Unit 1: Real Numbers – Exercise 1.1 (Solved)

Welcome to the complete 9th Class Mathematics Unit 1 – Real Numbers Exercise 1.1 solved notes according to the Punjab Curriculum & Textbook Board (PCTB) New Syllabus 2026.

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📖 Question 1

Identify each of the following as Rational or Irrational Numbers.

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(i) 2.353535…

🎯 Concept

A decimal that terminates or repeats is called a Rational Number.

🔍 Solution

The decimal

$$2.353535...$$

has a repeating pattern.

A repeating decimal can always be written as

$$\frac{p}{q}$$

where

$$q\ne0$$

✅ Final Answer

Rational Number

💡 Exam Tip

Every terminating and repeating decimal is a Rational Number.

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(ii)

$$0.\overline6$$

🎯 Concept

A repeating decimal is always rational.

🔍 Solution

Since 6 repeats forever,

it can be expressed as a fraction.

✅ Final Answer

Rational Number

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(iii)

$$2.236067...$$

🎯 Concept

A non-terminating and non-repeating decimal is irrational.

🔍 Solution

This decimal never ends and has no repeating pattern.

✅ Final Answer

Irrational Number

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(iv)

$$\sqrt7$$

🎯 Concept

The square root of a non-perfect square is irrational.

🔍 Solution

7 is not a perfect square.

Therefore,

$$\sqrt7$$

cannot be expressed as a fraction.

✅ Final Answer

Irrational Number

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(v)

$$e$$

🎯 Concept

Euler’s Number is irrational.

✅ Final Answer

Irrational Number

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(vi)

$$\pi$$

🎯 Concept

Pi is irrational.

✅ Final Answer

Irrational Number

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(vii)

$$5+\sqrt{11}$$

🎯 Concept

A rational number plus an irrational number remains irrational.

🔍 Solution

5 is rational.

$$\sqrt{11}$$

is irrational.

Therefore,

their sum is irrational.

✅ Final Answer

Irrational Number

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(viii)

$$\sqrt3+\sqrt{13}$$

🎯 Concept

The sum of these irrational numbers is irrational.

✅ Final Answer

Irrational Number

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(ix)

$$\frac{15}{4}$$

🎯 Concept

Every number written as

$$\frac{p}{q}$$

where

$$q\ne0$$

is rational.

✅ Final Answer

Rational Number

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(x)

$$(2-\sqrt2)(2+\sqrt2)$$

🎯 Concept

Use the identity

$$(a-b)(a+b)=a^2-b^2$$

🔍 Step 1

Compare

$$a=2,\qquad b=\sqrt2$$

🔍 Step 2

Apply the identity.

$$=(2)^2-(\sqrt2)^2$$

🔍 Step 3

Square each term.

$$=4-2$$

🔍 Step 4

Simplify.

$$=2$$

2 is an integer.

Therefore,

✅ Final Answer

Rational Number

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📖 Question 2

Represent the following numbers on the Number Line.

(i)

$$\sqrt2$$

Solution

$$\sqrt2=1.414...$$

Locate the point slightly after 1.4 on the number line.

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(ii)

$$\sqrt3$$

$$\sqrt3=1.732...$$

Locate it between 1 and 2.

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(iii)

$$4\frac13$$

Convert to an improper fraction.

$$=\frac{13}{3}$$

Convert into decimal.

$$=4.333...$$

Locate between 4 and 5.

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(iv)

$$-2\frac17$$

Convert.

$$=-\frac{15}{7}$$

$$=-2.143...$$

Locate between −3 and −2.

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(v)

$$\frac58$$

Convert.

$$=0.625$$

Locate between 0 and 1.

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(vi)

$$2\frac34$$

Convert.

$$=\frac{11}{4}$$

$$=2.75$$

Locate between 2 and 3.

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📖 Question 3

Express the following as Rational Numbers.

(i)

$$0.\overline4$$

🎯 Concept

Convert a repeating decimal into a fraction.

📝 Rule

• One repeating digit → Multiply by 10
• Two repeating digits → Multiply by 100
• Three repeating digits → Multiply by 1000

🔍 Step 1

Let

$$x=0.\overline4$$

🔍 Step 2

Multiply both sides by 10.

$$10x=4.\overline4$$

🔍 Step 3

Subtract the first equation.

$$10x-x = 4.\overline4-0.\overline4$$

$$9x=4$$

🔍 Step 4

Divide by 9.

$$x=\frac49$$

✅ Final Answer

$$\boxed{\frac49}$$

⚠️ Common Mistake

Students often multiply by 100. Since only one digit repeats, multiply by 10.

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(ii)

$$0.\overline{37}$$

Use the same method.

Since 2 digits repeat, multiply by 100.

$$x=0.\overline{37}$$

$$100x=37.\overline{37}$$

Subtract.

$$99x=37$$

$$x=\frac{37}{99}$$

✅ Final Answer

$$\boxed{\frac{37}{99}}$$

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(iii)

$$0.\overline{21}$$

Since 2 digits repeat, multiply by 100.

$$x=0.\overline{21}$$

$$100x=21.\overline{21}$$

Subtract.

$$99x=21$$

$$x=\frac{21}{99}$$

Simplify.

$$=\frac7{33}$$

✅ Final Answer

$$\boxed{\frac7{33}}$$

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📖 Question 4

Name the Property Used

Part	Property
(i)	Associative Property of Addition
(ii)	Commutative Property of Addition
(iii)	Additive Inverse Property
(iv)	Distributive Property
(v)	Additive Identity Property
(vi)	Multiplicative Identity Property
(vii)	Associative Property of Multiplication
(viii)	Commutative Property of Multiplication

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📖 Question 5

Name the Property Used

Part	Property
(i)	Addition Property of Inequality
(ii)	Reciprocal Property
(iii)	Addition Property of Inequality
(iv)	Division Property (Positive Number)
(v)	Division Property (Negative Number)
(vi)	Trichotomy Property

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📖 Question 6

Find Two Rational Numbers Between

(i)

$$\frac13 \quad\text{and}\quad \frac14$$

Convert both fractions to a common denominator.

$$\frac13=\frac{12}{36}$$

$$\frac14=\frac9{36}$$

Choose any fractions between them.

$$\frac{10}{36},\qquad\frac{11}{36}$$

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(ii)

Between 3 and 4

Two rational numbers are

$$3.25,\qquad3.5$$

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(iii)

$$\frac35 \quad\text{and}\quad \frac45$$

Two rational numbers are

$$\frac{13}{20},\qquad\frac7{10}$$

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📚 Exercise Summary

✅ Learned Rational Numbers

✅ Learned Irrational Numbers

✅ Number Line Representation

✅ Repeating Decimals to Fractions

✅ Properties of Real Numbers

✅ Properties of Inequalities

✅ Finding Rational Numbers Between Two Numbers