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Number System Class 9 Notes Maths Chapter 1

Introduction

Number system is a writing system used for expressing the numbers. A number is a mathematical object used to count, label and measure.

Types of Numbers

Natural numbers: All counting numbers are called natural numbers. It is denoted by “N”.

Example: N = {1, 2, 3, 4 ... .}

Natural numbers are represented on the number line as follows:

Number System Class 9 Notes Maths Chapter 1

Whole numbers: The group of natural numbers including zero is called whole numbers. It is denoted by “W”. Zero is a very powerful number because if we multiply any number with zero it becomes zero. All natural numbers are called whole numbers.

Example: W = {0, 1, 2, 3, 4 ... .}

Whole numbers are represented on number line as follows:

Number System Class 9 Notes Maths Chapter 1

Integers: The collection of all whole numbers and negatives of all natural numbers or counting numbers are called integers. They are denoted by “Z” or “I”. All whole numbers are integers, but all integers are not whole numbers

Example: Z or I = {... − 3, −2, −1, 0, 1, 2, 3 ... . .}

Integers represented on number line as follow:

Number System Class 9 Notes Maths Chapter 1

Real numbers

  • The collection of rational & irrational numbers together forms real numbers. The set of real numbers is denoted by symbol R. How do you know whether any number is real number or not?
  • If that number can be shown on number line then that number is a real number. So, any number which can be shown on the number line is real number.

Example: √2 , √3, −5, 0 , 1/5 5 etc.... All rational numbers are real numbers but all the real numbers are not rational numbers. Also, all irrational numbers are real numbers, but the reverse is not true

Rational numbers: Numbers that can be represented in the form of p/q where p & q are integers & q ≠ 0 are called rational numbers. The word rational came from the word ‘ratio’. It is denoted by letter Q and Q is taken from the word quotient. All integers are rational numbers.

Example: Q ={1/2, 3, -4, 3/2 etc ....}

Rational numbers also include natural numbers, whole numbers and integers. This can be explained using following example:

Example: -16 can also be written as -16/1 Here p = – 16 and q = 1. Therefore, the rational numbers also include the natural numbers, whole numbers and integers.

Number System Class 9 Notes Maths Chapter 1

Equivalent rational numbers: The equivalent rational numbers are numbers that have same value but are represented differently.

Example: If a/b is equivalent to c/d and a/b = x then c/d = x

Also, if a/b  = c/d, then a x d = b x c.

Irrational numbers: A number which can’t be expressed in the form of p/q and its decimal representation is non-terminating and non-repeating is known as irrational numbers. It is denoted by “S”.

Example: S = √2 = 1.4142135... . , √3 = 1.73205 ... . , etc

Methods to determine rational number between two numbers

We know that there are infinitely many rational numbers between any given rational numbers. Hence, for determining one or more than one rational number between two given rational numbers we use the following methods

(i) When one rational number is to be determined:
Let a and b be two rational numbers, such that b > a. Then, Number System Class 9 Notes Maths Chapter 1  is a rational number lying between a and b

Example: Find a rational number between 4 and 5  Here, 5 > 4

We know that, if a and b are two rational numbers, such that b > a. Then, Number System Class 9 Notes Maths Chapter 1  is a rational number lying between a and b.
So, a rational number between 4 and 5 = Number System Class 9 Notes Maths Chapter 1

(ii) When more than one rational number are to be determined:
Let a and b be two rational numbers, such that b > a and we want to find n rational numbers between a and b. Then, n rational numbers lying between a and b are
(a + b), (a + 2d ), (a + 3d), ..........(a + nd), where, d = ( b -a ) / (n + 1)
Here, a and b are two rational numbers n is the number of rational numbers between a and b

Example (i): Find six rational numbers between 3 and 4

Here, 4 > 3
So, let a = 3 and b = 4 and n = 6
Since, d = b -a / n + 1
Number System Class 9 Notes Maths Chapter 1
Now, Number System Class 9 Notes Maths Chapter 1
Number System Class 9 Notes Maths Chapter 1
Hence, the required six rational numbers lying between 3 and 4 are Number System Class 9 Notes Maths Chapter 1

Example (ii): Find four rational numbers between − 6 and– 7.

