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NCERT Solutions for Class 9 Maths Chapter 1 Number System

## NCERT Solutions Class 9 Maths Chapter 1 Number System - Free PDF

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems strictly based on the updated syllabus provided by CBSE, has been prepared by experts of Vedantu keeping in mind the problems and difficulties faced at the examinations. Greater emphasis has been given on the conceptual understanding, mathematical formulations and solutions of problems to endow you with the skills of solving problems adroitly. It contains ample of solved examples with key facts covering the entire syllabus. Well, I guess that’s all you need to ace your exams!!

So we at Vedantu provide you with an easy way through our __NCERT Solutions Class 9 Maths__ Chapter 1 Number System to learn tricks. Our __NCERT solution__ for class 9 maths chapter 1 will teach you about the characteristics of whole numbers, natural numbers, irrational numbers, and prime numbers, etc. Subjects like Science, Maths, English will become easy to study if you have access to NCERT Solution for __Class 9 Science__, Maths solutions, and solutions of other subjects that are available on Vedantu only.

## NCERT Solutions Class 9 Maths Chapter 1 Number System - Free PDF Download

You can opt for Chapter 1 - Number System NCERT Solutions for Class 9 Maths PDF for Upcoming Exams and also You can Find the Solutions of All the Maths Chapters below.

__NCERT Solutions for Class 9 Maths__

Number System Class 9

Symbols like 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are the ten simple digits called numbers. These digits have more influence in our life than we could ever imagine. From your phone number to the number in your address and from our weight to the number of stars in the sky, everything on Earth involves numbers. Therefore, the establishment of a number system and categorization of the numbers became important. In this chapter, we will learn about the number line, types of numbers, laws and operations on the numbers.

List of Exercises and topics they cover:

Exercise 1.1: Different Types of Numbers

Exercise 1.2: Irrational Numbers

Exercise 1.3: Real Numbers and their decimal expansions.

Exercise 1.4: Number Line

Exercise 1.5: Operation on Real Numbers

Exercise 1.6: Laws of exponent For Real Numbers

### Journey To Infinity - Number Line

What if I tell you there is a line (with numbers on it) extending in both directions equally forever till infinity just like space that goes on and on without an end? Well yeah, this line is called the number line and it contains all positive and negative numbers. The numbers are placed at equal intervals along its entire length. The middle of the line is marked as zero, to the left, the numbers are negative and get lesser as it proceeds. Similarly to the right, the numbers are positive and become bigger as it goes on.

### A Beautiful Whole is a Sum of Different Parts

Now that you know the number line, let's break it down into all its unique parts. Don't worry, I guess you know more about numbers than you think! Numbers can be divided into two - Real Numbers and Imaginary Numbers. All the numbers you can think of are real numbers. Imaginary numbers are the numbers that don't exist in reality but it is used in various calculations.

All the Real Numbers can fit into a number line.

### There Are Numbers That Are Not Real - Imaginary Numbers

Imaginary numbers have their own universe and do not fit in the Number Line or in real-world but as told you before, it does find its applications in many important calculations. The even roots of negative reals are called imaginary numbers. For example - the square root of minus one (\[\sqrt{-1}\]) is an imaginary number denoted by ‘i’. Every other imaginary number is just a multiple of ‘i’ like 3i, -6.52i, and π i. Forget not, 0i isn’t imaginary, because 0i = 0 which is a real number.

### Complex Numbers Are Complicated

Any number in the form of a + bi, where a and b are real (and b is not 0) is called complex numbers. In other words, we can say that a complex number is the sum of a real number and an imaginary number. The complex numbers play a significant role in physics, in the areas of electromagnetism, particle physics, and the list goes on.

### Into The Abyss - Real Numbers

Do you know there are infinite numbers between 0 and 1 like 0.000000001, 0.0965, 0.64 and many more? Now imagine how many numbers are there between all the negative and positive numbers? It's like a vast abyss of numbers with no end. Scary isn’t it? But no matter how scary it is, the fact is, these numbers exist. Thus, all these numbers together are called Real Numbers.

