Can u explain solving of rational numbers with appropriate property

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Is zero (0) a rational number ? justify your answer ?

Yes, zero is a rational number

zero can be expressed as etc

which are in the form of, where p and q are integers and q 0.

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Write the simplest form of a rational number

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Is a rational number or not ?

so it is rational number.

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Identify an irrational number among the following numbers: 0.13 , , , 0.3013001300013.....

0.13 is a terminating number. So, it is not an irrational number.

= 0.131515......, 15 is repeating continuously so it is not an irrational number.

= 0.13151315........is repeating continuously so it is not an irrational number.

0.3013001300013....., non terminating and non recurring decimal. Hence, it is an irrational number. So, 0.3013001300013 is an irrational number.

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Is the product of two irrational number is always an irrational number ?

No, it may be rational or irrational.

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Insert three rational numbers between and .

So three rational numbers are

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Find two rational numbers between 4 and 5 .

The numbers are and

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Express the rational numbers in the form , where p and q are integers and

Let X = 0.999............

10x = 9.999.......

10x - x = (9.999....) – (0.999....)

9x = 9

X = 1

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Calculate the irrational number 2 and 2.5

Since

hence, the irrationl number between2 and 2.5 is

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Simplify the number

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Find the rational numbers between 0.121221222122221...and 0.141441444144441... in the p form, where p and q integers and

Two rational numbers between0.121221222122221...and 0.141441444144441...are 0.13 and 0.14

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Find four rational numbers between and

Since LCM of 5 and 6 is 30

Hence , four rational numbers between and

are

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Find six rational numbers between 3 and 4.

let a = 3 , and b = 4

Here, we find six rational numbers i,e n = 6

1st rational number =

2nd rational number =

3 rd rational number =

4th rational number =

5th rational number =

6th rational number =

So six rational numbers are

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Insert three rational numbers between and and

LCM of 5 and 7 is 35

and

so

The required three rational numbers are

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is not a rational number as and are not rational.

State whether it is true or false. Justify your answer.

false

justification :

Which is a rational number

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simplify by rationalizing the denominator

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Find three rational numbers between and

Since LCM of 7 and 11 is 77

Hence three rational numbers between and are

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Rationalize the denominator

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Rationalize the denominator of

Alternative Method

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Give two rational numbers whose :

(1) Difference is a rational number

(2) Sum is a rational number

(3) Product is a rational number

(4) Division is a rational number Justify also.

Any example and verification of example

product = 4/5 x 9/2 = 36/10 (Rational number)

Division = 9/2 4/5 = 45/8 ( Rational number)

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Rationalize the denominator of

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Given two rational numbers whose

(1) difference is a rational number

(2) sum is a rational number

(3) product is a rational number

(4) division is a rational number

Any example and verification of example

let m = 4/5, n= 9/2

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Find any two irrational numbers between 0.1 and 0.12.

Required to irrational number are :

i) 0.10100100010000......

ii) 0.1020020002000.......

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Find the values of a and b when

Compairing the rational and irrational parts of both sides we get

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Rationalize the denominator of

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if then find the value of

Given

Now ,

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(i)Find the six rational numbers bbetween 3 and 4

(ii) Which mathematical concept is used in this problem

(iii) Which value is depticted in this question

(i) We known that between two rational numbers x and y such that x< y there is a rational number .

ie,

Now a rational number between 3 and is :

A rational nummber berween

Further a rational number between 3 and

A rational number between

A rational number between

Hence , six rational numbers between 3 and 4 are

(ii) Number syatem

(iii) Rationality is always welcomed

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Find the smallest natural number by which 1,200 should be multiplied so that the square root of the product is a rational number.

The smallest natural number is 3.

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What is the condition for the decimal expansion of a rational number to terminate? Explain with the help of an example.

The decimal expansion of a rational number terminates, if the denominator of rational no , When p and q are co-primes and q can be expressed as 2^{m} 5^{n }. where m and n are non- negative integers.

eg-

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Find the smallest positive rational number by which should be multiplied so that its decimal

expansion terminates after 2 places of decimal

Since

Thus smallest rational number is

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What type of decimal expansion does a rational number has? How can you distinguish it from decimal expansion of irrational number?

A rational number is either terminating or non- terminating repeating.

An irrational number is non-terminating and non- repeating.

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Show that is an irrational number.

Let be a rational number, which can be put in the form , where , a and b are co-prime

= rational

But, we know that is an irrational number.

Thus, our assumption is wrong.

Hence, 5 is an irrational number.

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Write the denominator of the rational number in the form 2^{m} x 5^{n }, where m and n are non- negative

integers. Hence write its decimal expansion without actual division.