Here, −6 > −7
Let a = − 7, b = − 6 and n = 4
Now, d = b -a / n + 1
= Number System Class 9 Notes Maths Chapter 1
So, four rational numbers between – 6 and − 7 are (a + d), (a + 2d), (a + 3d) and (a + 4d)
i.e., Number System Class 9 Notes Maths Chapter 1 and Number System Class 9 Notes Maths Chapter 1
= Number System Class 9 Notes Maths Chapter 1 and Number System Class 9 Notes Maths Chapter 1
= Number System Class 9 Notes Maths Chapter 1 and Number System Class 9 Notes Maths Chapter 1
The above rational numbers are the rational numbers which lie between – 6 and – 7.

Question for Chapter Notes: Number System
Try yourself:Which of the following is an example of a rational number?
 
View Solution

Irrational number

A number which can’t be expressed in the form of p/q and its decimal representation is non-terminating and non-repeating is known as irrational numbers. The set of irrational numbers is denoted by “S”.

Example: S = √2 ,√3 , π, etc. ..

Locate an irrational number on the number line:
We see how to locate an irrational number on number line with the help of following example:

Example: Locate √17 on the number line

Here, 17 = 16 + 1 = (4)2 + (1)(Sum of squares of two natural numbers)
So, we take a = 4 and b = 1
Now, draw OA = 4 units on the number line and then draw AB = 1 join OB.

Number System Class 9 Notes Maths Chapter 1

By using Pythagoras theorem, in ∆OAB
Number System Class 9 Notes Maths Chapter 1
Taking O as centre and radius equal to OB, draw an arc, which cuts the number line at C. Hence, OC represents √17.
Number System Class 9 Notes Maths Chapter 1

Real Numbers and their Decimal Expansion

Real numbers: The collections of rational & irrational numbers together form real numbers. They are denoted by R. Every point on the number line is a real number.

Number System Class 9 Notes Maths Chapter 1

Rational and Irrational numbers are Subsets of Real Numbers

Example: √2, √3, −5, 0, 1/5, 5 etc.... All rational numbers are real number but all real numbers are not rational numbers. Also, all irrational numbers are real number, but the reverse case is not true.

Real numbers and their decimal expansion: The decimal expansion of real numbers can be either terminating or non – terminating, repeating or non – terminating non – repeating. With the help of decimal expansion of real numbers, we can check whether it is rational or irrational.
(i) Decimal expansion of rational numbers:
Rational numbers are present in the form of p/q, where q ≠ 0, on dividing p by q, two main cases occur,
(a) Either the remainder becomes zero after few steps
(b) The remainder never becomes zero and gets repeating numbers.

Case I: Remainder becomes zero
On dividing p by q, if remainder becomes zero after few steps, and then the decimal expansion terminates or ends after few steps. Such decimal expansion is called terminating decimal expansion.

Example: Number System Class 9 Notes Maths Chapter 1

Number System Class 9 Notes Maths Chapter 1On dividing  Number System Class 9 Notes Maths Chapter 1 we get exact value 0.625 and remainder is zero.
So, we say that Number System Class 9 Notes Maths Chapter 1  is a terminating decimal expansion.
On dividing  Number System Class 9 Notes Maths Chapter 1 we get exact value 0.625 and remainder is zero. So, we say that  Number System Class 9 Notes Maths Chapter 1  is a terminating decimal expansion.

Case II: Remainder never becomes zero
On diving p by q, if remainder never becomes zero and the sets of digits repeats periodically or in the same interval, then the decimal expansion is called non – terminating repeating decimal expansion. It is also called non – terminating recurring decimal expansion.
Example (i): Number System Class 9 Notes Maths Chapter 1

Number System Class 9 Notes Maths Chapter 1Number System Class 9 Notes Maths Chapter 1= 0.333... . . or Number System Class 9 Notes Maths Chapter 1  = 0. 3= [The block of repeated digits is denoted by bar ‘– ‘over it]
On dividing Number System Class 9 Notes Maths Chapter 1 we get the repeated number 3 and remainder never becomes zero. Hence, 1 by 3 has a non – terminating repeating decimal expansion.

