Zero is a rational or an irrational number?

Is (x+y)/2 is an irrational number or a rational number?

How can we find the irrational numbers between two fractions

What is the multiplicative inverse of rational numbers

find two rationals between 0.5 and 0.55

Can u explain solving of rational numbers with appropriate property

## 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.

## Write the simplest form of a rational number

## Is     a rational number or not ?

so it is rational number.

## 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.

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

No, it may be rational or irrational.

## Insert three rational numbers between    and   .

So three rational numbers are

## Find two rational numbers between 4 and 5 .

The numbers are  and

## 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

## Calculate the irrational number 2 and 2.5

Since

hence, the irrationl number between2 and 2.5 is

## Simplify the number

## 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

## Find four rational numbers between      and

Since LCM of 5 and 6 is 30

Hence , four rational numbers between      and

are

## 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

## Insert three rational numbers between  and     and

LCM of 5 and 7 is  35

and

so

The required three rational numbers are

## 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

## simplify      by rationalizing the denominator

## Find three rational numbers between    and

Since LCM of 7 and 11 is 77

Hence three rational numbers between    and    are

## Rationalize the denominator

## Rationalize the denominator of

Alternative Method

## 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)

## Rationalize the denominator of

## 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

## Find any two irrational numbers between 0.1 and 0.12.

Required to irrational number are :

i)  0.10100100010000......

ii) 0.1020020002000.......

## Find the values of a and b when

Compairing the rational and irrational parts  of both sides we get

## Rationalize the denominator of

## if    then find the value of

Given

Now ,

## (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

## 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.

## 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 2m 5n . where m and n are non- negative integers.

eg-

## 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

## 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.

## 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.

## Write the denominator of the rational number   in the form 2m x 5n , where  m and n are non- negative integers. Hence write its decimal expansion without actual division.

Denominator  = 500

= 22 x 53

Decimal expansion,

Let

Substracting

## 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

## 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.

## 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  m2 and p  divides m . [ p divides pn2 ]

[ P is prime and p divides m2 p divides m]

Let              m = pq  for some integer q

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

pn2 = p2q2

n2 = pq2

p divides n2

and p divides n.

[ p is prime and p divides n 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.

## 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.

## 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 = 3b2

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 .

## Find the three rational numbers lying between 3 and 4.

LCM of 3 and 4 = 12

So rational nos. can be

## Find the 6 rational numbers between  and   ​

The rational numbers are  ,

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

3 rational numbers between  and  are

1

13

19

35

## 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

is irrational.

## 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 a2

2 divides a

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

This means 2 divides b2, 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.

## 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.