Number System, Class 9 Mathematics Detailed Chapter Notes PDF Download

INTRODUCTION

"God gave us the natural number, all else is the work of man". It was exclaimed by L. Kronecker (1823-1891), the reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers played in the evolution of human thought.
N : The set of natural numbers,
W : The set of whole numbers,

Z : The set of Integers, R 
Q : The set of rationals,
R : The set of Real Numbers.
In our day to life we deal with different types of numbers which can be broady classified as follows.

1. Natural Numbers (N) : The counting numbers 1, 2, 3, ..... are known as natural numbers. The collection of natural number is denoted by 'N' N = {1, 2, 3, 4...∞}

2. Whole numbers (W) : The number '0' together with the natural numbers 1, 2, 3, .... are known as whole numbers. The collection of whole number is denoted by 'W' W = {0, 1, 2, 3, 4...∞}

3. Integers (I or Z) : All natural numbers, 0 and negative of natural numbers are called integers. The collection of integers is denoted by Z or I. Integers (I or Z) : I or Z = {–∞, ..... –3, –2, –1, 0, +1, +2, +3 .... +∞}  Positive integers : {1, 2, 3...}, Negative integers : {.... –4, –3, –2, –1}

4. Rational numbers :– These are real numbers which can be expressed in the form of pq , where p and q are integers and q ≠ 0.

Ex. (2/3), (37/15), (-17/19), –3, 0, 10, 4.33, 7.123123123.........

5.  Irrational numbers :– A number is called irrational number, if it can not be written in the form p q , where p & q are integers and q ≠ 0. All Non-terminating & Non-repeating decimal numbers are Irrational numbers.

6.  Real numbers :– The totality of rational numbers and irrational numbers is called the set of real number i.e. rational numbers and irrational numbers taken together are called real numbers. Every real number is either a rational number or an irrational number.

FINDING RATIONAL NUMBERS BETWEEN TWO NUMBERS

(A) 1st method : Find a rational number between x and y then, is a rational number lying between x and y.

(B) 2nd method : Find n rational number between x and y (when x and y is non fraction number) then we use formula.

(C) 3rd method : Find n rational number between x and y (when x and y is fraction Number) then we use formula

then n rational number lying between x and y are (x + d), (x + 2d), (x + 3d) .....(x + nd)

Remark : x = First Rational Number, y = Second Rational Number, n = No. of Rational Number

Ex. Find 3 rational number between 2 and 5.

Sol. Let, a = first rational number. b = second rational number n = number of rational number Here a = 2, b = 5
A rational number between 2 and 5 =

Second rational number between 2 and 

Third rational number between  

Hence, three rational numbers between 2 and 5 are :  Ans.

RATIONAL NUMBER IN DECIMAL REPRESENTATION

Every rational number can be expressed as terminating decimal or non-terminating decimal.

1. Terminating Decimal : The word "terminate" means "end". A decimal that ends is a terminating decimal.

OR

A terminating decial doesn't keep going. A terminating decimal will have a finite number of digits after the decimal point.

Ex. Express 7/8 in the decimal form by long division method.

Sol. We have,

2. Non terminating & Repeating (Recurring decimal) :– A decimal in which a digit or a set of finite number of digits repeats periodically is called Non-terminating repeating (recurring) decimals.

Ex. Find the decimal representation of  (8/3)

Sol. By long division, we have

REPRESENTATION OF RATIONAL NUMBERS ON A NUMBER LINE

We have learnt how to represent integers on the number line. Draw any line. Take a point O on it. Call it 0(zero). Set of equal distances on the right as well as on the left of O. Such a distance is known as a unit length. Clearly, the points A, B, C, D represent the integers 1, 2, 3, 4 respectively and the point A', B', C' D' represent the integers –1, –2, –3, –4 respectively

Thus, we may represent any integer by a point on the number line. Clearly, every positive integer lies to the right of O and every negative integer lies to the left of O. Similarly we can represent rational numbers.

