RD Sharma Solutions: Number System- 4 Notes | EduRev

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Q.1. Define an irrational number.

Proof: An irrational number is a real number that cannot be reduced to any ratio between an integer p and a natural number q.

If the decimal representation of an irrational number is non-terminating and non-repeating, then it is called irrational number. For example RD Sharma Solutions: Number System- 4 Notes | EduRev 


Q.2. Explain, how irrational numbers differ from rational numbers?

Proof: 

Every rational number must have either terminating or non-terminating but irrational number must have non- terminating and non-repeating decimal representation.

A rational number is a number that can be written as simple fraction (ratio) and denominator is not equal to zero while an irrational is a number that cannot be written as a ratio.


Q.3. Examine, whether the following numbers are rational or irrational:

(i) √7

(ii) √4

(iii) 2 + √3

(iv)3 + 2

(v) 3 + 5

(vi) ( 2 – 2)2

(vii) (2−2) (2+2) 

(viii) (2 + 3–√)2

(ix) √5−2

(x) √23

(xi) √225

(xii) 0.3796

(xiii) 7.478478

(xiv) 1.101001000100001

Proof: (i) Let RD Sharma Solutions: Number System- 4 Notes | EduRev 

Therefore,

RD Sharma Solutions: Number System- 4 Notes | EduRev 

It is non-terminating and non-repeating

HenceRD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number

(ii) Let RD Sharma Solutions: Number System- 4 Notes | EduRev 

Therefore,

RD Sharma Solutions: Number System- 4 Notes | EduRev 

It is terminating.

HenceRD Sharma Solutions: Number System- 4 Notes | EduRevis a rational number.

(iii) Let RD Sharma Solutions: Number System- 4 Notes | EduRev be the rational 

Squaring on both sides

RD Sharma Solutions: Number System- 4 Notes | EduRev

Since, x is rational 

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational

But,RD Sharma Solutions: Number System- 4 Notes | EduRevis irrational

So, we arrive at a contradiction.

Hence RD Sharma Solutions: Number System- 4 Notes | EduRev  is an irrational number

(iv) Let RD Sharma Solutions: Number System- 4 Notes | EduRev be the rational number

Squaring on both sides, we get

RD Sharma Solutions: Number System- 4 Notes | EduRev

Since, x is a rational number

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational number

But RD Sharma Solutions: Number System- 4 Notes | EduRev  is an irrational number

So, we arrive at contradiction

Hence RD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number

(v) Let RD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number

Squaring on both sides, we get

RD Sharma Solutions: Number System- 4 Notes | EduRev

Now, x is rational number

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 4 Notes | EduRev is rational number

But
RD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number

So, we arrive at a contradiction

Hence RD Sharma Solutions: Number System- 4 Notes | EduRev  is an irrational number

(vi) Let RD Sharma Solutions: Number System- 4 Notes | EduRev be a rational number.

RD Sharma Solutions: Number System- 4 Notes | EduRev 

RD Sharma Solutions: Number System- 4 Notes | EduRev

Since, x is rational number,

⇒ x – 6 is a rational number

RD Sharma Solutions: Number System- 4 Notes | EduRevis a rational number

RD Sharma Solutions: Number System- 4 Notes | EduRevis a rational number

But we know that RD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number, which is a contradiction 

So RD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number

(vii) Let RD Sharma Solutions: Number System- 4 Notes | EduRev

Using the formula RD Sharma Solutions: Number System- 4 Notes | EduRev 

RD Sharma Solutions: Number System- 4 Notes | EduRev

RD Sharma Solutions: Number System- 4 Notes | EduRev

RD Sharma Solutions: Number System- 4 Notes | EduRevis a rational number

⇒is RD Sharma Solutions: Number System- 4 Notes | EduRev a rational number

But we know that RD Sharma Solutions: Number System- 4 Notes | EduRevis an irrational number 

So, we arrive at a contradiction

So RD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number.

(ix) Let x = √5−2 be the rational number

Squaring on both sides, we get

RD Sharma Solutions: Number System- 4 Notes | EduRev

Now, x is rational

x2 is rational.

So, x2−29 is rational

RD Sharma Solutions: Number System- 4 Notes | EduRev

But, √5 is irrational. So we arrive at contradiction

Hence x = √5−2 is an irrational number

(x) Let

RD Sharma Solutions: Number System- 4 Notes | EduRev

It is non-terminating or non-repeating

Hence RD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number

(xi) Let RD Sharma Solutions: Number System- 4 Notes | EduRev 

RD Sharma Solutions: Number System- 4 Notes | EduRev

Hence RD Sharma Solutions: Number System- 4 Notes | EduRev is a rational number

(xii) Given x= 0.3796.

