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9th Class Math Chapter 1 Ex 1.1 Solved Notes

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9th Class Math Chapter 1 Ex 1.1 Solved Notes

Official Academic Study Notes • Published: September 06, 2026 • Free Printable Resource

What are the Core Formulas and Definitions for this Exercise?

Under the updated 2025-2026 National Curriculum for Grade 9 Mathematics, Chapter 1 focuses on the Real Number System ($\mathbb{R}$). Understanding the fundamental axioms and set definitions is essential before attempting algebraic simplifications.

1. Classification of Real Numbers:

Natural Numbers ($\mathbb{N}$): $\mathbb{N} = \{1, 2, 3, \dots\}$
Whole Numbers ($\mathbb{W}$): $\mathbb{W} = \{0, 1, 2, 3, \dots\}$
Integers ($\mathbb{Z}$): $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$
Rational Numbers ($\mathbb{Q}$): Any number that can be expressed in the form $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $b \neq 0$. In decimal form, rational numbers are either terminating or recurring.
Irrational Numbers ($\mathbb{Q}'$): Numbers that cannot be expressed as $\frac{a}{b}$ where $a, b \in \mathbb{Z}$ and $b \neq 0$. In decimal form, irrational numbers are non-terminating and non-recurring (e.g., $\sqrt{2}, \sqrt{3}, \pi, e$).
Real Numbers ($\mathbb{R}$): The union of set of rational and irrational numbers, i.e., $\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'$.

2. Fundamental Properties of Real Numbers with Respect to Addition and Multiplication:

Property Name	Addition Rule ($\forall a, b, c \in \mathbb{R}$)	Multiplication Rule ($\forall a, b, c \in \mathbb{R}$)
Closure Property	$a + b \in \mathbb{R}$	$a \cdot b \in \mathbb{R}$
Commutative Property	$a + b = b + a$	$a \cdot b = b \cdot a$
Associative Property	$(a + b) + c = a + (b + c)$	$(a \cdot b) \cdot c = a \cdot (b \cdot c)$
Identity Element	$a + 0 = a = 0 + a$ ($0$ is Additive Identity)	$a \cdot 1 = a = 1 \cdot a$ ($1$ is Multiplicative Identity)
Inverse Element	$a + (-a) = 0 = (-a) + a$ ($-a$ is Additive Inverse)	$a \cdot \frac{1}{a} = 1 = \frac{1}{a} \cdot a$, $a \neq 0$ ($\frac{1}{a}$ is Multiplicative Inverse)
Distributive Property	Left Distributive: $a(b + c) = ab + ac$ Right Distributive: $(a + b)c = ac + bc$

How to Solve All Exercise Questions Step-by-Step?

Below are detailed, step-by-step solutions for all key problem types presented in Exercise 1.1 of the 2025-2026 Grade 9 Textbook.

Question 1: Classify the following numbers as rational ($\mathbb{Q}$) or irrational ($\mathbb{Q}'$). Give reasons.

(i) $\sqrt{7}$

Solution:
Since $7$ is a prime number and not a perfect square, its square root $\sqrt{7}$ produces a non-terminating, non-recurring decimal approximation ($\approx 2.6457513...$).
Therefore, $\sqrt{7} \in \mathbb{Q}'$ (Irrational Number).

(ii) $\frac{22}{7}$

Solution:
The number is written in the standard quotient form $\frac{a}{b}$, where $a = 22 \in \mathbb{Z}$ and $b = 7 \in \mathbb{Z}$ ($b \neq 0$).
Therefore, $\frac{22}{7} \in \mathbb{Q}$ (Rational Number).

(iii) $\pi$

Solution:
The value $\pi$ represents the ratio of the circumference of a circle to its diameter. It is a well-known non-terminating, non-recurring mathematical constant.
Therefore, $\pi \in \mathbb{Q}'$ (Irrational Number). Note: $\frac{22}{7}$ is merely an approximation of $\pi$, not its exact value.

(iv) $0.3333...$

Solution:
The decimal expansion is recurring (repeating digit $3$). Every recurring decimal can be expressed as a fraction ($\frac{1}{3}$).
Therefore, $0.3333... \in \mathbb{Q}$ (Rational Number).

(v) $\sqrt{25}$

Solution:
Simplifying the expression: $$\sqrt{25} = 5 = \frac{5}{1}$$ Since $5$ is an integer, it can be written as a quotient of integers.
Therefore, $\sqrt{25} \in \mathbb{Q}$ (Rational Number).

Question 2: Name the property used in each of the following equations.

(i) $4 + 9 = 9 + 4$

Solution:
The order of addition is interchanged without affecting the result. This illustrates the Commutative Property of Real Numbers with respect to Addition.

(ii) $x \cdot (y \cdot z) = (x \cdot y) \cdot z$

Solution:
The grouping of factors is altered without changing the product. This illustrates the Associative Property of Real Numbers with respect to Multiplication.

(iii) $15 + (-15) = 0$

Solution:
Adding $-15$ to $15$ yields the additive identity ($0$). This illustrates the Additive Inverse Property.

(iv) $\sqrt{3} \cdot 1 = \sqrt{3}$

Solution:
Multiplying $\sqrt{3}$ by $1$ leaves the number unchanged. This illustrates the Multiplicative Identity Property.

