Class 9 RD Sharma Solutions Chapter 1 Number System Exercise 1.1 (original) (raw)
Last Updated : 16 Sep, 2024
Exercise 1.1 in Chapter 1 "Number System" typically deals with irrational numbers and their properties. This exercise often focuses on identifying irrational numbers, understanding their characteristics, and performing basic operations with them.RD Sharma's Class 9 Mathematics textbook is a comprehensive guide designed to help students master fundamental mathematical concepts. Chapter 1, titled "Number System," is a crucial foundation for higher mathematics, introducing students to the complex world of numbers beyond the familiar integers and fractions they've encountered in earlier classes.
**Question 1: Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?
**Solution:
Yes, zero is a rational number.
It can be written in p/q form provided that q ≠ 0.
For Example: 0/5, 0/6, 0/7 etc.
**Question 2: Find five rational numbers between 1 and 2.
**Solution:
We know how to find a rational number between two numbers x and y = (x + y)/2
Now let's find 5 rational numbers between 1 and 2.
**Step 1: Rational number between 1 and 2
= (1 + 2)/2
= 3/2
**Step 2: Rational number between 1 and 3/2
= (1 + 3/2)/2
= 5/4
**Step 3: Rational number between 1 and 5/4
= (1 + 5/4)/2
= 9/8
**Step 4: Rational number between 3/2 and 2
= 1/2 [(3/2) + 2)]
= 7/4
**Step 5: Rational number between 7/4 and 2
= 1/2 [7/4 + 2]
= 15/8
Therefore, 5 rational numbers between 1 and 2 are **9/8, 5/4, 3/2, 7/4, 15/8
**Question 3: Find six rational numbers between 3 and 4.
**Solution:
To find n rational numbers between any two rational numbers we have to Multiply and divide both the numbers by n+1.
n = 6
So, n + 1 = 7
Multiplying and dividing 3 and 4 by 7,
3 × 7/7 = 21/7
4 × 7/7 = 28/7
Now we have to choose 6 rational numbers between 21/7 and 28/7
Therefore, 6 rational numbers between 3 and 4 are **22/7, 23/7, 24/7, 25/7, 26/7, 27/7
**Question 4: Find five rational numbers between 3/5 and 4/5.
**Solution:
To find n rational numbers between any two rational numbers we have to Multiply and divide both the numbers by n+1.
n = 5
So, n + 1 = 6
Multiplying and dividing 3/5 and 4/5 by 6,
3/5 × 6/6 = 18/30
4/5 × 6/6 = 24/30
Now we have to choose 5 rational numbers between 18/30 and 24/30
Therefore, 5 rational numbers between 3/5 and 4/5 are **19/30, 20/30, 21/30, 22,30, 23/30
**Question 5: Are the following statements true or false? Give a **reason for your answer.
****(i) Every whole number is a natural number.**
****(ii) Every integer is a rational number.**
****(iii) Every rational number is an integer.**
****(iv) Every natural number is a whole number,**
****(v) Every integer is a whole number.**
****(vi) Every rational number is a whole number.**
**Solution:
****(i)** False.
**Reason: because 0 is a whole number but not a natural number.
****(ii)** True
**Reason: because every integer can be represented in the form of a fraction n/1.
****(iii)** False.
**Reason: numbers such as 4/3, 2/9, 7/5 are rational numbers but not integers.
****(iv)** True.
Reason: every natural number is a whole number because whole numbers are positive integers including 0 and natural number are positive integers.
****(v)** False.
**Reason: Negative numbers are not whole numbers.
****(vi)** False.
**Reason: numbers such as 4/3, 2/9, 7/5 are rational numbers but not whole numbers.
Summary
Exercise 1.1 in RD Sharma's Class 9 textbook introduces students to the concept of irrational numbers, their properties, and operations involving them. It covers topics such as identifying irrational numbers, proving irrationality, and understanding the relationship between rational and irrational numbers. This exercise lays the foundation for more complex number theory concepts in later chapters.The exercise begins with fundamental concepts, such as identifying irrational numbers and understanding their decimal representations. It then progresses to more complex topics, including proving the irrationality of specific numbers like √2 and √3. Students learn to perform operations with irrational numbers, such as addition, subtraction, multiplication, and division, and understand how these operations can sometimes yield surprising results (e.g., the sum or product of irrational numbers can be rational in certain cases).