NCERT Solutions Class 9 Maths Chapter 1 Number Systems

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NCERT Solutions Class 9 Maths Chapter 1 – CBSE Free PDF Download

NCERT Solutions Class 9 Maths Chapter 1 provides comprehensive guidance for CBSE students, offering a free PDF download to help in their mathematical understanding. This resource covers fundamental concepts essential for building a strong foundation in mathematics. With detailed explanations and step-by-step solutions, students can clarify doubts and enhance their problem-solving skills. These NCERT solutions serve as a valuable tool for self-study, homework assistance, and exam preparation, ensuring students grasp key mathematical concepts effectively.

Download the Free NCERT Solutions for Class 9 Maths Chapter 1 PDF and other Chapter 1 Resources from the below link

• NCERT Solutions for Class 9 Maths Download PDF
• Number System Class 9 Notes Maths Chapter 1
• NCERT Exemplar Solutions Class 9 Maths Chapter 1
• CBSE Worksheet Class 9 Maths Chapter 1
• RD Sharma Class 9 Solutions Chapter 1

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems – Marking Scheme

As the Number System, the most important chapter in Maths, it has a weightage of 8 marks in Class 9 Maths CBSE Term I exams. At Least three questions are asked from this unit.

• One amongst three questions in part A (1 marks).

• One out of three questions in part B (2 marks).

• One out of three questions in part C (3 marks).

This chapter talks about:

• Introduction of Number Systems
• Irrational Numbers
• Real Numbers and their Decimal Expansions
• Representing Real Numbers on the Number Line.
• Operations on Real Numbers
• Laws of Exponents for Real Numbers
• Summary

List of Exercises in NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

• Exercise 1.1 Solutions 4 Questions ( 2 long, 2 short)
• Exercise 1.2 Solutions 4 Questions ( 3 long, 1 short)
• Exercise 1.3 Solutions 9 Questions ( 9 long)
• Exercise 1.4 Solutions 2 Questions ( 2 long)
• Exercise 1.5 Solutions 5 Questions ( 4 long 1 short)
• Exercise 1.6 Solutions 3 Questions ( 3 long)

NCERT Class 9 Maths Chapter 1 – Number Systems: Step-by-Step Solutions

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?

Step 1: Recall the definition of a rational number

• A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0

Step 2: Express zero in the form p/q

• Zero can be written as 0/1, 0/2, 0/3, etc.

Step 3: Verify the conditions

• In all these cases, p = 0 (integer) and q is any non-zero integer

Final Answer: Yes, zero is a rational number because it can be written as 0/q, where q is any non-zero integer.

Question 2: Find any six rational numbers between 3 and 4.

Step 1: Express 3 and 4 with the same denominator

• 3 = 3/1, 4 = 4/1

Step 2: To find multiple numbers between them, multiply numerator and denominator by 7

• 3 = 21/7, 4 = 28/7

Step 3: List six numbers with numerators between 21 and 28

• 22/7, 23/7, 24/7, 25/7, 26/7, 27/7

Final Answer: 22/7, 23/7, 24/7, 25/7, 26/7, 27/7

Question 3: Find five rational numbers between 1 and 2.

Step 1: Express 1 and 2 with the same denominator

• 1 = 6/6, 2 = 12/6

Step 2: List five numbers with numerators between 6 and 12

• 7/6, 8/6, 9/6, 10/6, 11/6

Final Answer: 7/6, 8/6, 9/6, 10/6, 11/6

Question 4: State whether the following are true or false. Give reasons:

(i) Every natural number is a whole number.

Step 1: Define natural numbers and whole numbers

• Natural numbers: 1, 2, 3, 4, …
• Whole numbers: 0, 1, 2, 3, 4, …

Step 2: Compare the sets

• All natural numbers are included in the set of whole numbers

Answer: True. Every natural number is a whole number because whole numbers include all natural numbers plus zero.

(ii) Every integer is a whole number.

Step 1: Define integers and whole numbers

• Integers: …, -2, -1, 0, 1, 2, …
• Whole numbers: 0, 1, 2, 3, …

Step 2: Identify the difference

• Integers include negative numbers, but whole numbers do not

Answer: False. Negative integers (like -1, -2, -3) are not whole numbers.

(iii) Every rational number is a whole number.

Step 1: Define rational numbers and whole numbers

• Rational numbers: All numbers that can be written as p/q (including fractions and negative numbers)
• Whole numbers: 0, 1, 2, 3, …

Step 2: Find counterexamples

• Rational numbers like 1/2, -3, 2.5 are not whole numbers

Answer: False. Many rational numbers (fractions, negative numbers) are not whole numbers.