Ex. Represent 1/2 and -1/2 on the number line.

Sol. Draw a line. Take a point O on it. Let it represent 0. Set off unit length OA and OA' to the right as well as to the left of O.
The, A represents the integer 1 and A' represents the integer –1.

Now, divide OA into two equal parts. Let OP be the first part out of these two parts. Then, the point P represents the rational number 1/2.
Again, divide OA' into two equal parts. Let OP' be the first part out of these 2 parts. Then the point P' represents the rational number -1/2

REPRESENTATION OF DECIMAL NUMBERS ON A NUMBER LINE

Termentaing Decimal :- Suppose we want to represent 2.3 on the number line. We know that it lies between 2 and 3. We therefore, look at the portion of the number line between 2 and 3. We divide this portion into 10 equal parts.

This first mark will represent 2.1, second mark will represent 2.2 and the third mark P will represent 2.3.

REPRESENTATION OF NUMBER ON THE NUMBER LINE BY MEANS OF MAGNIFYING GLASS

The process of visualization of numbers on the number line through a magnifying glass is known as successive magnification. Sometimes, we are unable to check the numbers like 3.765 and on the number line we seek the help of magnifying glass by dividing the part into subparts and subparts into again equal subparts to ensure the accuracy of the given number.

Represent 3.765 on the number line. This number lies between 3 and 4. The distance 3 and 4 is divided into 10 equal parts. Then the first mark to the right of 3 will represent 3.1 and second 3.2 and so on. Now, 3.765 lies between 3.7 and 3.8. We divide the distance between 3.7 and 3.8 into 10 equal parts. 3.76 will be on the right of 3.7 at the six^th mark, and 3.77 will be on the right of 3.7 at the 7^th mark and 3.765 will lie between 3.76 and 3.77 and soon.

To mark 3.765 we have to use magnifying glass.

Non terminating & repeating decimals :-
Visualize on the number line, up to 4 decimals places.
Suppose we want to represent on the number line, up to 4 decimals places by magnifying glass. This can be
done as follows :
We have, = 4.262626

This number lies between 4 and 5. The distance between 4 and 5 is divided into 10 equal parts. Then the first mark to the right of 4 will represent 4.1 and second 4.2 and soon.Now 4.2626 lies between 4.2 and 4.3. We divide the distance between 4.2 and 4.3 into 10 equal parts. Now, 4.2626 lies between 4.26 and 4.27. Again we divide the distance between 4.26 and 4.27 into 10 equal parts. The number 4.2626 lies between 4.262 and 4.263. The distance between 4.262 and 4.263 is again divided into 10 equal parts. Sixth mark from right to the 4.262 is 4.2626.

CONVERSION OF TERMINATING AND NON-TERMINATING DECIMAL NUMBERS INTO THE FORM OF p/q

Terminating decimal numbers :- When the decimal number is of terminating nature.
Algorithm :
Setp-1 : Obtain the rational number.
Setp-2 : Determine the number of digits in its decimal part.
Setp-3 : Remove decimal point from the numerator. Write 1 in the denominator and put as many zeros on the right side of 1 as the number of digits in the decimal part ofthe given rational number.
Setp-4 : Find a common divisor of the numerator and denominator and express the rational numbe to lowest terms by dividing its numerator and denominator by the common divisor.

Ex. Express each of the following numbers in the form p/q
1. 

2. 

3. 

Non-terminating repeating decimal numbers :- In a non terminating repeating decimal, there are two types of decimal representation.
(a) Pure recurring decimal (b) Mixed recurring decimal

(a) Pure recurring decimals : A decimal in which all the digit after the decimal point are repeated. These type of decimals are known as pure recurring decimals.
For example : are pure recurring decimals.

(b) Mixed recurring decimals : A decimals in which at least one of the digits after the decimal point is not repeated and then some digit or digits are repeated. This type of decimals are known as mixed recurring decimals.
For example, are mixed recurring decimals.

Conversion a pure recurring decimal to the form p/q.

