Rational and Irrational Numbers ICSE Class 9: Surds, Rationalisation and Proofs

What Rational and Irrational Numbers is about

Weightage: CISCE publishes no chapter weightage and no question-paper layout for ICSE Class 9 Mathematics. Class IX is examined by your school: one 80-mark paper of 3 hours plus 20 marks of internal assessment. Rational and Irrational Numbers is in the 1. Pure Arithmetic part of the 2028 syllabus.Source: CISCE ICSE 2028 Mathematics syllabus, Class IX

Key concepts

Real numbers: rational and irrational

The number system grows step by step: natural numbers $N$, whole numbers $W$, integers $Z$, rational numbers $Q$, and finally real numbers $R$, which are all the rational and irrational numbers together.

• A rational number has a decimal that either terminates ($\frac{3}{8}=0.375$) or recurs ($\frac{2}{11}=0.\overline{18}$).
• An irrational number has a decimal that is non-terminating and non-recurring ($\sqrt{2}=1.41421356\dots$).

A fraction in lowest terms terminates only when its denominator has no prime factors other than 2 and 5.

The sum or product of a non-zero rational number and an irrational number is always irrational. The sum or product of two irrational numbers may be rational: $\sqrt{2}\times \sqrt{2}=2$, and $(2+\sqrt{3})+(2-\sqrt{3})=4$.

Worked example

Classify as rational or irrational: (a) $\sqrt{49}$ (b) $0.1010010001\dots$ (c) $\frac{\sqrt{12}}{\sqrt{3}}$ (d) $3+\sqrt{5}$.

Try it first, then show the solutionSolution

1. (a) $\sqrt{49}=7$: rational.
2. (b) The pattern grows (one more 0 each time), so the decimal never repeats a fixed block: irrational.
3. (c) $\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{4}=2$: rational.
4. (d) Rational 3 plus irrational $\sqrt{5}$: irrational.

Answer: (a) rational (b) irrational (c) rational (d) irrational

Converting a recurring decimal to p/q

Let $x$ equal the decimal. Multiply by a power of 10 that shifts one full repeating block past the decimal point, then subtract so the repeating tails cancel.

• Pure recurring: $x=0.\overline{36}$. Block length 2, so $100x=36.\overline{36}$; subtract: $99x=36$, $x=\frac{4}{11}$.
• Mixed recurring: first move the non-repeating digits past the point, then repeat the trick.

This proves every recurring decimal is rational.

Worked example

Express $0.2\overline{3}$ in the form $\frac{p}{q}$.

Try it first, then show the solutionSolution

1. Let $x=0.2333\dots$ Then $10x=2.333\dots$ and $100x=23.333\dots$
2. Subtract: $100x-10x=23.333\cdots -2.333\cdots =21$.
3. $90x=21$, so $x=\frac{21}{90}=\frac{7}{30}$.

Answer: $\frac{7}{30}$

Surds and their simplification

A surd is an irrational root of a positive rational number, such as $\sqrt{6}$ or $\sqrt[3]{4}$. The number under the root sign is the radicand; the index of the root is the order ($\sqrt{6}$ is of order 2).

Simplify a surd by taking out the largest perfect-square factor: $\sqrt{72}=\sqrt{36\times 2}=6\sqrt{2}$.

Like surds have the same irrational part after simplifying ($2\sqrt{3}$ and $5\sqrt{3}$); only like surds can be added or subtracted. Multiplication uses $\sqrt{a}\times \sqrt{b}=\sqrt{ab}$ for positive $a$ and $b$.

To compare surds, square them (or raise them to a common order): $2\sqrt{3}=\sqrt{12}$ and $3\sqrt{2}=\sqrt{18}$, so $3\sqrt{2}>2\sqrt{3}$.

Worked example

Simplify $\sqrt{50}+\sqrt{18}-\sqrt{8}$.

Try it first, then show the solutionSolution

1. $\sqrt{50}=5\sqrt{2}$, $\sqrt{18}=3\sqrt{2}$, $\sqrt{8}=2\sqrt{2}$.
2. All are like surds: $5\sqrt{2}+3\sqrt{2}-2\sqrt{2}=6\sqrt{2}$.

Answer: $6\sqrt{2}$

Rationalising the denominator

To rationalise means to make the denominator rational by multiplying the numerator and the denominator by the same suitable surd, called the rationalising factor.

• For $\frac{1}{\sqrt{a}}$, multiply by $\frac{\sqrt{a}}{\sqrt{a}}$.
• For $\frac{1}{a+\sqrt{b}}$, multiply by the conjugate $a-\sqrt{b}$, because $(a+\sqrt{b})(a-\sqrt{b})=a^2-b$, which is rational.
• For $\frac{1}{\sqrt{a}-\sqrt{b}}$, the conjugate is $\sqrt{a}+\sqrt{b}$, giving $a-b$.