### Let us Understand the Different Types of Real Numbers:

Getting Hold Of The Two End Of The Stick - Rational And Irrational Numbers

Rational Numbers

Rational numbers are the numbers which can be written in the form of \[\frac{p}{q}\] where q is not equal to zero. It is usually denoted by Q. Every real number can be expressed as either a terminating decimal or a recurring decimal. Hence every decimal or recurring decimal is a rational number.

### Types of Rational Numbers:

Natural Numbers - It is a part of the ancient system of counting sheep and goats, just to keep a tract of them. It is also called counting numbers because you can literally count whole things using natural numbers. It starts at 1 and goes on and on… It is denoted by N. Examples: N = 1, 2, 3,…

Whole Numbers - Its natural numbers with an unnatural number zero. Zero is considered to be not so natural because it's sophisticated to count zero number of things. It is denoted by W. Examples: W = 0, 1, 2, 3, 4, 5, 6, …

Integers - It contains all the numbers from negative to positive including zero. All the numbers left and right to zero are integers. It is denoted by I. Examples: I = ...., -2, -1, 0, 1, 2, …

Fractions - It represents a part of the whole which may be a single object or a group of objects. A fraction can be written as p/q where p and q are whole numbers but q not equal to zero. Examples: ¾, ½, etc. Even integers can be expressed as fractions.

Examples: -9 = -9/1, 2.25 = 9/4, -2.836 = -156/55, etc.

Decimals - Numbers with a small dots in between the digits such as 7023.602 and 8.897 are decimal numbers.

Irrational Numbers

The numbers that are not expressible in the form p/q are called irrational numbers. Also, it covers the numbers with unpredictable digits after decimals. Many values containing pie are also irrational numbers. Examples: \[\sqrt{2}\], \[\sqrt{3}\], \[\pi\], 0.10110111011110…

### Spotting Rational and Irrational Numbers - Decimal Expansion

Ok, there is one nice thing about real numbers and that is, they all have decimal expansions. Therefore, to differentiate between rational and irrational numbers you can check their decimal expansions. When you divide a number p by a number q of a fraction p/q then the resultant will give you a decimal number. There are two possibilities, the remainder may or may not end in zero.

Case (i): When the remainder becomes zero.

If while dividing the numerator by denominator, the decimal expansion terminates or ends after a finite number of steps then we call the decimal expansion of such numbers as terminating.

Case (ii): When the remainder never becomes zero.

If while dividing the numerator by denominator, the remainder repeats itself after a certain stage, forcing the decimal expansion to go on forever. We call the decimal expansion of such numbers as non-terminating recurring.

Irrational numbers can be identified as having decimal expansions that are non-terminating and non-recurring.

### Representing Real Numbers on Number Line

All real numbers can be shown in a number line. As shown above, the number line has negative numbers to the left side of zero and positive numbers to the right side of zero. Every real number (integer and fraction) can be represented as a decimal number too. For example, 2 can be represented as 2.0 and -7 can be represented as -7.0. Fractions like 1/20 and 5/2 can be represented as 0.05 and 2.5 respectively. Therefore a number line is a line that can accommodate all the real numbers in the form of decimals.

There are always infinite numbers between any given two real numbers. Let us take numbers between 2 and 3. The divisions between 2 and 3 are - 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0. Well, this is not the end. There are more numbers between any of the given numbers. Let us take the numbers between 2.6 to 2.7 and that is 2.61, 2.62, 2.63,...2.69,2.70. Now, let us magnify more and check the numbers between 2.66 and 2.67. The numbers are 2.661, 2.662, 2.663, 2.664, 2.665, 2.666,..2.669 and 2.670. And this process goes on.

“The visualization process of representation of numbers on a number line, through a magnifying glass, as the **P****rocess of Successive Magnification**”.

Operations on Real Numbers

A lot of cool things can be done with Real numbers. We can do operations like addition, subtraction, multiplication and division on Real numbers. The following are the important observations:

The sum or difference of a rational number is irrational.

