Real Number

Rational number is a number which can be written in the form ,


where p and q both are integers but q is not equals to zero.

Examples =

(i) Given number = 5

First draw a line and mark the origin Zero. The given number 5 is positive so we are going to locate it on the right side of the zero.

(ii) Given = -3

First draw a line and mark the origin Zero. The given number is negative (-3) so we are going to locate it on the left side of the zero.

(iii) Given = 5/7

First draw a line and mark the origin Zero. The given number is positive so we are going to locate it on the right side of the zero.

(iv) Given = 8/3

First draw a line and mark the origin Zero. The given number is positive so we are going to locate it on the right side of the zero.

(v) Given = 1.3

First draw a line and mark the origin Zero. The given number 1.3 is positive so we are going to locate it on the right side of the zero.

(vi) Given = -2.4

First draw a line and mark the origin Zero. The given number -2.4 is negative so we are going to locate it on the left side of the zero.

(vii) Given = 23/6

First draw a line and mark the origin Zero. The given number is positive so we are going to locate it on the right side of the zero.

(i)

Let suppose,

As we can see that a ˂ b

So, the rational number lying between a and b

So, we can say that is a rational number lying between 1/4 and 1/3.

(ii)

Let’s take

As we can see a˂b

A rational number lying between

is the number lying between

(iii) 1.3 and 1.4

By taking a = 1.3 and b = 1.4

As we can see a ˂ b

The rational number between a and b,

1.35 is the number lying between 1.3 and 1.4

(iv) 0.75 and 1.2

Let’s take a = 0.75 and b = 1.2

As we can see a˂b,

The rational number between a and b,

0.975 is the number between 0.75 and 1.2

(v)

Let’s take

The rational number between a and b,

is the number between -1 and

(vi)

Let’s take

The rational number between a and b,

So, the rational number between

Let’s take and n = 3

n= numbers required to find out

So,

Thus, three rational numbers are:

(a + d), (a + 2d), and (a + 3d)

Hence, three rational numbers lying between 1/5 and 1/4 are

Let’s take

n= numbers required to be find out

So,

Thus, five rational numbers are:

(a + d), (a + 2d), (a + 3d), (a + 4d) and (a + 5d)

Hence, five rational numbers lying between

Let’s takea = 3, b = 4 and n = 6

n= numbers required to be find out

So,

Thus, six rational numbers are:

(a + d), (a + 2d), (a + 3d), (a + 4d), (a + 5d) and (a + 6d)

Hence, six rational numbers lying between

Let’s take

n= numbers required to be find out

So,

Thus, 16 rational numbers are:

(a + d), (a + 2d), (a + 3d), (a + 4d), (a + 5d), (a + 6d), (a + 7d), (a + 8d), (a + 9d), (a + 10d), (a + 11d), (a + 12d), (a + 13d), (a + 14d), (a + 15d) and (a + 16d)

So,

(a + d) = (2.1 + 0.005) = 2.105

(a + 2d) = [2.1 + (2 × 0.005)] = 2.110

(a + 3d) = [2.1 + (3 × 0.005)] = 2.115

(a + 4d) = [2.1 + (4 × 0.005)] = 2.120

(a + 5d) = [2.1 + (5 × 0.005)] = 2.125

(a + 6d) = [2.1 + (6 × 0.005)] = 2.130

(a + 7d) = [2.1 + (7 × 0.005)] = 2. 135

(a + 8d) = [2.1 + (8 × 0.005)] = 2. 140

(a + 9d) = [2.1 + (9 × 0.005)] = 2. 145

(a + 10d) = [2.1 + (10 × 0.005)] = 2. 150

(a + 11d) = [2.1 + (11 × 0.005)] = 2. 155

(a + 12d) = [2.1 + (12 × 0.005)] = 2. 160

(a + 13d) = [2.1 + (13 × 0.005)] = 2. 165

(a + 14d) = [2.1 + (14 × 0.005)] = 2. 170

(a + 15d) = [2.1 + (15 × 0.005)] = 2. 175

(a + 16d) = [2.1 + (16 × 0.005)] = 2. 180

Thus, the rational numbers between 2.1 and 2.2 are 2.105, 2.110, 2.115,
2.120, 2.125, 2.130, 2.135, 2.140, 2.145, 2.150, 2.155, 2.160, 2.165, 2.170, 2.175, 2.180,

First we have to know what is terminating decimal. Terminating decimal is the number which has digits that do not go on forever.

(i)

Denominator 80 has factors = 2 × 2 × 2 × 2 × 5

So, 80 has no prime factors other than 2 and 5, thus is terminating decimal.

