CBSE Class 9 Maths: Number Systems Exercise 1.1 (NCERT)

Understanding Rational Numbers and Their Properties

In mathematics, numbers are classified into different systems based on their properties and characteristics. Exercise 1.1 primarily focuses on rational numbers. A number 's' is called a rational number if it can be written in the form p/q, where 'p' and 'q' are integers and 'q' is not equal to zero (q ≠ 0). This definition is fundamental. Natural numbers (1, 2, 3,...), whole numbers (0, 1, 2, 3,...), and integers (..., -2, -1, 0, 1, 2,...) are all rational numbers because they can be expressed in the p/q form. For example, 5 can be written as 5/1, -3 as -3/1, and 0 as 0/1. A crucial property of rational numbers is that there are infinitely many rational numbers between any two given rational numbers. This means you can always find another rational number, no matter how close two numbers seem. This property is what allows us to find multiple rational numbers between two specified values using various methods, which we will explore in detail.

Key Definitions in Number Systems

Natural Numbers (N)

These are the counting numbers: {1, 2, 3, 4, ...}. They are positive integers and do not include zero.

Whole Numbers (W)

This set includes all natural numbers along with zero: {0, 1, 2, 3, 4, ...}. Every natural number is a whole number.

Integers (Z)

Integers comprise all whole numbers and their negative counterparts: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Every whole number is an integer.

Rational Numbers (Q)

A number 's' is rational if it can be written in the form p/q, where p and q are integers, and q ≠ 0. Every integer is a rational number.

Method to Find Rational Numbers Between Two Given Numbers

1. Method 1: Average Method (Finding one at a time) — To find a rational number between two distinct rational numbers 'a' and 'b', you can use their average: (a + b) / 2. This method can be repeated to find more rational numbers. For example, to find a rational number between 2 and 3, calculate (2+3)/2 = 2.5 (or 5/2). Then, to find another between 2 and 2.5, calculate (2+2.5)/2 = 2.25, and so on. This method can be a bit tedious if you need many numbers.
2. Method 2: Equivalent Fraction Method (For finding 'n' numbers) — This is often the preferred method for finding multiple rational numbers. If you need to find 'n' rational numbers between 'a' and 'b': 1. Convert 'a' and 'b' into equivalent fractions with a common denominator. If 'a' and 'b' are integers, write them as a/1 and b/1. 2. Multiply the numerator and denominator of both fractions by (n + 1). This step creates enough 'space' between the numbers. 3. The rational numbers between the new numerators (keeping the common denominator) are your required numbers. For example, to find 5 rational numbers between 3 and 4, we need n=5, so n+1=6. Write 3 as 3/1 and 4 as 4/1. Then multiply by 6/6: 3/1 = 18/6 and 4/1 = 24/6. The rational numbers are 19/6, 20/6, 21/6, 22/6, 23/6.

Worked Examples from Number Systems Ex 1.1

• Example 1: Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0? A: Step 1: Recall the definition of a rational number: it can be expressed as p/q, where p and q are integers and q ≠ 0. Step 2: Consider the number zero (0). Can we write it as a fraction? Step 3: Yes, 0 can be written as 0/1, 0/2, 0/3, 0/-5, etc. In all these cases, 'p' is 0 (an integer), and 'q' is a non-zero integer (1, 2, 3, -5, etc.). Final answer: Yes, zero is a rational number.
• Example 2: Find six rational numbers between 3 and 4. A: Step 1: We need to find n=6 rational numbers. Use the (n+1) method, so multiply by (6+1)=7. Step 2: Write 3 and 4 as fractions with denominator 1: 3/1 and 4/1. Step 3: Multiply the numerator and denominator of both fractions by 7: 3/1 = (3 × 7) / (1 × 7) = 21/7 4/1 = (4 × 7) / (1 × 7) = 28/7 Step 4: Now, identify the integers between the new numerators (21 and 28): 22, 23, 24, 25, 26, 27. Final answer: Six rational numbers between 3 and 4 are 22/7, 23/7, 24/7, 25/7, 26/7, and 27/7.
• Example 3: Find five rational numbers between 3/5 and 4/5. A: Step 1: We need to find n=5 rational numbers. Use the (n+1) method, so multiply by (5+1)=6. Step 2: The given numbers are already fractions with a common denominator: 3/5 and 4/5. Step 3: Multiply the numerator and denominator of both fractions by 6: 3/5 = (3 × 6) / (5 × 6) = 18/30 4/5 = (4 × 6) / (5 × 6) = 24/30 Step 4: Identify the integers between the new numerators (18 and 24): 19, 20, 21, 22, 23. Final answer: Five rational numbers between 3/5 and 4/5 are 19/30, 20/30, 21/30, 22/30, and 23/30.