The product or quotient of a non-zero rational number with an irrational number is also irrational.

If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.

Let us take two positive real numbers a and b, then the following identities hold:

\[\sqrt{ab}\] = \[\sqrt{a}\] \[\sqrt{b}\] \[\sqrt{\frac{a}{b}}\] = \[\frac{\sqrt{a}}{\sqrt{b}}\] \[\sqrt{(a + b)}\] \[\sqrt{(a - b)}\] = a - b (a + \[\sqrt{b}) (a - \sqrt{b}\]) = \[a^{2}\] - b \[(\sqrt{a} + \sqrt{b})^{2} = a + 2\sqrt{ab} + b\] |

### Laws of Exponents

You must have come across the expression like 3² where 3 is called the base and 2 is the exponent. Exponents are also known as Powers or Indices. The exponent of a number explains how many times the number is used in a multiplication. Let us study the laws of exponent as it is very important to understand how exponents work before we indulge in solving problems:

If a is a real number and greater than zero and p and q are rational numbers then,

\[a^p\] × \[a^{q}\] = \[a^{p + q}\]

\[(a^{p})^{q}\] = \[a^{pq}\]

\[\frac{a^{p}}{a^{q}}\] = \[a^{p - q}\]

\[a^{p} b^{p}\] = \[(ab)^{p}\]

1. Product Law

According to the product law of exponents when multiplying two numbers that have the same base then we can add the exponents

\[a^{p}\times a^{q} = a^{p + q}\]

where a, p and q all are natural numbers. Here the base must be the same in both the quantities. For example,

\[2^{3}\times 2^{4} = 2^{7}\]

2. Quotient Law

According to the quotient law of exponents, we are allowed to divide two numbers if they have the same base by simply subtracting the exponents. In order to divide two exponents that have the same base, subtract the power in the denominator from the power in the numerator.

\[\frac{a^{p}}{a^{q}} = a^{p - q}\]

where a, p and q all are natural numbers. Here the base must be the same in both the quantities. For example,

\[2^{5} \div 2^{3} = 2^{2}\]

3. Power Law

According to the power law of exponents, we have to multiply the exponents if a number raises a power to a power.

\[((a^{p})^{q}) = a^{pq}\]

Here there is one base a and two powers m and n. For example, \[(5^{3})^{2} = 5^{6}\]

4. Power of Product

The power of product rule states that: \[(ab)^{p} = a^{p} b^{q}\], a and b are positive real numbers and m is the rational number. For example \[(2\times 5)^{10} = 2^{10} 5^{10}\]

### The Real Voyage of Discoveries - Historical Facts

Pythagoras Discovered Irrational Numbers:

Pythagoras a famous mathematician and a philosopher was the first one to discover the numbers which were not rationals, around 400 BC. These numbers are called irrational because they cannot be written in the form of ratios of integers.

Hipparchus of Croton died for a secret:

There are many myths surrounding the discovery of irrational numbers by Pythagorean, Hipparchus of Croton. In all the myths, Hippacus has had an unfortunate end, either because he discovered that root 2 is irrational or because he disclosed the secret about root 2 to people outside the secret Pythagorean sect!

Proofs by Dedekind and Cantor:

The two German mathematicians, Cantor and Dedekind In the 1870s showed that: Corresponding to every real number, there is always a point on the real number line, and also corresponding to every point on the number line, there always exists a unique real number.

Value of pi by Archimedes:

Archimedes, the Greek genius was the first person to compute digits in the decimal expansion of pie. He showed that 3.140845 < pie< 3.142857. Aryabhatta (476-550 CCE), the great Indian mathematician and astronomer, found the value of pi correct to four decimal places (3.1416). Using high-speed computers and advanced algorithms, pie has been computed to over 1.24 trillion decimal places.

