Class 9 Maths Chapter 1
You have learnt about the number line and how to represent different types numbers on it. In class 9 maths chapter 1, we learnt about Natural number, whole number, rational number, irrational number, real number etc.
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The number like 1, 2, 3, 4, and so on, are called natural number.The natural number denoted by N.
And the number like 0, 1, 2, 3, and so on is called whole number.The whole number denoted by W. All natural numbers with zero (0) is called whole number.
The collection of all positive and negative numbers is called integers. Integers denoted by Z.
Numbers like 1/2, 2/3, with integers is called rational numbers. The collection of rational numbers is denoted by Q. ‘Rational’ comes from the word ‘ratio’ and Q comes from ‘quotient’
Definition of rational number: A number ‘r’ is called a rational number. if it can be written in the form of p/q .where p and q are integers and q does not equal to zero.
Every integer is a rational number.You must have realised that in fact there are infinitely many rational number between 1 and 2. In general, there are infinitely many rational numbers between any two given rational numbers.
Let us take a look at the number line again. Have you picked up all the numbers? Not, yet. The fact is that there are infinitely many more numbers left on the number line! There are gaps in between the places of the numbers you picked up, and not just one or two but infinitely many. The amazing thing is that there are infinitely many numbers lying between any two of these gaps too!
CBSE Class 9 Maths Chapter 1 Exercise 1.1 Solutions
You can download the pdf of Ncert Class 9 Maths Chapter 1 PDF Exercise 1.1 free of cost in english medium. From this facility the students can access the solutions at your comfort.
Every integer is a rational number.You must have realised that in fact there are infinitely many rational number between 1 and 2. In general, there are infinitely many rational numbers between any two given rational numbers.Let us take a look at the number line again. Have you picked up all the numbers? Not, yet. The fact is that there are infinitely many more numbers left on the number line! There are gaps in between the places of the numbers you picked up, and not just one or two but infinitely many. The amazing thing is that there are infinitely many numbers lying between any two of these gaps too!
Irrational Numbers:
We saw, in the previous section, that there may be numbers on the number line that are not rationals.The number ‘s’ is called irrational, if it cannot be written in the form p/q , where p and q are integers and q does not equal to zero. You already known that there are infinitely many rationals. It turns out that there are infinitely many irrational numbers too. Some example – √2, √3 , π, 0.01101110…
If we now put all irrational numbers with the rational numbers. will there be any numbers left on the number line? The answer is no! It turns out that the collection of all rational numbers and irrational numbers together make up what we call the collection of real numbers. Real number is denoted by R.Therefore real number is either rational or irrational .
So, we can say that every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number.This is why we call the number line, the real number line.
Class 9 Maths Chapter 1 Exercise 1.2 Solutions
Real Numbers And Their Decimal Expansions
In this section, we are going to learn rational and irrational numbers from a different point of view. We will look at the decimal expansions of real numbers and see if we can use the expansions to distinguish between rationals and irrationals. We will also visualise the representation of real numbers on the number line using their decimal expansions.
The decimal expansion of a rational number is either terminating or non terminating recurring. Moreover a number whose decimal expansion is terminating or non- terminating recurring is rational.
So, now we know what the decimal expansion of a rational number can be. What about the decimal expansion of irrational numbers? Because of the property above, we can conclude that their decimal expansions are non-terminating non-recurring.
The decimal expansion of an irrational number is non- terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.
Class 9 Maths Chapter 1 Exercise 1.3 Solutions
Representing Real Numbers on the Number Line : Suppose we want to locate 2.665 on the number line. We know that this lies between 2 and 3. So, let us look closely at the portion of the number line between 2 and 3. Suppose we divide this into 10 equal parts and mark each point of division as:
(i) Then the first mark to the right of 2 will represent 2.1, the second 2.2 and so on.You might be finding some difficulty in observing these points of division between 2 and 3.
(ii) Now, 2.665 lies between 2.6 and 2.7 . So, let us focus on the portion between 2.6 and 2.7. We imagine to divide this again into ten equal parts. The first mark will represent 2.61, the next 2.62, and so on. Again 2.665 lies between 2.66 and 2.67.
So, let us focus on this portion of the number line and imagine to divide it again into ten equal parts. We magnify it to see it better. The first mark represents 2.661, the next one represents 2.662 and so on. So, 2.665 is the 5th mark in these subdivisions. This process is called successive magnification.
Class 9 Maths Chapter 1 Exercise 1.4 Solutions
Operations on Real Numbers : You have learnt, in earlier classes, that rational numbers satisfy the commutative , associative and distributive laws for addition and multiplication.
Moreover, if we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number (that is , rational numbers are ‘closed’ with respect to addition, subtraction, multiplication and division), It turns out that irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication.
However the sum, difference, quotient and products of irrational numbers are not always irrational.
(i) The sum or difference of a rational number and an irrational number is irrational.
(ii) The product or quotient of a non-zero rational number with an irrational number is irrational number.
(iii) If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.
Solutions Exercise 1.5
Laws of Exponents for real numbers : Laws of exponents are given below.Here a, n and m are natural numbers. Remember, a is called the base and m and n are the exponents.
Solution Exercise 1.6
Ncert solutions for class 9 maths chapter 1 pdf
Class 9 Maths Chapter 1 Extra Questions with Solution
Here some extra questions for ‘class 9 maths chapter 1 extra questions with solution’ are given . The students can download in pdf of this extra solutions.
In this chapter, you have studied the following points:
(i) All the rational and irrational numbers make up the collection of real numbers.
(2) A number s is called a irrational number, if it cannot be written in the form p/q where p and q are integers and q ≠ 0.
(3) The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.
(4) The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number a number whose decimal expansion is non-terminating non- recurring is irrational.
(5) A number r is called a rational number, if it can be written in the form p/q , where p and q are integers and q ≠ 0
(6) There is a unique real number corresponding to every point on the number line. Also, corresponding to each real number, there is a unique point on the number line.
(7) If r is rational and s is irrational, then r + s and r – s are irrational numbers, and rs and r/s irrational numbers, r ≠ 0.
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