The identity at work is $(x+y)(x-y)=x^2-y^2$.

Worked example

Rationalise the denominator of $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$.

Try it first, then show the solutionSolution

1. Multiply numerator and denominator by $\sqrt{5}+\sqrt{3}$.
2. Denominator: $(\sqrt{5}{)}^2-(\sqrt{3}{)}^2=5-3=2$.
3. Numerator: $(\sqrt{5}+\sqrt{3}{)}^2=5+3+2\sqrt{15}=8+2\sqrt{15}$.
4. Result: $\frac{8+2\sqrt{15}}{2}=4+\sqrt{15}$.

Answer: $4+\sqrt{15}$

Representing √2, √3 and √5 on the number line

Use Pythagoras theorem to build a length equal to the root, then transfer it to the number line with compasses.

• $\sqrt{2}$: draw a unit square on $0$ to $1$; its diagonal from $O$ is $\sqrt{1^2+1^2}=\sqrt{2}$. With centre $O$ and this radius, cut the number line.
• $\sqrt{5}$: at the point 2, draw a perpendicular of length 1; the hypotenuse from $O$ is $\sqrt{2^2+1^2}=\sqrt{5}$.
• $\sqrt{3}$: at the end of a segment of length $\sqrt{2}$, draw a perpendicular of length 1; the hypotenuse is $\sqrt{2+1}=\sqrt{3}$ (the 'square-root spiral').

Rational numbers such as $\frac{3}{4}$ are marked by dividing a unit into equal parts.

Worked example

Describe how to locate $\sqrt{10}$ on the number line.

Try it first, then show the solutionSolution

1. $10=3^2+1^2$, so take $OA=3$ units on the number line.
2. At $A$ draw $AB\perp OA$ with $AB=1$ unit.
3. $OB=\sqrt{9+1}=\sqrt{10}$. With centre $O$ and radius $OB$, draw an arc to cut the number line at $P$; $OP=\sqrt{10}$.

Answer: P is at √10 (between 3 and 4, about 3.16).

Proof that √2 is irrational (contradiction)

A proof by contradiction assumes the opposite of what you want and shows that this leads to something impossible.

1. Suppose $\sqrt{2}=\frac{p}{q}$, where $p$ and $q$ are integers, $q\ne 0$, with no common factor (lowest terms).
2. Square: $2q^2=p^2$. So $p^2$ is even, and therefore $p$ is even (the square of an odd number is odd).
3. Write $p=2k$. Then $2q^2=4k^2$, so $q^2=2k^2$; hence $q$ is also even.
4. Now 2 divides both $p$ and $q$, contradicting 'no common factor'.
5. So the assumption is false: $\sqrt{2}$ is irrational.

For $\sqrt{3}$ and $\sqrt{5}$ the same steps work with 'divisible by 3' or 'divisible by 5', using the fact that if a prime divides $p^2$, it divides $p$.

Worked example

Using the irrationality of $\sqrt{3}$, show that $2+\sqrt{3}$ is irrational.

Try it first, then show the solutionSolution

1. Suppose $2+\sqrt{3}=r$, where $r$ is rational.
2. Then $\sqrt{3}=r-2$. A rational number minus 2 is rational.
3. This says $\sqrt{3}$ is rational, which is false. So $2+\sqrt{3}$ is irrational.

Answer: 2 + √3 is irrational.

Maths in real life: Rational and Irrational Numbers

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Rational and Irrational Numbers, Recurring Decimals and the Irrationality of √2
• Surds: Simplification, Rationalising the Denominator and the Number Line

Rational and Irrational Numbers formula sheet

Rational number

$\frac{p}{q},p,q\in Z,q\ne 0$

Decimal terminates or recurs.

Product of surds

$\sqrt{a}\times \sqrt{b}=\sqrt{ab}$

For a, b > 0.

Quotient of surds

$\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$

For a ≥ 0, b > 0.

Conjugate product

$(a+\sqrt{b})(a-\sqrt{b})=a^2-b$

Used to rationalise a + √b.

Two surds

$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$

Used to rationalise √a ± √b.

Square of a binomial surd

$(\sqrt{a}\pm \sqrt{b}{)}^2=a+b\pm 2\sqrt{ab}$

Never write a + b alone.

Rational and Irrational Numbers: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the school exam asks Rational and Irrational Numbers

The question patterns to expect, and the method that works for each.

Tricks to remember Rational and Irrational Numbers

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Ends or repeats: rational

  Rational decimals either Stop or Loop. Irrational decimals never stop and never loop.

  Use it when: Classifying a decimal.

• Flip the middle sign

  The conjugate is the same two terms with the middle sign flipped: 3 + √2 becomes 3 − √2.

  Use it when: Choosing the rationalising factor.