### It Is Better To Know Solutions Than All The Questions - NCERT Solutions By Vedantu

Mathematics plays a dominant role in the field of science and technology. It has become a fundamental of contemporary life and an indispensable tool in the modern generation. Vedantu aims at fabricating the solution of the chapters to help you get the concept without any hiccups. The detailed and step-wise explanations for all the answers to the exercise questions are considered to be very useful for those who wish to pursue Mathematics as a subject for their further studies in their higher classes. Vedantu focuses on the roots of education to deserve the fruits of hard work and determination.

Maths is the most scoring subject and is easy to solve when all the basic concepts are clear. Therefore it is better to practise some solutions properly rather than just know all the questions. The NCERT Solutions for Class 9 maths chapter 1 number inculcate sterling qualities of objectivity, precision and accuracy along with the sharpening of your intellect. The solutions are strictly according to the latest syllabus prescribed by CBSE.

### Key Features of NCERT Solutions By Vedantu :

NCERT Class 9 Maths Chapter 1 is the best draft by the experts of Vedantu to help you solve and revise the entire syllabus in the form of problems and their solutions.

It contains step-wise solutions and graded exercise to help you clear your concept and score more marks in your examination.

The solution is in accordance with the latest syllabus and exam specifications.

It covers the entire syllabus along with important points and formulas. You get the gist of the entire chapter of number system class 9 and also the concepts.

NCERT Class 9 maths chapter 1provides you ample examples for your practice. Practise is the only way to attain perfection and avoid making silly mistakes in the exam.

The number system class 9 is written keeping in mind the age group of the students so the solutions are in simple language and emphasis on basic facts, terms, principles and applications on various concepts.

NCERT Class 9 maths chapter 1 has complicated solutions broken down into simple parts and well-spaced to save the students from the unnecessary strain on their minds.

The answers in ch 1 maths class 9 are treated systematically and presented in a coherent and interesting manner.

The content is kept concise, brief and self-explanatory.

Some answers are incorporated with necessary images to facilitate the understanding of the concept.

The solutions are according to the latest syllabus and exam specifications.

### NCERT Solutions for Class 9 Maths Chapter 1 Number Systems - Free PDF

NCERT Solutions for Class 9 Chapter 1 by Vedantu will easily make you learn the decimal expression of rational numbers and real numbers. One of the first concepts that this lesson teaches you is how to find rational numbers between two given numbers. By referring to the solutions by Vedantu you can get the topic without any hiccups. Vedantu also aims at fabricating the Solutions of Maths in such a way that it is easy for the students to understand. The NCERT Solutions for Class 9 aims at equipping the students with detailed and step-wise explanations so that they can answer the questions given in the exercises of this Chapter. In Chapter 1 Number System of Class 9, students are introduced to a lot of important topics that are considered to be very important for those who wish to pursue Mathematics as a subject for their further studies in their higher classes. Based on these NCERT Solutions, students can practice and prepare for their upcoming exams as well as endow themselves with the basics of Class 10 for the board exams then. These Maths Solutions of NCERT Class 9 are helpful as they are prepared with respect to the NCERT Syllabus and Guidelines. NCERT Solutions for Class 9 Maths Chapter 1 Number System are provided by Vedantu because we understand that Maths is all about creating the numbers by the number system. So we at Vedantu provide you with an easy way through our NCERT Solutions for Class 9 Maths Chapter 1 Number System to learn tricks. Our NCERT solution for class 9 maths chapter 1 will teach you the characteristics of whole numbers as well as natural numbers but also take care of the irrational numbers, and prime numbers, etc. You can always download these solutions in PDF format for free. Learning fundamentals in maths are always essential. NCERT Solutions for Class 9 Maths Chapter 1 will strengthen the student’s fundamental concept regarding numbers and number systems. It will help students to score better marks in exams. Our expert teachers of maths are constantly working hard for the subject to give the best NCERT Solutions for Class 9 Maths Chapter 1 Number System.