Algorithm :
Step-1 : Obtain the repeating decimal and put it equal to x (say).
Step-2 : Write the number in decimal form by removing bar from the top of repeating digit and listing repeating digits at least twice.

Step-3: Determine the number of digits having bar on their heads.
Step-4 : If the repeating decimal has 1 place repetition, multiply by 10; a two place repetition, multiply by 100; a three place repetition, multiply by 1000 and so on.
Step-5 : Subtract the number in step (ii) from the number obtained in step (iv).
Step-6 : Divide both sides of the equation by the coefficient of x.
Step-7 : Write the rational number in its simplest form.

Ex. Express each of the following decimals in the form p/q

(i) (ii)  (iii) 

(i)   Then, ⇒ x = 0.66 ......(i)

Here we have only one repeating digit, so we multiply both sides of (i) by 10 to get
⇒ 10x = 6.66 ... .....(ii)

On subtracting (i) from (ii), we get ; 10x - x = (6.66 ...) - (0.66 ) ⇒ 9x = 6 ⇒ x = 6/9 ⇒ x = 2/3

Hence 

(ii)        ...(i)
Here we have two repeating digit, so we multiply both sides of (i) by 10² = 100 to get
⇒  100x = 35.3535 ...  ...(ii)

On subtracting (i) firm (ii), we get 100x - x = (35.3535 ...) - (0.3535 ...)⇒ 99x = 35 ⇒ x = 35/99

(iii)

Here we have three repeating digit, so we multiply both slides of (i) by 10³ = 1000 to get

⇒ 1000x = 585.585585 .......(ii)

On subtracting (i) from (ii), we get 1000x - x = (585.585585 ....) - (0.585585 ....) ⇒999x= 585

Ex. Convert the following decimal numbers in form p/q :

(i)  (ii)

Sol.(i) Let x = ⇒ x = 5.2222    ...(i)
Multiplying bdth sides of (i) by 10, we get
10 x = 52.222 . ....(ii)
Subtracting (i) from (ii) we get 10 x - x = (52.222....) - (5.222 ...) ⇒ 9x = 47 ⇒ x = 47/9

(ii) Let x = ⇒ x = 23.434343 ...(i)
Multiplying both sides of (i) by 100, we get
10 x = 2343.4343 .....    ... (ii)
Subtracting (i) frcm (ii) we get
100 x - x = (2343.4343 . ..) - (23.4343 ...)⇒ 99x = 2320 ⇒ x = 2320/99

Aliter method :

We have,    [Using the above rule, we have 

Conversion of a mixed recurring decimal to the form p/q
Step-1 : Obtain the mixed recurring decimal and write it equal to x (say).
Step-2 : Determine the number of digits after the decimal point which do not have bar on them. Let there be n digits without bar just after the decimal point.
Step-3 : Multiply both sides of x by 10n so that only the repeating decimal is on the right side of the decimal point.
Step-4 : Use the method of converting pure recurring decimal to the form p/q and obtain the value of x.

Ex. Express the following decimals in the form p/q : (i) 

(i) Let x =
Clearly, there is just one digit on the right side of the decimal point which is without bar. So, we multiply both sides of x by 10 so that only the repeting decimal is left on the right side of the decimal point.

COMPETITION WINDOW

ADDITION & SUBTRACTION OF RECURRING DECIMALS

Addition and subtraction of recurring decimals can be done in two ways – either by converting them into vulgar
fractions and then operating them or alternatively as follows :-
Add
Step-1 : Express the number without bar as 5.732323232 + 8.613613613613

Step-2 : Write the number as one above other i.e. 5.732323232

8.613613613

Step-3 : Divide the number into two parts. In the first part i.e., left side write as many digits as there will be integral value with non recurring decimal. In the right side write as many digits as the LCM of the number of recurring digits in the given decimal number e.g.,
5.7/323232 (Since 5.7 is the integral + non recurring part) .8.6/136136 (LCM of 2 and 3 is 6)

Step-4 : Now add or subtract as usual.

