2. Step 2: Take square roots: √n lies between their roots $3 < \sqrt{10} < 4$
3. Step 3: See which square it is nearer to; a calculator checks it $\sqrt{10} = 3.16\ldots$ So $\sqrt{10}$ is nearer to $3$ .
4. Step 4: Read off the pointThe point is Q.
Question 4Order rational and irrational numbers
Use a calculator to arrange the following numbers in ascending order: P $= \sqrt{15}$ , Q $= 3.87$ , R $= \dfrac{31}{8}$ , S $= 3.\dot{8}$ . Write down the letters in order, starting with the smallest.
Show the worked solution
1. Step 1: Write each number as a decimal, correct to 4 decimal placesP: $\sqrt{15} = 3.8730$ (4 d.p.)Q: $3.87 = 3.8700$ (4 d.p.)R: $\dfrac{31}{8} = 3.8750$ (4 d.p.)S: $3.\dot{8} = 3.8889$ (4 d.p.)
2. Step 2: Compare the decimals: smallest firstIn order: Q, P, R, S
Question 5Always true? Give a counterexample
Priya writes: "The product of two irrational numbers is always irrational." State whether the statement is true or false. If it is false, give a counterexample. Choose the answer with its reason.
Show the worked solution
1. Step 1: A statement that says always is false if ONE example breaks it (a counterexample); one example that works proves nothingThe statement is false.
2. Step 2: Check the counterexample $\sqrt{5}$ is irrational, but $\sqrt{5} \times \sqrt{5} = 5$ , which is rational.

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Questions

Question 1Rational or irrational?

State whether $\sqrt{75}$ is a rational number or an irrational number.

Show the worked solution

Step 1: A square root is rational only when the number under it is a perfect square $75$ is not a perfect square, so $\sqrt{75}$ is a non-terminating, non-recurring decimal.
Step 2: Irrational: it cannot be written as a fraction of two integersSo $\sqrt{75}$ is irrational.

Question 2Pick the odd one out

Which one of the following numbers is an irrational number?

Show the worked solution

Step 1: Decide for each number: can it be written as a fraction of two integers?A: $\sqrt{36}$ is rational.B: $\sqrt[3]{8}$ is rational.C: $\sqrt{20}$ is irrational.D: $\dfrac{22}{7}$ is rational.
Step 2: Choose the irrational numberThe answer is C.

Question 3Square roots on a number line

The points P, Q, R and S are marked on the number line. Which point represents $\sqrt{10}$ ?

Show the worked solution

Step 1: Find the perfect squares on either side $9 < 10 < 16$
Step 2: Take square roots: √n lies between their roots $3 < \sqrt{10} < 4$
Step 3: See which square it is nearer to; a calculator checks it $\sqrt{10} = 3.16\ldots$ So $\sqrt{10}$ is nearer to $3$ .
Step 4: Read off the pointThe point is Q.

Question 4Order rational and irrational numbers

Use a calculator to arrange the following numbers in ascending order: P $= \sqrt{15}$ , Q $= 3.87$ , R $= \dfrac{31}{8}$ , S $= 3.\dot{8}$ . Write down the letters in order, starting with the smallest.

Show the worked solution

Step 1: Write each number as a decimal, correct to 4 decimal placesP: $\sqrt{15} = 3.8730$ (4 d.p.)Q: $3.87 = 3.8700$ (4 d.p.)R: $\dfrac{31}{8} = 3.8750$ (4 d.p.)S: $3.\dot{8} = 3.8889$ (4 d.p.)
Step 2: Compare the decimals: smallest firstIn order: Q, P, R, S

Question 5Always true? Give a counterexample

Priya writes: "The product of two irrational numbers is always irrational." State whether the statement is true or false. If it is false, give a counterexample. Choose the answer with its reason.

Show the worked solution

Step 1: A statement that says always is false if ONE example breaks it (a counterexample); one example that works proves nothingThe statement is false.
Step 2: Check the counterexample $\sqrt{5}$ is irrational, but $\sqrt{5} \times \sqrt{5} = 5$ , which is rational.