(ii)

Denominator 24 has factors = 2 × 2 × 2 × 3

So, 24 has factors other than 2 and 5, thus is not a terminating decimal.

(iii)

Denominator 12 has factors = 2 × 2 × 3

So, 12 has factors other than 2 and 5, thus is not a terminating decimal.

(iv)

Denominator 35 has factors = 5 × 7

So, 35 has factors other than 2 and 5, thus is not a terminating decimal.

(v)

Denominator 24 has factors = 5 × 5 × 5

So, 125 has no prime factors other than 2 and 5, thus is a terminating decimal.

(i) Given

By actual division method, we get

Thus

(ii) Given

By actual division method we get:

Thus

(iii)

By actual division method we get:

Thus

(iv)

By actual division method we get:

Thus

(v)

By actual division method we get:

Thus

(i) Given

Let x equals to the repeating decimal =

As we can see the repeating digit is 3

x = 0.3333333……… (i)

10x = 3.333333……… (ii)

Subtracting (i) from (ii), we get

10x - x = 3.3333 – 0.3333

9x = 3

So, we can say that 0.3333333333…. is equals to the .

(i) Given

Let x equals to the repeating decimal =

As we can see the repeating digit is 3

x = 1.3333333……… (i)

10x = 13.333333……… (ii)

Subtracting (i) from (ii), we get

10x-x = 13.3333 – 1.3333

9x = 12

So, we can say that 1.3333333333…. is equals to the

(ii)

Let x equals to the repeating decimal =

As we can see the repeating digit is 34

X = 0.3434343434……… (i)

10x = 3.434343434……… (ii)

100x = 34.3434343434….. (iii)

Subtracting (i) from (iii), we get

100x-x = 0.34343434 – 34.34343434

99x = 34

So, we can say that 0.34343434343434…. is equals to the

(iii)

Let x equals to the repeating decimal =

As we can see the repeating digit is 14

X = 3.1414141414……… (i)

10x = 31.41414141414……… (ii)

100x = 314.1414141414….. (iii)

Subtracting (i) from (iii), we get

100x - x = 3.1414141414 – 314.14141414

99x = 311

So, we can say that 3.1414141414…. is equals to the

(iv)

Let x equals to the repeating decimal =

As we can see the repeating digit is 324

X = 0.324324324324324……… (i)

10x = 3.24324324324324……… (ii)

100x = 32.4324324324324….. (iii)

1000x = 324.324324324324…….(iv)

Subtracting (i) from (iv), we get

1000x - x = 0.324324324324 – 324.324324324324

999x = 324

So, we can say that 0.324324324324324324324…. is equals to the

(v)

Let x equals to the repeating decimal =

As we can see the repeating digit is 7

x = 0.17777777777……… (i)

10x = 1.777777777……… (ii)

100x = 17.77777777….. (iii)

Subtracting (ii) from (iii), we get

100x - 10x = 17.777777-1.777777

90x = 16

So, we can say that 0.17777777…. is equals to the

(vi)

Let x equals to the repeating decimal =

As we can see the repeating digit is 4

X = 0.5444444444……… (i)

10x = 5.44444444……… (ii)

100x = 54.444444….. (iii)

Subtracting (ii) from (iii), we get

100x-10x = 54.44444-5.44444

90x = 49

So, we can say that 0.5444444…. is equals to the

(vii)

Let x equals to the repeating decimal =

As we can see the repeating digit is 63

X = 0.16363636363……… (i)

10x =1.6363636363……… (ii)

100x = 16.363636363….. (iii)

1000x = 163.63636363…. (iv)

Subtracting (ii) from (iv), we get

1000x-10x = 163.636363-1.63636363

990x = 162

So, we can say that 0.16363636363…. is equals to the

(i) True: every natural number is the whole number because natural number starts with 1 and whole number start with 0. So, every natural number will automatically fall in the category of whole number.

(ii) False: every whole number can’t be natural number as natural number starts from 1 and whole number starts with 0.

(iii) True: Integers includes all whole numbers and their negative counterparts. Rational numbers can be expressed in the form of fractions where denominator is not equals to the zero but both, numerator and denominator are integers.

(iv) False: Rational number is the number which can be expressed in the form of fraction where denominator is not equals to zero. But whole numbers are natural numbers including zero and they can’t be written in fractional form.

(v) True: Rational number is the number which can be expressed in the form of fraction where denominator is not equals to zero and terminating decimal can also be written in fraction form.

(vi) True: Yes, every repeating decimal is also the rational number because it also written in the form of fraction.

(vii) True: yes, 0 is also the rational number because it can be written in the form of fraction.