^{m}x 5

^{n }, where m and n are non- negative

Denominator = 500

= 2^{2} x 5^{3}

Decimal expansion,

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Express the number in the form of rational number .

Let

Substracting

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Prove that is an irrational number

Let be a rational number

Where p and q are co-prime integers and

is divisible by 2.

P is divisible by 2.

Let p = 2r for some positive integer r

is divisible by 2.

q is divisible by 2.

From (i) and (ii), p and q are divisible by 2, which contradicts the fact that p and q are co-primes.

Hence, our assumption is false.

So, is irrational

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Prove that is an irrational number.

Let be a rational number

(a and b are integers and co-primes )

On squaring both the sides,

is divisible by 3

is divisible by 3

We can write a = 3c for some integer

is divisible by 3

b is divisible by 3.

From equations (i) ad (ii), we get 3 as a factor of "a" and "b"

Which is contradicting the fact that a and b are co-primes. Hence our assumption that is a rational number, is false. So is an irrational number.

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if p is a prime number, then prove that is irrational.

Let p be a prime number and if possible, let be rational.

Let where m and n are integers having no

common factor other than 1 and

Then,

Squaring on both sides, we get

P divides m^{2} and p divides m . [ p divides pn^{2} ]

[ P is prime and p divides m^{2} p divides m]

Let m = pq for some integer q

on putting m = pq [in eq.(i) we get ]

pn^{2} = p^{2}q^{2}

n^{2} = pq^{2}

p divides n^{2}

and p divides n.

[ p is prime and p divides n^{2 } p divides n ]

Thus p is a common factor of m and n but this contraficts the fact that m and n have no common factor other than 1.

The contradiction arises by assuming that is a rational .

Hence, is irrational.

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Prove that is an irrational number.

Let 3+ is a rational number

is irrational and is a rational

But the rational number cannot be equal to an irrational number.

is an irrational number.

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Prove that is an irrational number. Hence show that 7+2 is also an irrational number

If possible let be a rational number.

(i) , where a and b are integers and co -primes

squaring both sides, we have

is divisible by 3

a is divisible by 3 .............................(1)

We can write a = 3c for some c (integer)

(3c)^{2} = 3b^{2}

is divisible by 3.

is divisible by 3 ...........................(2)

From eq (i) and (ii) we have,

3 is a factor a and b which is contradicting the fact that 'a' and 'b' are co-prime.

Thus our assumption that is rational numbers is wrong.

Hence, is an irrational number.

(ii) Let us assume to the contrary that is a rational number.

p- 7q and 2q both are integers hence is a rational number.

But this contradicts the fact that is is irrational number. Hence is is an irrational number .

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Find the three rational numbers lying between 3 and 4.

LCM of 3 and 4 = 12

So rational nos. can be

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Find the 6 rational numbers between and

The rational numbers are ,

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Find 3 rational numbers between 1/4 and 1/2

3 rational numbers between and are

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The rational number not equivalent to is

a.

b.

c.

d.

1

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The product of two rational numbers is if one them is find the other.

13

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Insert 6 rational numbers b/3: and

19

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Fill in the blanks:

i. Cubes of all ........... national no's are odd.

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

iii. The product of any rational number with ............. is the rational number itself

iv. (425)^{2} - (425)^{2} = ................

^{2}- (425)

^{2}= ................

35

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Prove that is an irrational number and hence show that is also an irrational number.

Let be a rational number

(a, b are co -prime integers and )

5 is a factor of

5 is a factor of a

let a = 5c,

5 is a factor of

5 is a factor of b

5 is a common factor of a, b

But this contradicts the fact that a, b are co-primes

is irrational

Let be a rational

2 - a is rational, so is

But is not rational contradiction

is irrational.

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Prove that is an irrational number. Hence show that is alos an irrational number.

Let be a rational number

(a, b are co-prime integers and )

Squaring,

2 divides a^{2}

2 divides a

So we can write a = 2c for some integer c,, substitute for

This means 2 divides b^{2}, so 2 divides b.

a and b have '2' as a common factor

But this contradicts that a, b have no commom factor other than 1.

Our assumption is wrong.

Hence, is irrational.

Let be rational

where a and b are integers,

is rational but is not rational

Our assumption is wrong

is rational.

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Show that there is no positive integer n, for which is rational.

Let us assume that there is a positive integer n for which is rational and equal to ,where P and Q are positive integers ()

................(i)

................(ii)

Apply (i) + (ii) we get

.............(iii)

Apply (i) and (ii) we get

.................(iv)

From (iii) and (iv), we can say and both are rational because P and Q both are rational. But it is possible only when (n+1) and (n-1) both are perfect squares. But they differ by 2 and two perfect square never differ by 2. So both (n+1) and (n-1) cannnot be perfect squares, hence there is no positive integer n for which is rational.