Exam Tips and Common Mistakes to Avoid

When working with Number Systems, especially Exercise 1.1, students often make a few common errors. Firstly, always remember the condition 'q ≠ 0' in the definition of a rational number. Forgetting this can lead to incorrect reasoning. Secondly, when finding rational numbers between two fractions, ensure you have a common denominator before applying the (n+1) multiplication trick; if denominators are different, find the LCM first. Lastly, don't confuse rational numbers with integers or whole numbers; while all integers are rational, not all rational numbers are integers. Always provide proper justification for True/False questions, citing definitions or counter-examples. Practice helps solidify these concepts and avoids silly mistakes.

Practice Questions with Solutions

• Q: State whether the following statements are True or False. Give reasons for your answers. (i) Every natural number is a whole number. (ii) Every integer is a whole number. (iii) Every rational number is an integer. A: Step 1: (i) Natural numbers are {1, 2, 3, ...} and whole numbers are {0, 1, 2, 3, ...}. Since all natural numbers are present in the set of whole numbers, the statement is True. Step 2: (ii) Integers are {..., -2, -1, 0, 1, 2, ...} and whole numbers are {0, 1, 2, 3, ...}. Negative integers like -1, -2 are integers but not whole numbers. So, the statement is False. Step 3: (iii) Rational numbers are of the form p/q. Integers are {..., -2, -1, 0, 1, 2, ...}. A rational number like 1/2 is not an integer. So, the statement is False. Final answer: (i) True, (ii) False, (iii) False
• Q: Find five rational numbers between 1 and 2.
• Q: Find four rational numbers between 2/3 and 4/5.
• Q: Can every whole number be expressed as a rational number? Justify your answer.
• Q: Is 7 a rational number? Why or why not?
• A: Step 1: We need to find n=5 rational numbers between 1 and 2. Use the (n+1) method, so multiply by (5+1)=6. Step 2: Write 1 and 2 as fractions with denominator 1: 1/1 and 2/1. Step 3: Multiply the numerator and denominator of both fractions by 6: 1/1 = (1 × 6) / (1 × 6) = 6/6 2/1 = (2 × 6) / (1 × 6) = 12/6 Step 4: Identify the integers between the new numerators (6 and 12): 7, 8, 9, 10, 11. Final answer: Five rational numbers between 1 and 2 are 7/6, 8/6, 9/6 (or 3/2), 10/6 (or 5/3), and 11/6.
• A: Step 1: We need to find n=4 rational numbers. The denominators are different (3 and 5), so first find a common denominator, which is LCM(3, 5) = 15. Step 2: Convert the fractions to equivalent fractions with denominator 15: 2/3 = (2 × 5) / (3 × 5) = 10/15 4/5 = (4 × 3) / (5 × 3) = 12/15 Step 3: We need 4 rational numbers, so multiply by (n+1) = (4+1) = 5. Multiply the numerator and denominator by 5: 10/15 = (10 × 5) / (15 × 5) = 50/75 12/15 = (12 × 5) / (15 × 5) = 60/75 Step 4: Identify the integers between the new numerators (50 and 60): 51, 52, 53, 54, 55, 56, 57, 58, 59. We need only four. Final answer: Four rational numbers between 2/3 and 4/5 are 51/75, 52/75, 53/75, and 54/75.
• A: Step 1: Recall the definition of a rational number: a number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Step 2: Consider any whole number, for example, 0, 1, 2, 3. Step 3: We can write 0 as 0/1, 1 as 1/1, 2 as 2/1, 3 as 3/1, etc. In all these cases, the numerator (p) is an integer, and the denominator (q) is 1, which is a non-zero integer. Final answer: Yes, every whole number can be expressed as a rational number because any whole number 'w' can be written as w/1, satisfying the definition of a rational number.
• A: Step 1: Recall the definition of a rational number: it can be expressed as p/q, where p and q are integers and q ≠ 0. Step 2: Consider the number 7. Can we write it as a fraction? Step 3: Yes, 7 can be written as 7/1. Here, 'p' is 7 (an integer), and 'q' is 1 (a non-zero integer). Final answer: Yes, 7 is a rational number because it can be expressed in the form 7/1, where both 7 and 1 are integers and the denominator 1 is not zero.

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