Example (ii): Number System Class 9 Notes Maths Chapter 1

Number System Class 9 Notes Maths Chapter 1
Hence, Number System Class 9 Notes Maths Chapter 1 = 0.Number System Class 9 Notes Maths Chapter 1On dividing 4 by 13 we get the repeated numbers 0.30769230 again and again, and remainder never becomes zero. Hence, 4 by 13 has a non – terminating repeating decimal expansion.

Methods to Convert Non – Terminating Repeating Decimal Expansion in the form of p by q

Suppose the number is in the form of Number System Class 9 Notes Maths Chapter 1 ( and we have to convert the given number in the form of p by q. Follow the following steps:
Step I: Firstly, transform the non - repeated digits between decimal point and repeating number to left side of decimal by multiplying both sides by 10n
Where n = number of digits between decimal points and repeating numbers. i.e., Number System Class 9 Notes Maths Chapter 1. (In the above expression we see that one digit “b” exist between decimal point and repeating number. Hence,we multiply both side by 101. We get,
Number System Class 9 Notes Maths Chapter 1Step II: Count the number of digits in repeating number and then multiply equation (1) by that power of 10 and the equation becomes

Number System Class 9 Notes Maths Chapter 1Step III: Subtract equation (1) from equation (2) we get,

Number System Class 9 Notes Maths Chapter 1Example (i): Express Number System Class 9 Notes Maths Chapter 1 in the form of p by q 

Assume the given decimal expansion as x
Let,
x = Number System Class 9 Notes Maths Chapter 1 
x = 0.666 ... ... . . (i)
Here, only 1 digit is repeating. Hence, multiplying both side of equation (i) by 10 we get,
10x = 6.66... ... ... (ii)
Subtracting equation (i) from (ii) we get,
10x – x = 6.66 – 0.66
9x = 6.66 – 0.66
9x = 6
 x = 6/9 = 2/3
Hence, Number System Class 9 Notes Maths Chapter 1

Example (ii): Express 0.4Number System Class 9 Notes Maths Chapter 1 in the Number System Class 9 Notes Maths Chapter 1  form, where p and q are integers and q ≠ 0

Let, x = 0.43535 ......(i)
Here, we see that one digit exit between decimal point and recurring number
So, we multiply both sides of equation (i) by 10, we get
10x = 4.3535 ...... (ii)
Here we see that two digits are repeated in the recurring number
So, we multiply equation (ii) by 100, we get
1000x = 435.3535 ...... (iii)
Subtracting equation (ii) from equation (iii), we get
1000x − 10x = 435.3535 − 4.3535
990x = 431
x = Number System Class 9 Notes Maths Chapter 1
Hence,Number System Class 9 Notes Maths Chapter 1

Example (iii): Express 0.00232323.... in the Number System Class 9 Notes Maths Chapter 1  form, where p and q are integers and q ≠ 0

Let, x = 0.00232323 = 0.00Number System Class 9 Notes Maths Chapter 1 ......(i)
Here, we see that two digits exist between decimal point and recurring number
So, we multiply both sides of equation (i) by 100,
100x = 0.232323...... (ii)
Here we see that two digits are repeated in the recurring number
So, we multiply equation (ii) by 100, we get
10000x = 23.2323 ...... (iii)
Subtracting equation (ii) from equation (iii), we get
10000x − 100x = 23.2323 − 0.232329
990x = 23
x = Number System Class 9 Notes Maths Chapter 1
Hence, 0.002323 = Number System Class 9 Notes Maths Chapter 1

Decimal Expansion of Irrational Numbers


The decimal expansion of an irrational numbers is non-terminating non-recurring or a number whose decimal expansion is non - terminating and non-recurring is called irrational.
Example: √3 and π are the examples of irrational numbers because, the values of √3 = 1.7320508075688772.... and π = 3.14592653589793 are non-terminating non-recurring.

Example (i): Find the irrational number between Number System Class 9 Notes Maths Chapter 1 and Number System Class 9 Notes Maths Chapter 1

Number System Class 9 Notes Maths Chapter 1Number System Class 9 Notes Maths Chapter 1
Now,
Number System Class 9 Notes Maths Chapter 1Thus, Number System Class 9 Notes Maths Chapter 1
It means that the required rational numbers will lie between Number System Class 9 Notes Maths Chapter 1 and Number System Class 9 Notes Maths Chapter 1 . Also, we know that the irrational numbers have non-terminating non-recurring decimals. Hence, one irrational number between Number System Class 9 Notes Maths Chapter 1 and Number System Class 9 Notes Maths Chapter 1 is 0.20101001000... . .