Irrational number:- A number which can’t be expressed as an terminating decimal or recurring decimal and fractional form is called irrational.

Ex: π, (3.1415926535),

Irrational number is different from Rational number because Rational number is a number which can be expressed as fractional form or terminating decimal form is called rational number. It is exactly the opposite of Irrational number.

Ex:- 5 ,

(i)

=

∵ we can express 2 as which is the quotient of the integer 2 and 1

Hence, it is a rational number.

(ii)

=

∵ we can express 14 as which is the quotient of the integer 14 and 1

Hence, it is a rational number.

(iii)

=

∵we can not simplify in the form

Hence, it is an irrational number.

(iv)

We know that 43 is a prime number so we can not get prime factors of it and neither we can write in fractional form.

Hence, it is an irrational number.

(v)

∵ is irrational number and addition of an irrational number to any real number always gives irrational number.

Hence, it is an irrational number.

(vi)

∵ is irrational number and addition of an irrational number to any real number always gives irrational number.


Hence, it is an irrational number.

(vii)

=

∵ As, and √2 are irrational numbers and multiplication of an irrational number to a non zero rational number gives irrational number.

Hence, it is an irrational number.

(viii)

∵ we know that all repeating decimals are rational,

Hence, it is a rational number.

(ix) 1.232332333…

∵ The decimal expansion here is non terminating and non repeating,

Hence, it is an irrational number.

(x) 3.040040004…

∵ The decimal expansion here is non terminating and non repeating,

Hence, it is an irrational number.

(xi) 3.2576

∵ It is a terminating decimal fraction and can be expressed in form

Hence it is a rational number.

(xii) 2.3565656…

∵ it is a non terminating but repeating decimal form that can be written as 2.35.

Hence, it is a rational number.

(xiii)

∵ We know that π is a non terminating Decimal fraction,

Hence it is an irrational number.

(xiv)

∵ it is an fractional form,

Hence it is rational.

By using Pythagoras theorem,

By using Pythagoras theorem,

By using Pythagoras theorem,

By using Pythagoras theorem,

(i)

(ii)

(iii)

(i)

We cannot simplify

Hence 4 + is an irrational number.

(ii)

=

We can’t simplify

Hence it is an irrational number.

(iii)

We can’t simplify

Hence it is an irrational number.

(iv)

=

We can’t simplify

Hence it is an irrational number.

(v)

We cannot simplify

Hence it is an irrational number.

(vi)

We cannot simplify

Hence it is an irrational number.

(i) True: = always a rational number.

(ii) False: = , which is a rational number.

(iii) True: = always a rational number.

(iv) False: = which is a rational number.

(v) True : = , is always irrational.

(vi) False: = is always an irrational number.

(vii) False : As rational numbers are on number line and all numbers on number line is real. Hence, every rational number is also Real.

(viii) True: As both rational and irrational numbers can be presented at number line are real. Hence they may be rational or irrational.

(ix) True: π = 3.141592653…… non terminating decimal form and is a fractional form.

(i)

Adding by making pairs,

=

(ii)

Adding by making pairs,

=

(iii)

Adding by making pairs,

=

(i)

(ii)

=

=

(iii)

=

(iv)

=

(v)

=

(vi)

=

(i)

=

(ii)

=

(iii)

=

∵ (a + b) (a - b) = a² - b²

=

(ii)

∵ (a + b) (a - b) = a² - b²

(iii)

∵ (a + b) (a - b) = a² - b²

=

=

=

(v)

∵ (a - b)² = a² + b² - 2ab

=

(vi)

∵ (a - b)² = a² + b² - 2ab

=

Let’s draw a line AB = 3.2 units

Extend this line from B to C by 1 unit.

Now find the mid-point M of AC.

Take M as the center and MA as radius draw a semicircle.

Draw BD ⟘ AC intersecting the semi circle at D.

Then BD =

With B as center and BD as radius, draw an arc, meeting AC produced at E.

Then BE = BD =

Lets draw a line AB = 7.28 units

Extend this line from B to C by 1 unit.

Now find the mid-point M of AC.

Take M as the center and MA as radius draw a semicircle.

Draw BD ⟘ AC intersecting the semi circle at D.

Then BD =

With B as center and BD as radius, draw an arc, meeting AC produced at E.

Then BE = BD =

Closure property of addition of rational numbers:

The sum of two rational numbers is always a rational number.

If and are any two rational numbers, then is also a rational number.

Example:

Consider the rational numbers and Then,

, is a rational number

Commutative property of addition of rational numbers:

Two rational numbers can be added in any order.