• Square-root staircase

  Each new step: old hypotenuse as the base, 1 unit up. √1, √2, √3, √4 … climb the spiral.

  Use it when: Constructing √3, √5 on the number line.

• Assume, square, share a factor

  ASSF: Assume p/q in lowest terms, Square, Show both share a factor, contradiction Follows.

  Use it when: Irrationality proofs.

Rational and Irrational Numbers: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Rational and Irrational Numbers

The slips examiners see most often, and what to write instead.

• ✗Mistake: Squaring a binomial surd without the middle term

  Students write $(\sqrt{5}+\sqrt{3}{)}^2=5+3$, forgetting $2\sqrt{15}$.

  Wrong working

  $(\sqrt{5}+\sqrt{3}{)}^2=8$

  Right working

  $(\sqrt{5}+\sqrt{3}{)}^2=8+2\sqrt{15}$

  ✓Do this: Always use $(a+b{)}^2=a^2+b^2+2ab$.

  Usually costs: Every mark after that line.

• ✗Mistake: Multiplying only the denominator by the conjugate

  The value of the fraction changes unless numerator and denominator are both multiplied.

  ✓Do this: Write the conjugate as a fraction equal to 1, e.g. $\times \frac{2-\sqrt{3}}{2-\sqrt{3}}$.

  Usually costs: 1 to 2 marks.

• ✗Mistake: Adding unlike surds

  Students write $\sqrt{2}+\sqrt{3}=\sqrt{5}$, which is false ($1.414+1.732\ne 2.236$).

  Wrong working

  $\sqrt{2}+\sqrt{3}=\sqrt{5}$

  Right working

  $\sqrt{2}+\sqrt{3}$ cannot be simplified further

  ✓Do this: Only like surds combine; simplify each surd first.

  Usually costs: The whole simplification.

• ✗Mistake: Leaving out 'lowest terms' in the proof

  The contradiction depends on $p$ and $q$ having no common factor; without that assumption there is nothing to contradict.

  ✓Do this: Write 'where p and q are co-prime integers and q ≠ 0' in the first line.

  Usually costs: 1 mark of a 4 or 5-mark proof.

• ✗Mistake: Wrong multiplier for a mixed recurring decimal

  For $0.2\overline{3}$ students subtract $x$ from $10x$, but the tails do not cancel.

  ✓Do this: Shift so the repeating part starts right after the point in both numbers you subtract (here $100x$ and $10x$).

  Usually costs: The whole answer.

Practice questions on Rational and Irrational Numbers

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   The decimal form of $\frac{7}{40}$ is:
   (a) recurring
   (b) terminating
   (c) non-terminating non-recurring
   (d) none of these

   Hint

   Factorise 40.

   Answer

   (b): $40=2^3\times 5$, so $\frac{7}{40}=0.175$.

2. Q2 · 2 marks · Short answerExam level

   Express $0.\overline{27}$ as a fraction in lowest terms.

   Hint

   Two-digit block: multiply by 100.

   Answer

   $99x=27$, $x=\frac{27}{99}=\frac{3}{11}$.

3. Q3 · 2 marks · Short answerExam level

   Arrange in ascending order: $3\sqrt{2}$, $2\sqrt{5}$, $4$.

   Hint

   Square each number.

   Answer

   Squares: 18, 20, 16. So $4<3\sqrt{2}<2\sqrt{5}$.

4. Q4 · 1 mark · Assertion-reasonHarder exam level

   Assertion (A): $(3+\sqrt{2})(3-\sqrt{2})$ is rational.
   Reason (R): The product of two irrational numbers is always rational.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   Try √2 × √3.

   Answer

   (a). The product is $9-2=7$, rational; but $\sqrt{2}\times \sqrt{3}=\sqrt{6}$ shows (R) is false.

5. Q5 · 3 marks · Short answerHarder exam level

   Express $0.1\overline{36}$ as a fraction in lowest terms.

   Hint

   Multiply by 10 and by 1000.

   Answer

   $1000x-10x=136.\overline{36}-1.\overline{36}=135$; $990x=135$, $x=\frac{135}{990}=\frac{3}{22}$.

6. Q6 · 3 marks · Short answerHarder exam level

   Rationalise the denominator of $\frac{2\sqrt{3}-\sqrt{5}}{2\sqrt{2}+3\sqrt{3}}$.

   Hint

   Conjugate: $2\sqrt{2}-3\sqrt{3}$; denominator $8-27$.

   Answer

   Numerator $(2\sqrt{3}-\sqrt{5})(2\sqrt{2}-3\sqrt{3})=4\sqrt{6}-18-2\sqrt{10}+3\sqrt{15}$; denominator $=-19$. Result $\frac{18-4\sqrt{6}+2\sqrt{10}-3\sqrt{15}}{19}$.