Example (ii): Find the two irrational numbers between Number System Class 9 Notes Maths Chapter 1 and Number System Class 9 Notes Maths Chapter 1

If Number System Class 9 Notes Maths Chapter 1 = 0.333 (Given)
We have, Number System Class 9 Notes Maths Chapter 1 = 0.333 (Given)
Hence, Number System Class 9 Notes Maths Chapter 1 = 2 × Number System Class 9 Notes Maths Chapter 1 = 2 × 0.333 = 0.666
So, the two rational numbers between Number System Class 9 Notes Maths Chapter 1  and Number System Class 9 Notes Maths Chapter 1   may be 0.357643... and 0.43216 (In this solution we can write infinite number of such irrational numbers)

Example (iii): Find two irrational numbers between √2 and √3 .

We know that, the value of
√2 = 1. 41421356237606 and
√3 = 1.7320508075688772
From the above value we clearly say that √2 and √3 are two irrational numbers because the decimal representations are non-terminating non-recurring. Also, √3 > √2
Hence, the two irrational numbers may be 1.501001612 and 1.602019

Question for Chapter Notes: Number System
Try yourself: Which of the following numbers is an irrational number?
View Solution

Representation of Real Numbers on the Number Line

We know that, any real numbers has a decimal expansion. There are many decimal numbers or real numbers present between two integers. Every real number is represented by a unique point on number line. Also every point on a number line represents one & only one real number.

  • “The process of visualization of representation of decimal number on number line through a magnifying glass, is known as successive magnification”
  • The process of representation of real numbers on the number line can be understood with the help of an example:

Example: Visualise the representation of 4.36% on the number line up to 4 decimal places.

Here, we can understand representation of 4.36% on the number line up to 4 decimal places with the help of following steps:
Step I: Here, we know that, the number 4.36% lies between 4 and 5. Hence, first draw the number line and look at the portion between 4  and 5 by a magnifying glass.

Number System Class 9 Notes Maths Chapter 1

Step II: Divide the above part into ten equal parts and mark first point to the right of 4 as 4.1, the second as 4.2 and so on.

Number System Class 9 Notes Maths Chapter 1

Step III: Now 4.36 lies between 4.3 and 4.4. So, divide this portion again into ten equal parts and mark first point to the right of 4.3 as 4.31, second 4.32 and so on.

Number System Class 9 Notes Maths Chapter 1

Step IV: Now, 4.366 lies between 4.36 and 4.37. So, divide this portion again into ten equal parts and mark first point to the right of 4.36 as 4.361, second 4.362 and so on.

Number System Class 9 Notes Maths Chapter 1

Step V: To visualize 4.36 more accurately, again divide the portion between 4.366 and 4.367 into 10 equal parts and visualize the representation of 4.36% as in the figure given below:
We can proceed endlessly in this manner. Thus, 4.3666 is the 6th mark in this subdivision.

Number System Class 9 Notes Maths Chapter 1

Example: Visualise 2.565 on the number line, using successive magnification.

We know that, 2.565 lies between 2 and 3. So, divide the part of the number line between 2 and 3 into 10 equal parts and look at the portion between 2.5 and 2.6 through a magnifying glass.

Now, 2.565 lies 2.5 and 2.6 hence first draw the number line and look at the portion between 2.5 and 2.6 by a magnifying glass.

Number System Class 9 Notes Maths Chapter 1Now, we imagine and divide this again into 10 equal parts. The first mark will represent 2.51, the next 2.52 and so on. To see this clearly we magnify this as shown in the following figure,

Number System Class 9 Notes Maths Chapter 1

Again 2.565 lies between 2.56 and 2.57 so, now focus on this portion of the number line and imagine to divide it again into 10 equal parts as shown in the following figure

Number System Class 9 Notes Maths Chapter 1

This process is called visualization of representation of number on the number line through a magnifying glass.
Thus we can visualise that 2.561 is the first mark and 2.565 is the fifth mark in these subdivision.

