Get the best Rbse Solutions for Class 9 Maths Chapter 1 Exercise 1.1. Master rational numbers, integers, and whole numbers with detailed, step-by-step answers
The Number Systems chapter is the cornerstone of Class 9 Maths. It introduces students to the rigorous definitions and properties of the number line, differentiating between basic number types like natural, whole, and rational numbers.
Understanding these concepts thoroughly, particularly the properties of rational numbers and irrational numbers, is non-negotiable for success in subsequent classes. Our guide walks you through the core ideas and provides solutions to challenging problems from NCERT Book Exercise 1.1.
1. Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$?
Answer:
Yes, zero (0) is a rational number.
A rational number is defined as any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
We can write zero in the form $\frac{p}{q}$ in multiple ways:
$$\frac{0}{1}, \frac{0}{2}, \frac{0}{-5}, \text{etc.}$$
Here, $p=0$ (which is an integer) and $q$ can be any non-zero integer.
2. Find six rational numbers between 3 and 4.
Answer:
To find $n$ rational numbers between two numbers, we can multiply both numbers by $\frac{n+1}{n+1}$. Here, $n=6$, so we multiply by $\frac{6+1}{7} = \frac{7}{7}$.
- Write 3 and 4 as equivalent fractions with a denominator of 7:$$3 = \frac{3 \times 7}{7} = \frac{21}{7}$$$$4 = \frac{4 \times 7}{7} = \frac{28}{7}$$
- The six rational numbers between $\frac{21}{7}$ and $\frac{28}{7}$ are the numbers with numerators between 21 and 28:$$\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}$$
Alternative Method (Averaging): You can also find rational numbers by averaging: $\frac{3+4}{2} = \frac{7}{2}$. Then find numbers between 3 and $\frac{7}{2}$, and so on.
3. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.
Answer:
To find $n=5$ rational numbers, we multiply the fractions by $\frac{n+1}{n+1} = \frac{6}{6}$.
- Write $\frac{3}{5}$ and $\frac{4}{5}$ as equivalent fractions with a larger denominator:$$\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$$$$\frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}$$
- The five rational numbers between $\frac{18}{30}$ and $\frac{24}{30}$ are:$$\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30}$$
4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
Answer: True.
Reason:
- Natural Numbers (N) start from 1: $\{1, 2, 3, 4, \dots\}$
- Whole Numbers (W) start from 0: $\{0, 1, 2, 3, 4, \dots\}$Since the set of whole numbers includes all natural numbers (plus zero), every natural number is indeed a whole number.
(ii) Every integer is a whole number.
Answer: False.
Reason:
- Integers (Z) include negative numbers: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$
- Whole Numbers (W) do not include negative numbers.Therefore, negative integers (like $-2, -5, -100$) are integers but not whole numbers.
(iii) Every rational number is a whole number.
Answer: False.
Reason:
- Rational Numbers (Q) include fractions and decimals that are not whole numbers.A whole number must be a non-negative integer ($\{0, 1, 2, 3, \dots\}$).A rational number like $\frac{1}{2}$ or $-3$ is a rational number but not a whole number.
Rational Numbers Unlocked: Class 9 Maths Chapter 1 Guide
💡 Key Concepts from Exercise 1.1
Is Zero a Rational Number? (The FAQ)
This is one of the most frequently asked conceptual questions.
A number is rational if it can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Since zero can be represented as $\frac{0}{1}$, $\frac{0}{5}$, or $\frac{0}{-100}$, zero is definitively a rational number. The only constraint is that the denominator ($q$) cannot be zero.
Finding Rational Numbers Between Two Values
A key skill tested in NCERT Class 9 Maths Chapter 1 is the ability to insert multiple rational numbers between any two given numbers.
Fact: There are infinitely many rational numbers between any two given rational numbers.
To find a specific count (e.g., five or six), the simplest method is to convert both numbers into equivalent fractions with a common denominator larger than the required count ($n+1$).
- To find 6 numbers, use a denominator of 7.
- To find 5 numbers, use a denominator of 6.
This technique ensures that you can always find sequential integers in the numerator, providing the required quantity of distinct rational numbers.
❓ FAQs on Number Systems
Q: What is the difference between natural numbers and whole numbers?
A: The difference is only the number zero. Natural numbers start from 1 ($\{1, 2, 3, \dots\}$). Whole numbers include all natural numbers plus zero ($\{0, 1, 2, 3, \dots\}$). Every natural number is a whole number, but zero is a whole number that is not natural.
Q: Is every integer a rational number?
A: Yes. Every integer (positive, negative, or zero) can be written as a fraction with a denominator of 1. For example, $5 = \frac{5}{1}$ and $-3 = \frac{-3}{1}$. Since it satisfies the $\frac{p}{q}$ condition, every integer is also a rational number.
Q: Are all rational numbers also whole numbers?
A: No, this is false. Rational numbers include fractions ($\frac{1}{3}$), non-terminating but repeating decimals ($0.333\dots$), and negative numbers ($-2$). None of these are whole numbers, which must be non-negative integers ($\{0, 1, 2, \dots\}$).