It is terminating

Hence it is a rational number

(xiii) Given number RD Sharma Solutions: Number System- 4 Notes | EduRev 

RD Sharma Solutions: Number System- 4 Notes | EduRev 

It is repeating 

Hence it is a rational number

(xiv) Given number is RD Sharma Solutions: Number System- 4 Notes | EduRev 

It is non-terminating or non-repeating

Hence it is an irrational number


Q.4. Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:

(i) (√4)

(ii) 3√18

(iii) √1.44

(iv) √927

(v) −√64

(vi) √100

Proof: (i) Given number is x =RD Sharma Solutions: Number System- 4 Notes | EduRev 

x = 2, which is a rational number

(ii) Given number is RD Sharma Solutions: Number System- 4 Notes | EduRev 

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it is an irrational number

(iii) Given number is RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to check whether it is rational or irrational

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it is a rational

(iv) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev

Now we have to check whether it is rational or irrational

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it is an irrational number

(v) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to check whether it is rational or irrational

Since, RD Sharma Solutions: Number System- 4 Notes | EduRev 

So it is a rational number

(vi) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to check whether it is rational or irrational

Since, RD Sharma Solutions: Number System- 4 Notes | EduRev 

So it is rational number


Q.5. In the following equations, find which variables x, y, z etc. represent rational or irrational numbers:

(i) x2 = 5

(ii) y2 = 9

(iii) z2 = 0.04

(iv) u2 = 17/4

(v) v2 = 3

(vi) w2 = 27

(vii) t2 = 0.4

Proof: (i) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to find the value of x

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it x is an irrational number

(ii) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to find the value of y

RD Sharma Solutions: Number System- 4 Notes | EduRev

So y is a rational number

(iii) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to find the value of z

RD Sharma Solutions: Number System- 4 Notes | EduRev

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it is rational number

(iv) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to find the value of u

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it is an irrational number

(v) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to find the value of v

RD Sharma Solutions: Number System- 4 Notes | EduRev

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it is an irrational number

(vi) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to find the value of w

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it is an irrational number

(vii) Given that RD Sharma Solutions: Number System- 4 Notes | EduRev 

Now we have to find the value of t

RD Sharma Solutions: Number System- 4 Notes | EduRev

So it is an irrational number


Q.6. Give two rational numbers lying between 0.232332333233332... and 0.212112111211112.

Proof: Let a = 0.232332333233332

b = 0.212112111211112

Here the decimal representation of a and b are non-terminating and non-repeating. So we observe that in first decimal place of a and b have the same digit but digit in the second place of their decimal representation are distinct. And the number a has 3 and b has 1. So a > b.

Hence two rational numbers are RD Sharma Solutions: Number System- 4 Notes | EduRev  lying between 0.232332333233332... and 0.212112111211112...


Q.7. Give two rational numbers lying between 0.515115111511115...0.5353353335...

Proof: Let a = 0.515115111511115... and b = 0.535335333533335...

Here the decimal representation of a and b are non-terminating and non-repeating. So we observe that in first decimal place a and b have the same digit  but digit in the second place of their decimal representation are distinct. And the number a has 1 and b has 3. So a < b.

Hence two rational numbers are RD Sharma Solutions: Number System- 4 Notes | EduRev lying between 0.515115111511115.. and 0.535335333533335...


Q.8. Find one irrational number between 0.2101 and 0.222... = 0.2¯.

Proof: Let 

RD Sharma Solutions: Number System- 4 Notes | EduRev

Here a and b are rational numbers .Since a has terminating and b has repeating decimal. We observe that in second decimal place a has 1 and b has 2. So a < b.

Hence one irrational number is RD Sharma Solutions: Number System- 4 Notes | EduRev lying between 0.2101 and 0.2222...


Q.9. Find a rational number and also an irrational number lying between the numbers 0.3030030003 ... and 0.3010010001 ...

Proof: Let 

RD Sharma Solutions: Number System- 4 Notes | EduRev

Here decimal representation of a and b are non-terminating and non-repeating. So a and b are irrational numbers. We observe that in first two decimal place of a and b have the same digit but digit in the third place of their decimal representation is distinct.

Therefore, a > b.

Hence one rational number is RD Sharma Solutions: Number System- 4 Notes | EduRev lying between  0.3030030003... and 0.3010010001...