Thus for any two rational numbers and , we have

=

Example:

=

=

=

Existence of additive identity property of addition of rational numbers:

0 is a rational number such that the sum of any rational number and 0 is the rational number itself.

Thus,

= for every rational number

0 is called the additive identity for rationals.

Example:

=

Existence of additive inverse property of addition of rational numbers:

For every rational number , there exists a rational number

such that = = = 0 and similarly, = 0.

Thus,

= is called the additive inverse of

Example:

= and similarly,

Thus, and are additive inverses of each other.

Associative property of addition of rational numbers:

While adding three rational numbers, they can be grouped in any order.

Thus, for any three rational numbers and , we have

= + =

Example:

Consider three rational numbers, Then,

=

=

=

=

Closure property of multiplication of rational numbers:

The product of two rational numbers is always a rational number.

If and are any two rational numbers then is also a rational number.

Example:

Consider the rational numbers and . Then,

Commutative property of multiplication of rational numbers:

Two rational numbers can be multiplied in any order.

Thus, for any rational numbers and , we have:

= =

Example:

Let us consider the rational numbers and Then,

= and

Therefore,

Associative property of multiplication of rational numbers:

While multiplying three or more rational numbers, they can be grouped in any order.

Thus, for any rationals we have:

=

Example:

Consider the rationals , we have

=

=

=

Existence of multiplicative identity property:

For any rational number , we have

1 is called the multiplicative identity for rationals.

Example:

Consider the rational number . Then, we have

=

Existence of multiplicative inverse property:

Every nonzero rational number has its multiplicative inverse .

Thus,

= is called the reciprocal of .

Clearly, zero has no reciprocal.

Reciprocal of 1 is 1 and the reciprocal of (-1) is (-1)

By rationalization the denominator, we get,

=

By rationalization the denominator, we get,

=

By rationalization the denominator, we get,

=

By rationalization the denominator, we get,

=

By rationalization the denominator, we get,

=

By rationalization the denominator, we get,

By rationalization the denominator, we get,

By rationalization the denominator, we get,

By rationalization the denominator, we get,

By rationalizing the L.H.S we get,

Putting LHS = RHS, we get,

= 2 + √3 = a + b√3

Clearly, a = 2 and b = 1.

By rationalizing the L.H.S we get,

=

Putting LHS = RHS, we get,

=

=

Clearly, a =

By rationalizing the L.H.S we get,

=

Putting LHS = RHS, we get,

Clearly,

By rationalizing the L.H.S we get,

Putting LHS = RHS, we get,

= 11 - 6√3 = a - b√3

Clearly a = 11 and b = 6.

By taking LCM,

By taking LCM,

=

Given that,

By rationalizing we get,

=

Hence,

Given that,

By Rationalizing

=

Now,

= 7 + 7 = 14

By rationalizing LHS we get,

=

=

= 3 + √8 - √8 - √7 + √7 + √6 - √6 - √5 + √5 + 2 = 3 + 2 = 5

Clearly, LHS = RHS,

Hence, proved.

(i)

We know that powers get added in multiplication, so,

(ii)

We know that powers get added in multiplication, so,

=

(iii)

We know that powers get added in multiplication, so,

=

(i)

We know that powers get subtracted in dividing, so,

(ii)

We know that powers get subtracted in dividing, so,

(iii)

We know that powers get subtracted in dividing, so,

=

(i)

We know that when the powers are same then only numbers get multiplied, so,

(ii)

We know that when the powers are same then only numbers get multiplied, so,

(iii)

We know that when the powers are same then only numbers get multiplied, so,

(i)

=

(ii)

=

(iii)

=

(i)

(ii)

(iii)

(i)

(ii)

(iii)

(i)

(ii)

(iii)

A number which cannot be written as simple fraction is called Irrational Number.

3.141141114… is an irrational number because in this decimal is going forever without repeating.

= 7 =

=

= = = 2 =

= 2.236067977…..

is an irrational number because in this decimal is going forever without repeating.

0.3232232223… is an irrational number because decimal is going forever without repeating.

= 15 =

So, is a rational number.

Every rational number is a real number.

Between any two rational numbers there are infinitely many rational numbers.

The decimal representation of a rational number is neither terminating nor repeating

The decimal representation of an irrational number is non-terminating and non-repeating. Ex. Value of pie.

= 1.41421356…….

So, Decimal expansion of is a non-terminating and non-repeating decimal

The product of two irrational number is sometimes rational and sometimes irrational

Case 1: X = 2 (this is rational)

Case 2: (this is irrational)

Every real number is either rational or irrational. If any real number can be written as fraction then it would be rational number otherwise it will be irrational number.

is irrational and is rational

= 3.14159265358979…….. (this is non-terminating and non-repeating)

= 3.142857142857142857 (this is repeating)

= 1.4142…

= 1.7321…..