Question 5: Classify the following numbers as rational or irrational: 2 – √5

Step 1: Identify the nature of √5

• √5 is an irrational number (cannot be expressed as p/q)

Step 2: Apply the property of irrational numbers

• When a rational number is added to or subtracted from an irrational number, the result is irrational

Final Answer: 2 – √5 is irrational.

Question 6: Find: 64^(1/2)

Step 1: Express 64 as a perfect square

• 64 = 8²

Step 2: Apply the exponent rule

• 64^(1/2) = (8²)^(1/2)

Step 3: Simplify using the power rule

• (8²)^(1/2) = 8^(2×1/2) = 8¹ = 8

Final Answer: 8

Question 7: Find: 32^(1/5)

Step 1: Express 32 as a perfect fifth power

• 32 = 2⁵

Step 2: Apply the exponent rule

• 32^(1/5) = (2⁵)^(1/5)

Step 3: Simplify using the power rule

• (2⁵)^(1/5) = 2^(5×1/5) = 2¹ = 2

Final Answer: 2

Question 8: Find: 125^(1/3)

Step 1: Express 125 as a perfect cube

• 125 = 5³

Step 2: Apply the exponent rule

• 125^(1/3) = (5³)^(1/3)

Step 3: Simplify using the power rule

• (5³)^(1/3) = 5^(3×1/3) = 5¹ = 5

Final Answer: 5

Question 9: Find five rational numbers between 3/5 and 4/5.

Step 1: Multiply numerator and denominator by 6 to create more options

• 3/5 = 18/30, 4/5 = 24/30

Step 2: List five numbers with numerators between 18 and 24

• 19/30, 20/30, 21/30, 22/30, 23/30

Final Answer: 19/30, 20/30, 21/30, 22/30, 23/30

Question 10: What is an irrational number? Give an example.

Step 1: Define an irrational number

• An irrational number is a number that cannot be written as p/q, where p and q are integers and q ≠ 0

Step 2: Provide characteristics

• Irrational numbers have non-terminating, non-repeating decimal expansions

Step 3: Give an example

• √2 is an irrational number

Final Answer: An irrational number cannot be expressed as a fraction p/q. Example: √2

Question 11: What is a real number?

Step 1: Define real numbers

• Real numbers include all rational and irrational numbers

Step 2: Explain representation

• All real numbers can be represented on the number line

Step 3: Give examples

• Examples include: 5 (natural), 0 (whole), -3 (integer), 2/3 (rational), √2 (irrational)

Final Answer: A real number is any number that can be found on the number line, including all rational and irrational numbers.

Question 12: Write the decimal expansion of 1/3. Is it terminating or non-terminating?

Step 1: Perform the division

• 1 ÷ 3 = 0.333…

Step 2: Observe the pattern

• The digit 3 repeats indefinitely

Final Answer: 1/3 = 0.333… (non-terminating, repeating decimal)

Question 13: Write the decimal expansion of 1/4. Is it terminating or non-terminating?

Step 1: Perform the division

• 1 ÷ 4 = 0.25

Step 2: Observe the result

• The decimal stops after two digits

Final Answer: 1/4 = 0.25 (terminating decimal)

Question 14: Represent √2 on the number line.

Step 1: Draw a number line and mark points O (0) and A (1)

Step 2: From point A, draw a perpendicular line of 1 unit length and mark point B

Step 3: Join O and B

• By Pythagoras theorem: OB = √(1² + 1²) = √2

Step 4: Use compass to draw an arc

• With O as center and OB as radius, draw an arc to intersect the number line at point C

Final Answer: Point C represents √2 on the number line.

Question 15: If a rational number’s decimal expansion terminates, what can you say about its denominator?

Step 1: Consider the condition for terminating decimals

• A rational number p/q (in lowest terms) has a terminating decimal if and only if the denominator q has no prime factors other than 2 and 5

Step 2: Explain why this is true

• Our decimal system is base 10 = 2 × 5, so only denominators with factors of 2 and/or 5 can produce terminating decimals

Final Answer: The denominator (when the fraction is in lowest terms) must have only 2 and/or 5 as its prime factors.

Question 16: Simplify 9^(3/2).

Step 1: Express 9 as a perfect square

• 9 = 3²

Step 2: Apply the exponent rule

• 9^(3/2) = (3²)^(3/2)

Step 3: Simplify using the power rule

• (3²)^(3/2) = 3^(2×3/2) = 3³ = 27

Final Answer: 27

Question 17: What is the value of 27^(2/3)?