And irrational number is RD Sharma Solutions: Number System- 4 Notes | EduRev lying between RD Sharma Solutions: Number System- 4 Notes | EduRev and RD Sharma Solutions: Number System- 4 Notes | EduRev 


Q.10. Find three different irrational numbers between the rational numbers RD Sharma Solutions: Number System- 4 Notes | EduRev and RD Sharma Solutions: Number System- 4 Notes | EduRev .

Proof: Let RD Sharma Solutions: Number System- 4 Notes | EduRev and RD Sharma Solutions: Number System- 4 Notes | EduRev 

Here we observe that in the first decimal x has digit 7 and y has 8. So x < y. In the second decimal place x has digit 1. So, if we considering irrational numbers

a = 0.72072007200072..

b = 0.73073007300073..

c = 0.74074007400074....

We find that

x<a<b<c<y

HenceRD Sharma Solutions: Number System- 4 Notes | EduRev are required irrational numbers.


Q.11. Give an example of each, of two irrational numbers whose:

(i) difference is a rational number.

(ii) difference is an irrational number.

(iii) sum is a rational number.

(iv) sum is an irrational number.

(v) product is an rational number.

(vi) product is an irrational number.

(vii) quotient is a rational number.

(viii) quotient is an irrational number.

Proof: (i) Let RD Sharma Solutions: Number System- 4 Notes | EduRev

And, so RD Sharma Solutions: Number System- 4 Notes | EduRev

Therefore,RD Sharma Solutions: Number System- 4 Notes | EduRevand RD Sharma Solutions: Number System- 4 Notes | EduRevare two irrational numbers and their difference is a rational number

(ii) Let RD Sharma Solutions: Number System- 4 Notes | EduRevare two irrational numbers and their difference is an irrational number

BecauseRD Sharma Solutions: Number System- 4 Notes | EduRev is an irrational number

(iii) LetRD Sharma Solutions: Number System- 4 Notes | EduRevare two irrational numbers and their sum is a rational number

That isRD Sharma Solutions: Number System- 4 Notes | EduRev

(iv) LetRD Sharma Solutions: Number System- 4 Notes | EduRevare two irrational numbers and their sum is an irrational number 

That isRD Sharma Solutions: Number System- 4 Notes | EduRev 

(v) Let RD Sharma Solutions: Number System- 4 Notes | EduRevare two irrational numbers and their product is a rational number

That isRD Sharma Solutions: Number System- 4 Notes | EduRev

(vi) LetRD Sharma Solutions: Number System- 4 Notes | EduRevare two irrational numbers and their product is an irrational number

That isRD Sharma Solutions: Number System- 4 Notes | EduRev

(vii) LetRD Sharma Solutions: Number System- 4 Notes | EduRev are two irrational numbers and their quotient is a rational number

That isRD Sharma Solutions: Number System- 4 Notes | EduRev

(viii) LetRD Sharma Solutions: Number System- 4 Notes | EduRevare two irrational numbers and their quotient is an irrational number

That is RD Sharma Solutions: Number System- 4 Notes | EduRev


Q.12. Find two irrational numbers between 0.5 and 0.55.

Proof: Let 

a = 0.5

b = 0.55

Here a and b are rational number. So we observe that in first decimal place a and b have same digit .So a < b.

Hence two irrational numbers areRD Sharma Solutions: Number System- 4 Notes | EduRevandRD Sharma Solutions: Number System- 4 Notes | EduRev

lying between 0.5 and 0.55.


Q.13. Find two irrational numbers lying between 0.1 and 0.12.

Proof: Let 

a = 0.1

b = 0.12

Here a and b are rational number. So we observe that in first decimal place a and b have same digit. So a < b.

Hence two irrational numbers areRD Sharma Solutions: Number System- 4 Notes | EduRevandRD Sharma Solutions: Number System- 4 Notes | EduRevlying between 0.1 and 0.12.


Q.14. Prove that 3 + 5 is an irrational number.

Proof: Given that √3 + √5 is an irrational number

Now we have to prove √3 + √5  is an irrational number 

Let x = √3 + √5 is a rational

Squaring on both sides

RD Sharma Solutions: Number System- 4 Notes | EduRev

Now x is rational

RD Sharma Solutions: Number System- 4 Notes | EduRevis rational

RD Sharma Solutions: Number System- 4 Notes | EduRevis rational

RD Sharma Solutions: Number System- 4 Notes | EduRevis rational

But,RD Sharma Solutions: Number System- 4 Notes | EduRevis an irrational

Thus we arrive at contradiction thatRD Sharma Solutions: Number System- 4 Notes | EduRevis a rational which is wrong.

HenceRD Sharma Solutions: Number System- 4 Notes | EduRevis an irrational

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