You have to be willing to think outside the box, or at least of the number line if you want to find numbers that aren’t real. We, at Vedantu, try to develop a conceptual understanding of this chapter so that all the upcoming chapters will be easily understandable. In this chapter, there are questions related to rational numbers and their properties. The solutions provided by Vedantu of class 9th maths chapter 1 will make you understand what the properties are and how they can be used in solving questions. In this NCERT Solutions for Class 9 Maths Chapter 1, you will unearth easy solutions for real numbers and their decimal expressions. In this section, you will get to know how the real numbers are represented on a number line. You will also get to learn how to simplify an expression which consists of both rational and irrational numbers. This chapter will also teach you how to represent numbers on a number line by using successive magnification.

We provide NCERT Solutions that give step by step solutions for all numerical and conceptual problems. These solutions will help students to frame a better understanding of all the topics. We also place the cherry on the cake by providing free pdf downloads. You get the facility of free online classes, free video lectures, and of course free doubt clearing sessions. Our Maths teachers are gifted with high capability and experience.

### Reading The Scoreboard - Weightage of the Chapter

NCERT Solutions for Class 9 Maths Chapter 1 Number System creates a strong basic fundamental understanding of numbers and their properties. The Number System is useful in our day to day life also for performing various operations related to numbers. NCERT Solutions of maths class 9 chapter 1 gives a self-explanatory solution for every question of the chapter and will also help you to complete your homework without any external help. As the Number System is one of the important topics in Maths, it has a weightage of 6 marks in class 9 Maths exams. On an average three questions are mostly asked from this unit.

Part A - One question. (1 marks).

Part B - One question. (2 marks).

Part C - One question. (3 marks).

NCERT Solutions for Class 9 Maths Chapter 1 is the first chapter of class 9 Maths which discusses the Number Systems and their important applications. It starts with the introduction of Number Systems (types of numbers in number line) in section 1.1 followed by two very important topics in section 1.2 and 1.3 (i.e, rational & irrational numbers and their decimal expansions). We learn about two types of decimal expansions that are, terminating and non-terminating recurring. Section 1.4 helps us to visualise the representation of real numbers in the number line whereas section 1.5 deals with the operation on real numbers. The last section 1.6 entirely works on the laws of exponents for Real Numbers. For further information you can download NCERT Solutions for Class 9 Maths Chapter 1 Number Systems - PDF Download.

### Summary

There are two basic divisions of numbers in the Number System - Real and Imaginary Numbers.

Real Numbers again have two broad divisions - Rational and Irrational Numbers.

Natural Numbers, Whole Numbers, Integers and Fractions are all Rational Numbers.

Every Real Number has a decimal expansion which may be terminating or non-terminating recurring.

Natural numbers are counting numbers starting from 1.

Whole numbers are natural numbers and 0.

Whole numbers together with the negative numbers make up the Integers

Rational numbers are numbers which can be expressed in the form of p/q, where p and q are integers but q is not equal to 0.

Irrational numbers are numbers which cannot be expressed in the form of p/q, where p and q are integers but q is not = 0.

There are infinite numbers of rational numbers between any two given rational numbers.

Every point on the number line demonstrates a unique real number. Irrational numbers cannot be found on the number line.

If a and b are irrational numbers then all the operations that are addition(a+b), subtraction (a-b), multiplication(a*b) and division (a/b) are all irrational too only if b is not equal to zero.

For the positive real numbers like a and b, the following identities hold:

\[\sqrt{ab}\] = \[\sqrt{a}\] \[\sqrt{b}\]

\[\sqrt{\frac{a}{b}}\] = \[\frac{\sqrt{a}}{\sqrt{b}}\]

\[\sqrt{(a + b)}\] \[\sqrt{(a - b)}\] = a - b

(a + \[\sqrt{b}) (a - \sqrt{b}\]) = \[a^{2}\] - b

\[(\sqrt{a} + \sqrt{b})^{2} = a + 2\sqrt{ab} + b\]

To rationalise the denominator of \[\frac{1}{\sqrt{a} + b}\], we multiply it with \[\frac{\sqrt{a} - b}{\sqrt{a} - b}\], where a and b are integers.