Step-5 : Put the bar over the digits which are on the right side in the resultant value

MULTIPLICATION & DIVISION OF RECURRING DECIMALS

If can be done as usual. Just convert the decimals into vulgar fractions and then operate as required.

IRRATIONAL NUMBERS
A number is called irrational numbe if it canot be written in the form p q , where P and q are integer and q ≠ 0.

OR

A non-terminating and non repeating decimal is called an irractional numbers 0.10100100010000 ...... etc.
 etc are all irrational numbers.

HISTORICAL FACT ABOUT IRRATIONALS

We saw, in the previous section, that there may be numbers on the number line that are not rationals. In this section, we are going to investigate these numbers. So far, all the numbers you have come across, are of the form p/q where p and q are integers and q ≠ 0. So, you may ask : are there numbers which are not of this form?  There are indeed such numbers. 

The Pythagoreans in Greece, followers of the famous mathematician and philosopher Pythagoras, where the first to discover the numbers which were not rationals, around 400 BC. These numbers are called irrational numbers (irrationals), because they cannot be written in the form of a ratio of integers. There are many myths surrounding the discovery of irrational numbers by the Pythagorean, Hippacus of Croton. In all the myths Hippacus has an unfortunate end, either for discovering that √2 is irrational or for disclosing the secret about √2 to people outside the secret Pythagorean sect! 

Pythogoream proved that √2 is irrational. Later in approximately 425 BC. Theoderus of cyrene showed that
 are also irrationals. π was known to various century for thousand of years. If was proved to be irrational by lambest and Legendre only in the late 1700 s.

1. π is approximately equal to 22/7. So π is irrational and 22/7 is rational number.
2. There are infinitely many irrational number between two irrationals.

PROPERTIES OF IRRATIONAL NUMBERS

(i)  Negative of ail irrational number is an irraticnal number.

(ii) Sum and difference of a rational and an irrational number is an irrational mauter.

 are irrational number
 are irrational number
(iii) Sum and difference of two irrational number is not neaessarily an irrational number.

 is an irrational and
 is an irrational number
  which is rational.
  a rational number.

(iv) Product of a rational number with an irrational number is not always irrational.

(v) Product of a nan-zero rational number with an irrational muter is always irrational.

(vi) Product of mi irrational with on irrational is not always irrational.

  an irrational nurrioer.

REAL NUMBERS

Rational numbers together with irrational numbers are said to be real numbers. That is, a real number is either rational or irrational.

This was proved in different ways in the 1870s by to German mathematicians. Cantor and Dedekind that corresponding to every real number there is a point on the real number line and corresponding; to every point on the number line. there exists a unique real number

COMPLEX NUMBER

A number Z of the form Z = a +ib, where a and fc are real numbers and i = √-1 is called a complex number. 

i² = - 1, i³ = i² x i = (-1) x i =-i, i⁴ = i² x i²= (-1) x (-1) = 1

 it is not an irrational number.
 are called imaginary numbers.

REPRESENTATION OF IRRATIONAL NUMBERS ON A NUMBER LINE

Method-Ist :- Plot  on a number line Constructing the 'square root spiral' :- Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point O and draw a line segment OP₁ of unit length. Draw a line segment P₁P₂ perpendicular to OP₁ of unit length [see figure]. Now draw a line segment P₂P₃ perpendicular to OP₂. The draw a line segment P₃P₄ perpendicular to OP₃.

Continuing in this manner, we can get the line segment P_{n – 1}P_n by drawing a line segment of unit length perpendicular to OP_{n – 1}. In this manner, we will have created the points : P₁, P₂, P₃, ....., P_n, ...., and joined them to create a beautiful spiral depicting

Method : II

To represent. √2 on the real number line :
let ℓ be a real nunber line and 0 be a point representing 0 (zero) . Take OA = 1 unit. Draw AB ⊥ QA such that AB = 1 unit. 

with O as a centre and OB radius, drawT an arc, meeting line l at P.
Then, OB = OP = √2 unit    
Thus, the point P represent √2 on the number line .

