Class 9 Maths Ch 1 Number Systems Ex 1.3

Class 9 Maths Ch 1 Number Systems Ex 1.3 focuses on the concept of irrational numbers and how to locate them on the number line using the Pythagoras Theorem. This part of the chapter helps students understand how irrational numbers differ from rational numbers and how they can still be represented geometrically.

In this solution set, each question is explained clearly with step-by-step methods to help you understand the logic behind locating square roots like √2, √3, √5, and so on. These types of questions are important as they build a strong foundation in understanding the real number system and their graphical representation.

Class 9 Maths Ch 1 Number Systems Ex 1.3 – Textbook

Question 1:
Write the following in decimal form and say what kind of decimal expansion each has:

(i) 36 / 100
Solution:
36 ÷ 100 = 0.36
Type of Decimal Expansion: Terminating

(ii) 1 / 11
Solution:
1 ÷ 11 = 0.090909…
Type of Decimal Expansion: Non-terminating, repeating

(iii) 4 1/8
Solution:
Convert to improper fraction: (4×8 + 1)/8 = 33/8
33 ÷ 8 = 4.125
Type of Decimal Expansion: Terminating

(iv) 3 / 13
Solution:
3 ÷ 13 = 0.230769230769…
Type of Decimal Expansion: Non-terminating, repeating

(v) 2 / 11
Solution:
2 ÷ 11 = 0.181818…
Type of Decimal Expansion: Non-terminating, repeating

(vi) 329 / 400
Solution:
329 ÷ 400 = 0.8225
Type of Decimal Expansion: Terminating.

Question 2:
You know that 1/7 = 0.142857 (repeating).
Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, and 6/7 are, without actually doing the long division? If so, how?

Answer:
Yes, we can predict the decimal expansions without doing long division.

Since
1/7 = 0.142857 (repeating),

all the other fractions with denominator 7 (like 2/7, 3/7, etc.) will have the same digits in a different order — they follow a cyclic pattern of the digits 142857.

So,

2/7 = 0.285714 (repeating)
3/7 = 0.428571 (repeating)
4/7 = 0.571428 (repeating)
5/7 = 0.714285 (repeating)
6/7 = 0.857142 (repeating)

Each of these is a non-terminating, repeating decimal and uses the same digits as 1/7 in a rotated order.

Ex 1.3 Class 9 Maths Question 3:
Express the following in the form p/q, where p and q are integers and q ≠ 0.

(i) 0.6̅
Let x = 0.666…
Multiply both sides by 10:
10x = 6.666…

Now subtract:
10x − x = 6.666… − 0.666…
9x = 6
x = 6/9 = 2/3

Answer: 2/3

(ii) 0.47̅
Let x = 0.474747…
Multiply both sides by 100:
100x = 47.474747…

Now subtract:
100x − x = 47.474747… − 0.474747…
99x = 47
x = 47/99

Answer: 47/99

(iii) 0.001̅
Let x = 0.001001001…
This is a repeating decimal with 3 repeating digits.

Multiply both sides by 1000:
1000x = 1.001001…

Now subtract:
1000x − x = 1.001001… − 0.001001…
999x = 1
x = 1/999

Answer: 1/999

Question 5.
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 117? Perform the division to check your answer.
Solution:

Question 5:
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.

Answer:
To find the maximum number of digits in the repeating block of a fraction of the form 1/p (where p is a prime number), we use this rule:

The maximum number of digits in the repeating block of 1/p is (p − 1).

So for 1/17:

Maximum number of digits in the repeating block = 17 − 1 = 16

Now, perform the division of 1 ÷ 17:

1 ÷ 17 = 0.0588235294117647 (this repeats)

So,
1/17 = 0.0588235294117647 (repeating block of 16 digits)

Question 6.
Look at several examples of rational numbers in the form p/q (q ≠ 0). Where, p and q are integers with no common factors other that 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Answer:
Yes. If a rational number p/q has a terminating decimal expansion, then the denominator q (in its simplest form) must have only 2 or 5 or both as prime factors.

That means q must be of the form:
q = 2ᵐ × 5ⁿ,
where m and n are non-negative integers (they can be zero).

Examples:

1/2 = 0.5 → q = 2
3/5 = 0.6 → q = 5
7/8 = 0.875 → q = 8 = 2 × 2 × 2
9/20 = 0.45 → q = 20 = 2 × 2 × 5

In all these examples, the denominator has only the prime factors 2 and/or 5, and the decimal terminates.rminates, and the denominator’s prime factors are only 2 or 5.

Ex 1.3 Class 9 Maths Question 7.
Write three numbers whose decimal expansions are non-terminating non-recurring.
Solution:
√2 = 1.414213562 ………..
√3 = 1.732050808 …….
√5 = 2.23606797 …….

Question:
Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Answer:
First, convert both rational numbers to decimal form for easier comparison:

5/7 ≈ 0.714285…
9/11 ≈ 0.818181…

Now, choose any three irrational numbers that lie between these two values.

Examples of irrational numbers between 0.714 and 0.818:

√0.52 ≈ 0.7211… (irrational)
√0.55 ≈ 0.7416… (irrational)
π/4 ≈ 0.7853… (irrational)

All of these numbers are non-terminating, non-repeating decimals, and they lie between 5/7 and 9/11.

Class 9 Maths Ch 1 Number Systems Ex 1.3 – NCERT Answers

Class 9 Maths Ch 1 Number Systems Ex 1.3 – Textbook

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