Operations on Real Numbers

We know that, “The collection of rational & irrational numbers together forms Real numbers”. 

Number System Class 9 Notes Maths Chapter 1

Both Rational & irrational numbers satisfy commutative law, associative law, and distributive law for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always irrational. If we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number. But this statement is not true for irrational numbers. We can see the example of this one by one

Rational Number + Rational Number = Rational Number

Let, a = Number System Class 9 Notes Maths Chapter 1 (rational) and b = Number System Class 9 Notes Maths Chapter 1  (rational),
= Number System Class 9 Notes Maths Chapter 1
= Number System Class 9 Notes Maths Chapter 1
= Number System Class 9 Notes Maths Chapter 1
= Number System Class 9 Notes Maths Chapter 1 (rational number)

Rational Number - Rational Number = Rational Number

Let, a = Number System Class 9 Notes Maths Chapter 1 (rational) and b = Number System Class 9 Notes Maths Chapter 1 (rational)
= Number System Class 9 Notes Maths Chapter 1
= Number System Class 9 Notes Maths Chapter 1
Number System Class 9 Notes Maths Chapter 1  (rational number)

Rational Number X  Rational Number =  Rational Number

Example: 

Let, a =  Number System Class 9 Notes Maths Chapter 1 (rational) and b = Number System Class 9 Notes Maths Chapter 1 (rational)
Hence, Number System Class 9 Notes Maths Chapter 1 × Number System Class 9 Notes Maths Chapter 1= Number System Class 9 Notes Maths Chapter 1 (rational)

Rational Number / Rational Number = Rational Number

Let, a = Number System Class 9 Notes Maths Chapter 1 (rational) and b = Number System Class 9 Notes Maths Chapter 1  (rational)
Number System Class 9 Notes Maths Chapter 1  divided by Number System Class 9 Notes Maths Chapter 1 
i.e.,Number System Class 9 Notes Maths Chapter 1 
Hence,  Number System Class 9 Notes Maths Chapter 1  ÷ Number System Class 9 Notes Maths Chapter 1 = Number System Class 9 Notes Maths Chapter 1 x Number System Class 9 Notes Maths Chapter 1 =  Number System Class 9 Notes Maths Chapter 1 x 3 Number System Class 9 Notes Maths Chapter 1

The sum and difference of a rational number and an irrational number is an irrational number.

Example:

Let, a = Number System Class 9 Notes Maths Chapter 1  (rational) and b = √3 (irrational) then,
a + b = Number System Class 9 Notes Maths Chapter 1+ √3 =Number System Class 9 Notes Maths Chapter 1 (irrational)
a − b = Number System Class 9 Notes Maths Chapter 1 − √3 =Number System Class 9 Notes Maths Chapter 1 (irrational)

The multiplication or division of a non-zero rational number with an irrational number is an irrational number.

Example:

Let, a =Number System Class 9 Notes Maths Chapter 1 (rational) and b = √2 (irrational) then,

ab =  Number System Class 9 Notes Maths Chapter 1 × √2 =Number System Class 9 Notes Maths Chapter 1 (irrational)

Number System Class 9 Notes Maths Chapter 1  = Number System Class 9 Notes Maths Chapter 1 =  Number System Class 9 Notes Maths Chapter 1 ×  Number System Class 9 Notes Maths Chapter 1   =Number System Class 9 Notes Maths Chapter 1 (irrational)

If we add, subtract, multiply or divide two irrational numbers, we may get an irrational number or rational number.

Example:

Let two irrational numbers be
a = 3 + √2 and b = 3 − √2 then
a + b = ( 3 + √2 ) + ( 3 − √2 )
= 3 + √2 + 3 − √2
= 3 + 3
= 6 (rational)
Let two irrational numbers be
a = √3 + 1 and b = √3 − 1 then
A + b = (√3 + 1 ) + (√3 − 1)
= √3 + 1 + √3 − 1
= 2√3 (irrational)

Examples: Write which of the following numbers are rational or irrational.

(a) π − 2 

(b) (3 + √27 ) − (√12 + √3)
(c) Number System Class 9 Notes Maths Chapter 1

(a) π − 2

We know that the value of the π = 3.1415
Hence, 3.1415 – 2 = 1.1415
This number is non-terminating non-recurring decimals.