To represent √3 on the real number line :

  [By Pythagorus theorem]

Then, OC = OQ = √3 unit
Thus, the point Q represent √3 on the number line.

To represent √5 on the real ranter line :

  [By Pythagorus theorem]

Then, OB = OP = √5
Thus, the point P represent √5 on the mirtoer line.

To represent. √6 on the real number line :

 [By Pythagorus theorem]

Then, OC = OQ = √6 unit
Thus, the point Q represent √6 on the number line.

Ito represent √7 on the real number line :

Then, OD = OQ = √7 unit
Thus, the point Q represent √7 on the number line.

To represent √8 on the real ranter line :

Then, OB = OP = √8 unit
Thus, the point P represent √8 on the number line

To represent √12 on the real number line :

Thus, the point R represent √12on the number line.

Ex. Insert a rational and an irrational number between 2 and 3.
Sol.

If a, b are rational numbers, then 
2 is a rational number between them.
∴ A rational number between 2 and 3 is  Ans.
If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then is an irrational number lying between a and b.
Hence, irrational number between 2 and 3 is Ans.

Ex. Find two irrational numbers between 2 and 2.5.
Sol. If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then  is an irrational number lying between a and b.

∴ Irration number between 2 and 2.5 is 
Similarly, irrational number between
So, required numbers are 

GEOMETRICAL REPRESENTATION OF SQUARE ROOT OF A REAL NUMBERS ON A NUMBER LINE

THEOREM :- Prove that every positive real number x, x is also a positive real number (i.e., x exists) which
can be represented geometrically on the real line.
ALGORITHM
Step-1 : Let x be a positive real number. Take AB = x units and BC = 1 unit on the real line l.
Step-2 : Find the mid point O of AC and draw a semicircle with centre O and radius OA or OC.
Step-3 : At B, draw a line BD ⊥ AC, where D is a point on the semicircle.
Step-4 : Join OD.
Step-5 : Further, with centre B and radius BD, draw an arc intersecting the real line l at P.
Therefore, BP = BD = √x .
Justification : We have, In right triangle OBD,

 units (radius of the semicircle)

In right ΔOBD, we have OD² = OB² + BD²
BD² = OD² - OB²

and  [By Pythagoras theorem]
[∴ A² - B² = (A + B) (A - B)]

Tiros, √x exists for all positive real numbers.
Hence, the point P represents √x on the real line.

Ex. Represent   geometrically on the number line.

Sol. Let l be the number line. 
Draw a line segment AB = 3.28 units and BC = 1 unit. Find the mid point O of AC.

Draw a semicircle with centre O and radius OA or OC.
Draw BD ⊥ AC intersecting the semicircle at D. Then BD =  units. Now, with centre B and radius BD, draw an arc intersecting the number line ℓ at P.
Hence, BE = BP =

Represent the following lumbers geometrically on the nunber line.

EXPONENTS OF REAL NUMBERS

exponents OR index or index number or power

If a number is multiplied by itself a number of times, then it can be written in the exponential form 3 × 3 = 3² O R x × x × x × .... n times = xⁿ

   x = any rational number

   n = Positive Integer

We read as

 and xⁿ then number   are called bases and 2, 4, 3, n are called exponents or index.

Zero Exponent : For any non zero rational number x we define a° = 1

Positive integral Eiwer : - let a be a real number and nbe a pcsitix^e integer. Then we define aⁿ as aⁿ = a x a x a x a^x ....^x a (n times)
Where aⁿ is called the n^th power of a. The real number a is called the base and n is called the exponent of the n^th power of a.