If a is a real number and greater than zero and p and q are rational numbers then,

\[a^{p}\times a^{q} = a^{p + q}\]

\[((a^{p})^{q}) = a^{pq}\]

\[\frac{a^{p}}{a^{q}} = a^{p - q}\]

\[a^{p} b^{q} = (ab)^{p}\]

## Formulas for Exponents:

Law | Example |

\[x^{1}\] = x | \[6^{1}\] = 6 |

\[x^{0}\] = 1 | \[7^{0}\] = 1 |

\[x^{-1} = \frac{1}{x}\] | \[4^{-1} = \frac{1}{4}\] |

\[x^{m}x^{n} = x^{m+n}\] | \[x^{2}x^{3} = x^{2} + 3 = x^{5}\] |

\[\frac{x^{m}}{x^{n}} = x^{m - n}\] | \[\frac{x^{6}}{x^{2}} = x^{6-2} = x^{4}\] |

\[(x^{m})^{n} = x^{mn}\] | \[(x^{2})^{3} = x^{2\times 3} = x^{6}\] |

\[(xy)^{n} = x^{n}y^{n}\] | \[(xy)^{3} = x^{3}y^{3}\] |

\[(\frac{x}{y})^{n} = \frac{x^{n}}{y^{n}}\] | \[(\frac{x}{y})^{2} = \frac{x^{2}}{y^{2}}\] |

\[x^{-n} = \frac{1}{x^{n}}\] | \[x^{-3} = \frac{1}{x^{3}}\] |

### Important Facts

Natural numbers are a part of whole numbers

Every integer is a rational number.

Every rational number is not an integer.

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1. What is number system?

Answer: Number system is a systematic way of writing or representing all numbers of a particular type. A number system is also called a numeral system. There are various types of number systems, such as the decimal number system, the octal number system, the binary number system, the hexadecimal number system. We can express each and every number in a unique way on a number system. The base of the number system, the digit, and the position of the digit in a number are the three factors determining the value of a number in a number system. A number can have different values in different number systems.

2. What is the difference between a rational number and an irrational number?

Answer: Any number is said to be a rational number when it can be expressed as a ratio of two integers, and the denominator of the ratio is not equal to zero. A number that cannot be expressed as a ratio of two integers is said to be an irrational number. The rational numbers are called the real numbers, whereas the irrational numbers consist of an imaginary part, that is, √-1. Any non-repeating decimal or non-terminating decimal can be called irrational, whereas all rational numbers are either a terminating decimal or a repeating decimal.

3. How many sums are there in the NCERT Class 9 Chapter 1 Number System?

Answer: There are six exercises in the NCERT Class 9 Chapter 1 Number System. In the first exercise, Ex-1.1, there are 4 sums and in the second exercise, Ex-1.2, there are 3 sums. These first two exercises deal with the basic concepts of the number system, such as identifying the features of a rational number or an irrational number and locating them on the number line. In the third exercise, Ex-1.3, there are 9 sums, and most of them have sub-questions. The fourth exercise, Ex-1.4, comprises 2 sums, that deal with successive magnification for locating a decimal number on the number line. The fifth exercise, Ex-1.5, consists of 5 sums, on the concept of rationalization. The sixth exercise, Ex-1.6, consists of 3 sums, that have sub-questions. The sums in this exercise will require you to find the various roots of numbers.

4. Will the NCERT Solutions for Class 9 Maths Chapter 1 Number System help me to understand the concept of rational and irrational numbers?

Answer: Yes, the NCERT Solutions for Class 9 Maths Chapter 1 Number System will definitely help you to understand the concept of rational and irrational numbers. These solutions cover all the sums that are given in the exercises. They are prepared by the highly experienced teachers at Vedantu, in a step by step manner. So by following these solutions you will be able to understand the proper problem-solving process for every sum of this chapter. You will be able to understand various concepts such as locating a number on the line, the process of rationalization, and so on. Also, you can download the PDF for these NCERT solutions for free from Vedantu and consult them to develop a good grip on the concepts of rational and irrational numbers.