Tag: irrational numbers

Questions Related to irrational numbers

State true or false:
The three rational numbers between $\displaystyle \sqrt{3}$ and $\displaystyle \sqrt{5}$ are 1.6, 1.8, 2.2

irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

We know that, $ \sqrt {3} = 1.732 $ and $ \sqrt {5} = 2.236 $





Hence three rational numbers between $ 1.732 $ and $ 2.236 $  can be $

1.8( = \frac {18}{10}) ; 2 $ and $ 2.2 (= \frac {22}{10}) $


State the following statement is true or false:
$\dfrac{5}{12}$  lies between $\cfrac{1}{3}$ and $\cfrac{1}{2}$.

irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

The average of the two numbers will be in between the two numbers.
$ \cfrac { \left( \dfrac { 1 }{ 3 }  \right) +\left( \dfrac { 1 }{ 2 }  \right)  }{ 2 } $

$= \cfrac { \left( \dfrac { 5 }{ 6 }  \right)  }{ 2 } $

$= \dfrac { 5 }{ 12 } $ is a rational number which lies between $\dfrac{1}{3}\ and\ \dfrac{1}{2}$. 




State whether the given statement is true/false.
An irrational number between two numbers $\dfrac{1}{7}$ and $\dfrac{2}{7}$ is $0.1501500 15000...$ .

irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
Let us first find the decimal forms of the given numbers as follows: 
 
$\dfrac { 1 }{ 7 } =0.\overline { 142857 } ,\dfrac { 2 }{ 7 } =0.\overline { 285714 }$

We find a number which is non-terminating non-recurring lying between them.
So, we can find infinite many such numbers. For example, $0.150150015000...$ and $0.20200200020000....$

Hence, an irrational number between two numbers $\dfrac {1}{7}$ and $\dfrac {2}{7}$ is $0.150150015000...$

$A,B,C$ and $D$ are all different digits between $0$ and $9$. If $AB+DC=7B\ (AB,DC$ and $7B$ are two digit numbers), then the value of $C$ is

maths number systems existence of irrational numbers irrational numbers properties of irrational numbers

  1. $0$

  2. $1$

  3. $2$

  4. $3$

  5. $5$

Show answer
Correct Option: A

If $\sqrt{a}$ is an irrational number, what is a? 

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Rational

  2. Irrational

  3. $0$

  4. Real

Show answer
Correct Option: A
Explanation:

Consider the given irrational number$\sqrt{a}$ ,

Definition  of rational number- which number can be write in the form of $\dfrac{p}{q}$ but $q\ne 0$ is called rational number.

Hence, $a=\dfrac{a}{1}$

That why  $a$ is rational number

 

Hence, this is the answer.

Which of the following is irrational

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $\sqrt {\dfrac{4}{9}} $

  2. $\dfrac{4}{5}$

  3. $\sqrt 7 $

  4. $\sqrt {81} $

Show answer
Correct Option: C
Explanation:
A $=\sqrt{\dfrac{4}{9}}=\dfrac{2}{3}$         Rational

B $=\dfrac{4}{5}$                       Rational

C $=\sqrt7$                     Irrational

D $=\sqrt{81}=9$          Rational

The number $23+\sqrt{7}$ is

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Natural number

  2. Irrational number

  3. Rational number

  4. None of these

Show answer
Correct Option: B
Explanation:

As we've $\sqrt{7}$ is an irrational number and $23$ is a rational number then the sum of an irrational number and a rational number is again an irrational number.

Which of the following rational number represents a terminating decimal expansion?

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $

    \dfrac { 77 } { 210 }

    $

  2. $

    \dfrac { 13 } { 125 }

    $

  3. $

    \dfrac { 2 } { 15 }

    $

  4. $

    \dfrac { 17 } { 18 }

    $

Show answer
Correct Option: B
Explanation:
Any rational number its denominator is in the form of $2^m\times 5^n$, where $m,n$ are positive integer s are terminating decimals.

Solution is $B$ as $A$ is non terminating decimals.
$A =\dfrac{77}{210}= 0.366......$

$B =\dfrac{13}{125}= 0.104$

$C =\dfrac{2}{15}= 0.133.....$

$D =\dfrac{17}{18}=  0. 9444....$

Say true or false:

$87, 54, 0, -13, \sqrt{16}$ are integers 

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

The real value of $\sqrt { 16 } =4$

All other numbers are integers.
So, the given statement is true.

Read out each of the following numbers carefully and specify the natural numbers in it.
$87, 54, 0, -13, -4.7, \sqrt{7}, 2{1}{7}, \sqrt{15}, -{8}{7}, 3\sqrt{7}, 4.807, 0.002, \sqrt{16}$ and $2+\sqrt{3}.$

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $0,87,54,\sqrt{16}$

  2. $87, 54,$  $\sqrt{16}$, $217$

  3. $0, -13, -4,7, 217, 54, 87$

  4. $\sqrt{7}$, $\sqrt{15}$, $3 \sqrt{7}$, $\sqrt{16}$, $2 + \sqrt{3}$,

Show answer
Correct Option: B
Explanation:

Natural numbers from the given list are 87, 54,  $\sqrt { 16 } =4$ and 217