Example : 

Negative integral Power : Let a be any non-zero real number and n be a positive integer. Then, we define a^-n =

Example. : 

LAWS OF INTEGRAL EXPONENTS

First law (Product Law) :- Let a be any real number and m, n are positive integers, then

Second Law (Quotient- Law): Let a he a non zero real number and m, n are positive integers, then

Example : 

Third Law (Power Law) : - Let a be- a positive real number and m, n are positive integers, then

Example :

Fourth Law : - Let a, b be two real number and n is a positive integer, then

Example :

RATIONAL EXPONENTS OF A REAL NUMBER

Principal n^th root of a Positive real number :- Let a be a positive real number and n be a positive integer. Then, the principal n^th root of a is the unique positive real number x such that xⁿ = a.
The principal n^th root of a positive real number a is denoted by

Principal n^th Root of a Negative Real Number :- Let a be a negative real number and n be an odd positive integer.

Then, the principal n^th root of a is defined as 

i. e. the principal n^th root of a is minus of the principal n^th root of | a |,

Examples :

Remark : - n^th root of a is not defined. If a is negative real number and n is an even positive integer because an even power of a real number is always positive.

Examples :   is meaningless quantity .

Justification :  which is not possible as x2 should always 'be positive.

LAWS OF EXPONENTS

Let a, b > 0 be a real number, and let m and n be rational numbers.

Then, we have

We can also generalise the laws of exponent for the n^th root of a nurrber. These are given below : -

Ex. Evaluate each of the following :-

Sol. 4 We have 2s x 5² = (2 x 2 x 2 x 2 x 2) x (5 x 5) = 32 x 25 = 800
(ii) We have (2³)2 = (2)^{3 x 2} = 2⁶ = 64

If a is rational number and n is positive integer such that n^th root of a is an irrational number then is called surd or radical where the symbol is called the radical sign and index n is called order of the surd.  is real as n^th root of 'a' and can also be written as a^1/n.

O R

If the root of a number cannot exactly obtained, the root is called a surd or an irrational number.

Every surd is an irrational number but every irrational number is not surd.

OPERATION OF SURDS
(a) Addition and subtraction of surds: Addition & subtraction of surds are possible only when order and radicand are same i.e. only for like surds.
Ex. Simplify :-

Sol.  [Using distributive law]

 [Using distributive law]

First we reduce each term to its simplest form

First we reduce each term to its simplest form

(b) Multiplication and division of surds:

Multiplication of Surds :- Surds of the same order can be multiplied according to the following law :-

RATIONALIZATION OF SURDS

The process of changing an irrational number into rational number is called 'rationalisation' and the factor by which we multiply and Divide the number is called 'rationalising factor'

SOME EXAMPLE OF RATIONALIZING FACTOR :–

(i) Rationalizing factor of is a rational
(ii) Rationalizing factor of  is a rational
(iii) Rationalizing factor of    is a rational
(iv) Rationalizing factor of   is a rational
(v) Rationalizing factor of 

 which is rational.

(vi) Rationalizing factor of 
(vii) Rationalizing factor of  which is a rational number.

CONJUGATE SURD

Two binomial surds which differ only in sign (+ or –) between the terms connecting them, are called conjugate

surds.   are conjugate surds. Also   are conjugate surds.

Ex. Fiiii the raticiializing factors of following :

Sol.    = 10 as 10 is rational number. ∴Rational fector of    is 

 Rationalizing factor of √2 is √2

Hence, Rationalizing factor of 

 Hence, Rationalizing factor of
 Hence rationalizing fatctor of   . Hence, Rationalizing factor of 

 [changing into simplest term]

RATIONALISATION OF MONOMIAL SURDS

Ex. Rationalise the denominator in each of the following :

Sol. Ths rationalising factor of the dencminator is 

 The rationalising factor of the denaninator is 3^1/3..

RATIONALISATION OF BINOMIAL SURDS

Sol. We have  [Miltiply and divide by]

(ii) We have,

RATIONALISATION OF TRINOMIAL SURDS

Ex. Rationalise the dencninator 

SQUARE ROOTS OF BINOMIAL QUADRATIC SURDS

(a) Since 
(b) ∴square root of
(c) square root of 
(d) square root of
(e) and square root of