7. Q7 · 3 marks · Short answerHarder exam level

   If $\frac{\sqrt{2}+1}{\sqrt{2}-1}=a+b\sqrt{2}$, find $a$ and $b$. Hence, using $\sqrt{2}=1.414$, find the value of the fraction.

   Hint

   Multiply by $\sqrt{2}+1$.

   Answer

   $\frac{(\sqrt{2}+1{)}^2}{1}=3+2\sqrt{2}$, so $a=3$, $b=2$; value $=3+2.828=5.828$.

8. Q8 · 4 marks · Short answerHarder exam level

   If $x=\frac{1}{3-2\sqrt{2}}$, find $x^2+\frac{1}{x^2}$.

   Hint

   Rationalise x; 1/x = 3 − 2√2.

   Answer

   $x=3+2\sqrt{2}$, $\frac{1}{x}=3-2\sqrt{2}$, $x+\frac{1}{x}=6$, so $x^2+\frac{1}{x^2}=36-2=34$.

9. Q9 · 5 marks · Long answerHarder exam level

   Prove that $\sqrt{3}$ is irrational. Hence show that $5-2\sqrt{3}$ is irrational.

   Hint

   Contradiction twice.

   Answer

   Assume $\sqrt{3}=\frac{p}{q}$ in lowest terms; $p^2=3q^2$, so 3 divides $p$; $p=3k$ gives $q^2=3k^2$, so 3 divides $q$: contradiction. If $5-2\sqrt{3}=r$ (rational), then $\sqrt{3}=\frac{5-r}{2}$ would be rational: contradiction.

10. Q10 · 3 marks · Short answerHarder exam level

    Describe, with a sketch, how to locate $\sqrt{3}$ on the number line starting from a unit square on 0 to 1.

    Hint

    First get √2, then build on it.

    Answer

    Diagonal $OB=\sqrt{2}$; transfer it to the line at A ($OA=\sqrt{2}$). At A draw $AC\perp OA$, $AC=1$. $OC=\sqrt{2+1}=\sqrt{3}$; arc from O with radius OC cuts the line at $\sqrt{3}$.

11. Q11 · 4 marks · Long answerHOTS / competency

    Rationalise $\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$.

    Hint

    Group as $(\sqrt{2}+\sqrt{3})+\sqrt{5}$; rationalise twice.

    Answer

    Multiply by $\sqrt{2}+\sqrt{3}-\sqrt{5}$: denominator $(\sqrt{2}+\sqrt{3}{)}^2-5=2\sqrt{6}$. Then multiply by $\frac{\sqrt{6}}{\sqrt{6}}$: $\frac{(\sqrt{2}+\sqrt{3}-\sqrt{5})\sqrt{6}}{12}=\frac{2\sqrt{3}+3\sqrt{2}-\sqrt{30}}{12}$.

12. Q12 · 4 marks · Long answerHOTS / competency

    Find rational $p$ and $q$ such that $\frac{1}{2+\sqrt{3}}+\frac{2}{\sqrt{5}-\sqrt{3}}+\frac{1}{2-\sqrt{5}}=p+q\sqrt{3}$.

    Hint

    Rationalise each term separately.

    Answer

    $\frac{1}{2+\sqrt{3}}=2-\sqrt{3}$; $\frac{2}{\sqrt{5}-\sqrt{3}}=\sqrt{5}+\sqrt{3}$; $\frac{1}{2-\sqrt{5}}=\frac{2+\sqrt{5}}{-1}=-2-\sqrt{5}$. Sum $=0$, so $p=0$, $q=0$.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Define a rational number.Show answer

   A number that can be written as p/q, where p and q are integers and q ≠ 0.

2. 2.What kind of decimal does an irrational number have?Show answer

   Non-terminating and non-recurring.

3. 3.What is the conjugate of 3 − √7?Show answer

   3 + √7.

4. 4.Simplify √48.Show answer

   4√3.

5. 5.Is the product of two irrational numbers always irrational?Show answer

   No: √2 × √2 = 2.

6. 6.When does p/q (lowest terms) give a terminating decimal?Show answer

   When q has no prime factors other than 2 and 5.

7. 7.Express 0.4̅ as a fraction.Show answer

   4/9.

8. 8.What is the rationalising factor of 1/√6?Show answer

   √6.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you tell rational from irrational by the decimal or by simplifying?
• ☐Can you convert 0.2̅3̅ type and 0.23̅ type decimals to fractions?
• ☐Do you simplify each surd before adding?
• ☐Do you multiply both numerator and denominator by the conjugate?
• ☐Can you find x² + 1/x² from x = a + √b?
• ☐Can you locate √2, √3, √5 on a number line?
• ☐Can you write the √2, √3 and √5 proofs from memory?

Rational and Irrational Numbers: frequently asked questions