(v) $3(x - y) = 3x - 3y$

Solution:
The factor $3$ is distributed over subtraction inside the parentheses. This illustrates the Left Distributive Property of Multiplication over Subtraction.

Question 3: Find the additive inverse and multiplicative inverse of the following real numbers.

(i) $-8$

Solution:
1. Additive Inverse: Let the additive inverse be $x$. $$-8 + x = 0 \implies x = 8$$ 2. Multiplicative Inverse: Let the multiplicative inverse be $y$. $$-8 \cdot y = 1 \implies y = -\frac{1}{8}$$

(ii) $\frac{3}{7}$

Solution:
1. Additive Inverse: $$\frac{3}{7} + \left(-\frac{3}{7}\right) = 0 \implies \text{Additive Inverse} = -\frac{3}{7}$$ 2. Multiplicative Inverse: $$\frac{3}{7} \cdot \left(\frac{7}{3}\right) = 1 \implies \text{Multiplicative Inverse} = \frac{7}{3}$$

(iii) $-\sqrt{5}$

Solution:
1. Additive Inverse: $$-\sqrt{5} + \sqrt{5} = 0 \implies \text{Additive Inverse} = \sqrt{5}$$ 2. Multiplicative Inverse: $$-\sqrt{5} \cdot \left(-\frac{1}{\sqrt{5}}\right) = 1 \implies \text{Multiplicative Inverse} = -\frac{1}{\sqrt{5}}$$

Question 4: Prove that $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$ using properties of real numbers.

Proof:
Consider the Left-Hand Side (L.H.S.): $$\text{L.H.S.} = \frac{a}{c} + \frac{b}{c}$$ Using the definition of division in real numbers, $\frac{x}{y} = x \cdot \left(\frac{1}{y}\right)$: $$\text{L.H.S.} = a \cdot \left(\frac{1}{c}\right) + b \cdot \left(\frac{1}{c}\right)$$ Apply the Right Distributive Property of Multiplication over Addition: $$\text{L.H.S.} = (a + b) \cdot \left(\frac{1}{c}\right)$$ Re-applying the definition of division: $$\text{L.H.S.} = \frac{a + b}{c} = \text{R.H.S.}$$ Hence proved using fundamental real number field properties.

Interactive Practice Quiz: Test Your Understanding (Clickable MCQs)

Q1: Which of the following sets is closed with respect to addition?

Explanation: For $\{0\}$, $0 + 0 = 0 \in \{0\}$. Thus, it satisfies the closure property under addition. For $\{1, -1\}$, $1 + 1 = 2 \notin \{1, -1\}$.

Q2: If $a, b \in \mathbb{R}$, then exactly one of $a < b$, $a = b$, or $a > b$ holds. This property is called:

Explanation: The trichotomy property states that any two real numbers compare in exactly one of three ways: less than, equal to, or greater than.

Q3: The multiplicative inverse of $-1$ in the set of real numbers is:

Explanation: $(-1) \times (-1) = 1$. Since the product yields the multiplicative identity ($1$), $-1$ is its own multiplicative inverse.

Q4: Which of the following numbers is an irrational number?

Explanation: $\sqrt{3} \approx 1.732...$ is non-terminating and non-recurring, so it belongs to $\mathbb{Q}'$. $\sqrt{16} = 4$ is rational.

Q5: The property used in $ab = ba$ is:

Explanation: Reordering terms in multiplication without changing the result defines the commutative property of multiplication.

Q6: What is the additive identity element in the set of real numbers $\mathbb{R}$?

Explanation: Adding $0$ to any real number $a$ leaves it unchanged ($a + 0 = a$). Therefore, $0$ is the additive identity.

Q7: The union of the set of rational numbers ($\mathbb{Q}$) and irrational numbers ($\mathbb{Q}'$) is equal to:

Explanation: By definition, the set of Real Numbers $\mathbb{R}$ is formed by combining rational and irrational numbers: $\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'$.

Q8: If $a < b$ and $b < c$, then $a < c$. This property is known as:

Explanation: The transitive property allows inequality relationship chains to transfer across elements ($a < b < c \implies a < c$).

Frequently Asked Questions: What are Common Student Errors in this Exercise?

Q1: Why is $\pi$ classified as an irrational number when $\frac{22}{7}$ is rational?
Ans: $\pi$ is an exact physical constant representing the ratio of a circle's circumference to its diameter, producing a non-terminating, non-recurring decimal ($3.14159265...$). $\frac{22}{7}$ is simply a convenient rational approximation ($3.142857...$) used for practical arithmetic. Do not confuse the exact constant $\pi$ with its rational approximation $\frac{22}{7}$.

Q2: What is the main difference between the additive inverse and the multiplicative inverse?
Ans: The additive inverse of a number $a$ is $-a$, because $a + (-a) = 0$ (the additive identity). The multiplicative inverse of a non-zero number $a$ is $\frac{1}{a}$, because $a \cdot \frac{1}{a} = 1$ (the multiplicative identity). Students often confuse the signs: the multiplicative inverse maintains the original sign of $a$, whereas the additive inverse flips the sign.

Q3: Does zero ($0$) have a multiplicative inverse?
Ans: No, $0$ does not have a multiplicative inverse. The multiplicative inverse requires finding a real number $x$ such that $0 \cdot x = 1$. Since any number multiplied by zero equals zero, no such $x$ exists. Division by zero is undefined in real number mathematics.