1.5 is a rational number between.

Others would be 1.45, 1.55, 1.6 etc.

= =

= = 2

Let x = then

X = 0.32222 ………………(i)

10x = 3.2222 …………….(ii)

100x = 32.222 ………..(iii)

On subtracting (ii) from (iii) we get,

90x = 29

x=

= 14

(When only)

= 0.709

(When only)

= 5

= 0.414

Let x = then

X = 1.666 …………………(i)

10x = 16.666 …………….(ii)

On subtracting (i) from (ii) we get,

9x = 15

x=

x=

Let x = then

X = 0.545454………………(i)

10x = 5.45454 …………….(ii)

100x = 54.545454 ………..(iii)

On subtracting (i) from (iii) we get,

99x = 54

x=

x=

Let x = then

X = 0.123333 ………………(i)

10x = 1.23333 …………….(ii)

100x = 12.3333 ………..(iii)

1000x = 123.3333………(iv)

On subtracting (iii) from (iv) we get,

900x = 111

x=

x =

= 5.477225575051…..

= 3.316624790……..

The decimal representation of is non-terminating and non-repeating.

So, is an irrational number between 5 and 6.

Now, >

And, >

And, = 1.49534878122

And, = 1.56508458007

Both the options C & D are non-terminating and non-ending. So, both could be the answer.

= 0.142857142857

= 0.2857142857142857

= 0.2020305089104421……. It is non-terminating and non-ending.

A rational number between : =

Thus, the rational number between : =

Thus, the rational number between : =

Thus, three rational numbers between

The square roots of numbers that are not a perfect square are members of the irrational numbers.

Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

e may or may not be .

Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

The square roots of 3 is not a perfect square. So, it is an irrational number.

Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(a)-(r)

rational number.

(b)-(s)

an irrational number.

(c)-(q)

The value of is 0.142857142857142857…….. So length of period is 6.

(d)-(p)

If then

(a)-(r)

(b)-(s)

x -2 = - x + 4

x + x = 4 + 2

2x = 6

X = 3

(c)-(p)

then

(d)-(q)

=

=

Let two irrational numbers be 10 + 2√5 and 5 - 2√5
(i) sum is rational
= 10 + 2√5 + 5 - 2√5

= 15 (a rational number)
(ii) product is rational are

= (10 + 2√5) (10 - 2√5)
= (10)² - (2√5)²
= 100 - 20 = 80 (a rational number)

Let x = 10 and y = 2

X + y = is irrational number.

No

If you multiply any irrational number by the rational number zero, the result will be zero, which is rational.

is rational number.

Let x =

Then is an irrational number

And is a rational number.

Let x = then

X = 4.17171717………………(i)

10x = 41.7171717 …………….(ii)

100x = 417.171717 ………..(iii)

On subtracting (i) from (iii) we get,

99x = 413

X=

=

So, a =2 and b =1

=

So, a = and b =

=3

So, a = 3 and b=0.

So, a = and b =

Now,

= 98

Then, x

Then,

Ans,

Now,

=

= 3

We have,

Then,

= 34

Now,

If x and y are two rational numbers such that x<y then is a rational number between x and y.

So, rational numbers will be:

are two rational numbers lying between

If x and y are two rational numbers such that x<y then is a rational number between x and y.

So, rational numbers will be:

are two rational numbers lying between

Four irrational numbers are

0.1010010001…,

0.1212212221…,

0.13113313331…, and

0.1414414441…

As they all have non terminating and non repeating decimal

is given as:

Let x = then

X = 0.123232323………………(i)

10x = 1.23232323 …………….(ii)

100x = 123.23232323 ………..(iii)

On subtracting (ii) from (iii) we get,

90x = 122

X=

X =

So, a = 3 and b =2

0.1401401400014….. is an irrational number because it is non-ending and non-terminating.

There are infinitely many irrational numbers between two rational numbers.

Decimal representation of an irrational number is always non-terminating and non-repeating decimal.

We have,

Then,

=

We have,

Now, we have,

= 3 + 5

= 8

can be written as:

Equalizing powers by multiplying and multiplying and dividing by 12, we get,

Now, in ascending order,

So, is the smallest number.

(a)-(q)

is an irrational number.

(b)-(p)

is a rational number.

(c)-(s) = 0.23232323…… =

(d)-(r) = 0.23333… =

_{We have,}

On rationalizing we get,

Now,

Now,

Now,

We have,

= 15.7

We have,

= 1

Hence, Proved.