(b) (3 + √27 ) − (√12 + √3)

On simplification, we get
( 3+ Number System Class 9 Notes Maths Chapter 1 ) - (Number System Class 9 Notes Maths Chapter 1  +  Number System Class 9 Notes Maths Chapter 1)
= 3 + 3√3 − 2√3 − √3
= 3 + √3 − √3
= 3, which is a rational number.

(c) Number System Class 9 Notes Maths Chapter 1

Here, 4 is a rational number and √5 is an irrational number. Now, we know that division of rational number and irrational number is always an irrational number.

Example: Add:  3 √2 + 6 √3   and √2 - 3√3
= (3 √2 + 6 √3 ) + (√2 – 3 √3 )
= (3 √2 + √2 ) + (6 √3 – 3 √3 )
= (3 + 1) √2 + (6 − 3) √3
= 4√2 + 3√3

Example: Multiply: 5√3 x 3√3

5√3  x  3√3 
= 5 x 3 x √3 x √3 
= 15 x  3 = 45

Question for Chapter Notes: Number System
Try yourself:What is the value of the product of two irrational numbers?
View Solution

Representation of √x for any positive integer x on the number line geometrically

We understand this method with the help of following steps. This construction shows that √x exists for all real numbers x > 0

Step I: Firstly mark the distance x from fixed point on the number line i.e. PQ = x

Step II: Mark a point R at a distance 1 cm from point Q and take the mid-point of PR.

Step III: Draw a semicircle, taking O as centre and OP as a radius.

Step IV: Draw a perpendicular line from Q to cut the semi-circle to find √x

Step V: Take the line QR as a number line with Q as zero.

Step VI: Draw an arc having centre Q and radius QS to represent √x on number line.

Number System Class 9 Notes Maths Chapter 1

We can see this method with the help of example

Example: let us find it for x = 4.5, i.e., we find √4.5

(i) Firstly, draw a line segment AB = 4.5 units and then extend it to C such that BC = 1 unit.
(ii) Let O be the Centre of AC. Now draw the semi- circle with centre O and radius OA.
(iii) Let us draw BD from point B, perpendicular to AC which intersects semi-circle at point D.
Number System Class 9 Notes Maths Chapter 1Hence, the distance BD represents √4.5 ≈ 2.121 geometrically. Now take BC as a number line, draw an arc with centre B and radius BD from point BD, meeting AC produced at E. So, point E represents √4.5  on the number line.

Radical Sign: Let a > 0 be a real number and n be a positive integer, such that

(a) Number System Class 9 Notes Maths Chapter 1 = Number System Class 9 Notes Maths Chapter 1 is a real number, then n is called exponent, and a is called radical and “√ ” is called radical sign.
The expression Number System Class 9 Notes Maths Chapter 1  is called surd.

Example: If n = 2 then (4)  Number System Class 9 Notes Maths Chapter 1 = Number System Class 9 Notes Maths Chapter 1 is called square root of 2.

Identities

Now we will list some identities which are related to square roots. You are familiar with these identities, which hold good for positive real number a and b. Let a and b be positive real numbers. Then,

Let’s solve some examples on the basis some of identities:

Examples: Simplify each of the following

(a) Number System Class 9 Notes Maths Chapter 1 xNumber System Class 9 Notes Maths Chapter 1   
(b) Number System Class 9 Notes Maths Chapter 1
(c) ( √2 + √3 ) (√2  - √3)
(d) (5  +  √5 )  (5  - √5 ) 

(a) Number System Class 9 Notes Maths Chapter 1 xNumber System Class 9 Notes Maths Chapter 1   

We know that,
Number System Class 9 Notes Maths Chapter 1 x  Number System Class 9 Notes Maths Chapter 1 =  Number System Class 9 Notes Maths Chapter 1
= Number System Class 9 Notes Maths Chapter 1
= Number System Class 9 Notes Maths Chapter 1 = Number System Class 9 Notes Maths Chapter 12
= (25)Number System Class 9 Notes Maths Chapter 1
= 21
= 2

b) Number System Class 9 Notes Maths Chapter 1

We know that,
Number System Class 9 Notes Maths Chapter 1 =  Number System Class 9 Notes Maths Chapter 1
=  Number System Class 9 Notes Maths Chapter 1Number System Class 9 Notes Maths Chapter 1
=(34)Number System Class 9 Notes Maths Chapter 1
= 3 