Step 1: Express 27 as a perfect cube

• 27 = 3³

Step 2: Apply the exponent rule

• 27^(2/3) = (3³)^(2/3)

Step 3: Simplify using the power rule

• (3³)^(2/3) = 3^(3×2/3) = 3² = 9

Final Answer: 9

Question 18: State the law of exponents for real numbers: a^m × a^n.

Step 1: State the law

• When multiplying powers with the same base, add the exponents

Step 2: Write the formula

• a^m × a^n = a^(m+n)

Final Answer: a^m × a^n = a^(m+n)

Question 19: What is the decimal expansion of √3? Is it rational or irrational?

Step 1: Calculate √3

• √3 = 1.732050807…

Step 2: Observe the decimal pattern

• The decimal is non-terminating and non-repeating

Step 3: Classify the number

• Numbers with non-terminating, non-repeating decimals are irrational

Final Answer: √3 = 1.732050807… (non-terminating, non-repeating, therefore irrational)

Question 20: Rationalize the denominator of 1/√2.

Step 1: Multiply numerator and denominator by √2

• (1/√2) × (√2/√2)

Step 2: Simplify the numerator and denominator

• Numerator: 1 × √2 = √2
• Denominator: √2 × √2 = 2

Step 3: Write the final answer

• √2/2

Final Answer: √2/2

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

NCERT Solutions for Class 9 Maths Chapter 1 Number System is the first chapter of class 9 Maths. Number System is explained in detail in this chapter. The chapter helps to understand the Number Systems and their applications. The introduction of the chapter includes whole numbers, integers and rational numbers. To enhance the learning experience for Class 9 students, we have created a comprehensive set of multiple-choice questions (MCQs) covering various topics from the curriculum. These Class 9 Maths MCQs serve as additional resources to complement the NCERT Solutions

The chapter starts with the introduction of Number Systems in section 1.1 followed by two very important topics in sections 1.2 and 1.3

• Irrational Numbers – The numbers which cannot be written in the form of p/q.
• Real Numbers and their Decimal Expansions – Here you study the decimal expansions of real numbers and see whether it can help in distinguishing between rational and irrationals.

Next, it discusses the following topics.

• Representing Real Numbers on the Number Line – In this the solutions for 2 problems in Exercise 1.4.
• Operations on Real Numbers – Here you explore some of the operations like addition, subtraction, multiplication and division on irrational numbers.
• Laws of Exponents for Real Numbers – Use these laws of exponents to solve the questions.

Explore more about Number Systems and learn how to solve various kinds of problems only on NCERT Solutions For Class 9 Maths. It is also one of the best academic resources to revise for your term wise exams.

Advantages of Using NCERT Solutions Class 9 Maths Chapter 1 Number System

Key advantages of NCERT Solutions for Class 9 Maths Chapter 1- Number Systems

• These NCERT Solutions for Class 9 Maths helps you solve and revise the whole term wise syllabus of Class 9.
• After going through the step-wise solutions given by our subject expert teachers, you will be able to score more marks.
• It follows NCERT guidelines.
• It contains all the important questions from the examination point of view.
• It helps in scoring well in Maths in first term exams.

The faculty have quoted the solutions in a easy and simple manner to improve the problem-solving abilities among the students. For a more clear idea about Number Systems students can refer to the study materials available at INFINITY LEARN.

Class 9 Maths Chapter 1 Number System FAQs

Why should we download NCERT Solutions for Class 9 Maths Chapter 1?

The presentation of each solution in the Chapter 1 of NCERT Solutions for Class 9 Maths is described in a unique way by the INFINITY LEARN experts in Maths. The solutions are explained in easy language which improves grasping abilities among students. To score good marks, practising NCERT Solutions for Class 9 Maths can help students to solve the paper. This chapter can be used as a model of reference by the students to improve their conceptual knowledge and understand the different ways used to solve the problems.

Is INFINITY LEARN website providing answers to NCERT Solutions for Class 9 Maths Chapter 1 in a detailed way?

Yes, INFINITY LEARN website provides answers to NCERT Solutions for Class 9 Maths Chapter 1 in step by step manner. This allows the students to comprehend all the concepts in detail and also they can clear their doubts as well. Regular practising makes them score high in Maths Term I exams.

Give an outline of concepts mentioned in NCERT Solutions for Class 9 Maths Chapter 1.

NCERT Solutions for Class 9 Maths Chapter 1 comprises 3 exercises. The concepts of this chapter are mentioned below. Irrational Numbers Real Numbers and their Decimal Expansions Representing Real Numbers on the Number Line Operations on Real Numbers Laws of Exponents for Real Numbers