Class 9 Maths 1 provides a thorough exploration of essential mathematical concepts, including rational and irrational numbers, operations on real numbers, and laws of exponents. This guide is designed for students preparing for their Class 9 mathematics exams, offering clear explanations and numerous exemplar problems. Key topics include locating irrational numbers on the number line and rationalizing denominators. The content is structured to enhance understanding and problem-solving skills, making it an invaluable resource for learners aiming to excel in mathematics.

Key Points

  • Explains rational and irrational numbers with examples and definitions.
  • Covers operations on real numbers, including addition, subtraction, multiplication, and division.
  • Includes laws of exponents relevant for Class 9 mathematics curriculum.
  • Provides multiple choice questions and exemplar problems for practice and assessment.
Alvina Akhlaq
12 pages
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Alvina Akhlaq
12 pages
Biology
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(A) Main Concepts and Results
Rational numbers
Irrational numbers
Locating irrational numbers on the number line
Real numbers and their decimal expansions
Representing real numbers on the number line
Operations on real numbers
Rationalisation of denominator
Laws of exponents for real numbers
• A number is called a rational number, if it can be written in the form
p
q
, where p
and q are integers and q ≠ 0.
• A number which cannot be expressed in the form
p
q
(where p and q are integers
and q ≠ 0) is called an irrational number.
• All rational numbers and all irrational numbers together make the collection of real
numbers.
• Decimal expansion of a rational number is either terminating or non-terminating
recurring, while the decimal expansion of an irrational number is non-terminating
non-recurring.
NUMBER SYSTEMS
CHAPTER 1
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2 EXEMPLAR PROBLEMS
• If r is a rational number and s is an irrational number, then r+s and r-s are irrationals.
Further, if r is a non-zero rational, then rs and
r
s
are irrationals.
• For positive real numbers a and b :
(i)
ab a b
=
(ii)
aa
b
b
=
(iii)
(
(
ab
abab
=−
+−
(iv)
(
)
(
)
2
ab
a ba b
=−
+−
(v)
()
2
2
a ab b
ab
=+ +
+
• If p and q are rational numbers and a is a positive real number, then
(i) a
p
. a
q
= a
p + q
(ii) (a
p
)
q
= a
pq
(iii)
p
pq
q
a
a
a
−
=
(iv) a
p
b
p
= (ab)
p
(B) Multiple Choice Questions
Write the correct answer:
Sample Question 1 : Which of the following is not equal to
1
–
1
6
5
5
6








?
(A)
11
–
56
5
6



(B)
1
1
6
5
1
5
6







(C)
1
30
6
5



(D)
1
–
30
5
6



Solution : Answer (A)
EXERCISE 1.1
Write the correct answer in each of the following:
1. Every rational number is
(A) a natural number (B) an integer
(C) a real number (D) a whole number
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NUMBER SYSTEMS 3
2. Between two rational numbers
(A) there is no rational number
(B) there is exactly one rational number
(C) there are infinitely many rational numbers
(D) there are only rational numbers and no irrational numbers
3. Decimal representation of a rational number cannot be
(A) terminating
(B) non-terminating
(C) non-terminating repeating
(D) non-terminating non-repeating
4. The product of any two irrational numbers is
(A) always an irrational number
(B) always a rational number
(C) always an integer
(D) sometimes rational, sometimes irrational
5. The decimal expansion of the number
2
is
(A) a finite decimal
(B) 1.41421
(C) non-terminating recurring
(D) non-terminating non-recurring
6. Which of the following is irrational?
(A)
4
9
(B)
12
3
(C)
7
(D)
81
7. Which of the following is irrational?
(A) 0.14 (B)
0.1416
(C)
0.1416
(D) 0.4014001400014...
8. A rational number between
2
and
3
is
(A)
23
2
+
(B)
23
2
⋅
(C) 1.5 (D) 1.8
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FAQs

What are rational and irrational numbers?
Rational numbers can be expressed as the quotient of two integers, where the denominator is not zero. Examples include fractions like 1/2 and whole numbers like 3. In contrast, irrational numbers cannot be expressed in this form; they have non-terminating, non-repeating decimal expansions, such as the square root of 2 or pi. Understanding these definitions is crucial for solving various mathematical problems.

How do you locate irrational numbers on the number line?
To locate an irrational number on the number line, you can approximate it using rational numbers. For example, the square root of 2 is approximately 1.414. You can find this point by identifying the integers it lies between, which are 1 and 2, and then estimating its position more precisely. This method helps visualize the placement of irrational numbers among rational numbers.

What operations can be performed on real numbers?
Real numbers can undergo various operations, including addition, subtraction, multiplication, and division. When performing these operations, it's essential to consider whether the numbers involved are rational or irrational, as this can affect the outcome. For instance, adding a rational number to an irrational number results in an irrational number, while multiplying two irrational numbers can yield either a rational or an irrational result.

What is the significance of rationalizing the denominator?
Rationalizing the denominator is a mathematical technique used to eliminate irrational numbers from the denominator of a fraction. This process simplifies calculations and makes the expression easier to work with. For example, to rationalize a denominator like √2, you would multiply both the numerator and denominator by √2, resulting in a rational denominator. This technique is particularly useful in algebra and higher-level mathematics.