(c) (√2 + √3 ) (√2  - √3)

Number System Class 9 Notes Maths Chapter 1

(d) (5  +  √5 )  (5  - √5 )

We know that,
Number System Class 9 Notes Maths Chapter 1

Rationalising the Denominators
Looking at the value Number System Class 9 Notes Maths Chapter 1  can you tell where this value will lie on the number line? It is a little bit difficult. Because the value containing square roots in their denominators and division is not easy as addition, subtraction, multiplication and division are convenient if their denominators are free from square roots. To make the denominators free from square roots i.e. they are whole numbers, we multiply the numerator and denominators by an irrational number. Such a number is called a rationalizing factor.
Note: Conjugate of Number System Class 9 Notes Maths Chapter 1 is , Number System Class 9 Notes Maths Chapter 1  and conjugate of , Number System Class 9 Notes Maths Chapter 1
Let’s solve some examples on rationalizing the denominators:
Examples: Rationalise the denominator of the following
(a) Number System Class 9 Notes Maths Chapter 1
(b) Number System Class 9 Notes Maths Chapter 1
(c) Number System Class 9 Notes Maths Chapter 1
(d) Number System Class 9 Notes Maths Chapter 1

(a) Number System Class 9 Notes Maths Chapter 1

Rationalization factor for Number System Class 9 Notes Maths Chapter 1
Here, we need to rationalise the denominator i.e., remove root from the denominator. Hence, multiplying and dividing by Number System Class 9 Notes Maths Chapter 1
Number System Class 9 Notes Maths Chapter 1
Number System Class 9 Notes Maths Chapter 1

(b) Number System Class 9 Notes Maths Chapter 1

We know that the conjugate of 4 + √2 = 4 - √2
Number System Class 9 Notes Maths Chapter 1
Number System Class 9 Notes Maths Chapter 1

(c) Number System Class 9 Notes Maths Chapter 1

We know that the conjugate of √3 - √5 = √3 + √5
Number System Class 9 Notes Maths Chapter 1
Number System Class 9 Notes Maths Chapter 1

(d) Number System Class 9 Notes Maths Chapter 1

We know that the conjugate of 5 + 3√2 = 5 - 3√2
Number System Class 9 Notes Maths Chapter 1
Number System Class 9 Notes Maths Chapter 1

Laws of Exponent for Real Numbers

Now we will list some laws of exponents, out of these some you  have learnt in your earlier classes. Let a (> 0) be a real number and m, n be rational numbers.

(i) am X an = am+n

(ii) (am)n = amn

(iii) Number System Class 9 Notes Maths Chapter 1 = am-n

(iv) am X bm = (ab)m

(v) a-m = Number System Class 9 Notes Maths Chapter 1

(vi) (Number System Class 9 Notes Maths Chapter 1) -m = (Number System Class 9 Notes Maths Chapter 1)m

(vii) (a)Number System Class 9 Notes Maths Chapter 1 = Number System Class 9 Notes Maths Chapter 1 where m and n ∈ N

Note: The value of zero exponent i.e. a° =1

Let us now discuss the application of these laws in simplifying expression involving rational exponents of real numbers.
Examples: Simplify each of the following
(i) (2)5 x  (2)3 
(ii) (43)2
(iii) Number System Class 9 Notes Maths Chapter 1
(iv) 72 × 62
(v) 6-2
(vi) Number System Class 9 Notes Maths Chapter 1
(vii) 33/2

(i) (2)5 x  (2)3

We know that,
am x  an  =  am+n
Hence,
(2)5 × (2)3 = (2)5+3 = (2)8

(ii) (43)2

We know that,
(am)n = amn
(43)2 = (4)3 ×2 = (4)6

(iii) Number System Class 9 Notes Maths Chapter 1

We know that,
am/an = am-n
= Number System Class 9 Notes Maths Chapter 1 =  53-2  =  51

(iv) 72 × 62

We know that,
am  x  bm  =  (ab)m
(7)2 × (6)2 = (7 × 6)2 = (42)2

(v) 6-2

We know that,
a-m  =1/am
6-2  = 1/62 =  1/36

(vi) Number System Class 9 Notes Maths Chapter 1

We know that
Number System Class 9 Notes Maths Chapter 1

(vii) 33/2

We know that
 Number System Class 9 Notes Maths Chapter 1

Question for Chapter Notes: Number System
Try yourself:What is the decimal representation of a non-terminating repeating decimal?
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Summary

1. Rational Numbers:

  • A number is classified as rational if it can be expressed as a fraction p/q where p and q are integers and q ≠ 0.
  • The decimal expansion of a rational number is either terminating or non-terminating but recurring.

2. Irrational Numbers:

  • A number is classified as irrational if it cannot be expressed as p/q, where both p and q are integers and q ≠ 0.
  • The decimal expansion of an irrational number is non-terminating and non-recurring.

3.  Real Numbers:

  • The set of all rational and irrational numbers combined forms the collection of real numbers.

4. Operations with Rational and Irrational Numbers:

  • If r is rational and s is irrational, then:
  • r + s and r - s are irrational.
  • r × s and r / s (given r≠ 0) are irrational.

5. Identities for Positive Real Numbers:

   For any positive real numbers a and b:

Number System Class 9 Notes Maths Chapter 1

6. Rationalizing the Denominator:

To rationalize the denominator in terms like 1 / a + b multiply by the conjugate a - b / a - b.

7. Exponential Properties:

   Let  a > 0  be a real number and p and q be rational numbers. Then:

Number System Class 9 Notes Maths Chapter 1

The document Number System Class 9 Notes Maths Chapter 1 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on Number System Class 9 Notes Maths Chapter 1

1. What is an irrational number and how is it different from a rational number?
Ans. An irrational number is a type of real number that cannot be expressed as a fraction of two integers, meaning it cannot be written in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). Examples of irrational numbers include \( \sqrt{2} \) and \( \pi \). In contrast, a rational number can be expressed as a fraction, such as \( \frac{1}{2} \) or \( -3 \). The decimal expansion of irrational numbers is non-terminating and non-repeating, while rational numbers either terminate or repeat in their decimal form.
2. How are real numbers classified based on their decimal expansion?
Ans. Real numbers are classified into two main categories based on their decimal expansion: rational numbers and irrational numbers. Rational numbers have decimal expansions that either terminate after a finite number of digits (like 0.75) or repeat indefinitely (like 0.333...). Irrational numbers, on the other hand, have decimal expansions that are non-terminating and non-repeating. This means their decimal representation goes on forever without forming a repeating pattern.
3. What are the basic operations that can be performed on real numbers?
Ans. The basic operations that can be performed on real numbers include addition, subtraction, multiplication, and division. When performing these operations, it is important to note that the result of the operation on two real numbers will also be a real number. For example, adding two real numbers, such as 2.5 and 3.1, will yield another real number, 5.6. However, division by zero is undefined within the real number system.
4. What are the laws of exponents for real numbers?
Ans. The laws of exponents for real numbers include several key rules: 1. \( a^m \times a^n = a^{m+n} \) (Product of powers) 2. \( \frac{a^m}{a^n} = a^{m-n} \) (Quotient of powers) 3. \( (a^m)^n = a^{mn} \) (Power of a power) 4. \( a^0 = 1 \) (Any non-zero number raised to the power of zero is one) 5. \( a^{-n} = \frac{1}{a^n} \) (Negative exponent rule) These laws help simplify expressions involving exponents and are essential for solving various mathematical problems involving real numbers.
5. How can I summarize the key concepts of the number system for easy revision?
Ans. To summarize the key concepts of the number system for easy revision, focus on the following points: 1. Real numbers include both rational (fractions, integers) and irrational numbers (non-repeating, non-terminating decimals). 2. Real numbers can be represented on the number line, where rational numbers can be found at specific points, while irrational numbers fill in the gaps. 3. The basic operations (addition, subtraction, multiplication, division) are applicable to real numbers, and their results are also real numbers. 4. Familiarize yourself with the laws of exponents as they are crucial for simplifying expressions. 5. Remember the classification of numbers based on their properties, which will help in solving problems related to the number system effectively.
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