Tag: properties of irrational numbers

Questions Related to properties of irrational numbers

$A,B,C$ 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

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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

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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

There can be a pair of irrational numbers whose sum is irrational 

Such as: $\displaystyle \sqrt{3}+2$ and $\displaystyle 5+\sqrt{2}$

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:

To get the sum as irrational, the numbers need to have an irrational part as well which are different from each other.

Example, the pair of numbers $ \sqrt{3} + 2 $ and $ 5 + \sqrt {2} $ have the sum $ \sqrt{3} + 2 + 5 + \sqrt {2} = 7 + \sqrt {2} + \sqrt {3} $ which is an irrational number too.

State true or false:

$\sqrt3$ is an irrational number

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:

Decimal form of $\sqrt { 3 } $ is non terminating and non repeating, So, it is irrational number.

Simplify : 

$\displaystyle \sqrt{2}\times \sqrt[3]{3} \times \sqrt[4]{4}$.

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

  1. $\sqrt[3]{12}$

  2. $\sqrt[3]{24}$

  3. $\sqrt[3]{20}$

  4. $\sqrt[3]{25}$

Show answer
Correct Option: B
Explanation:

$ \sqrt{2} \times \sqrt[3] {3} \times \sqrt[4]{4}$
$=2^{ \frac { 1 }{ 2 }  } \times 3^{ \frac { 1 }{ 3 }  }\times 2^{ \frac { 2 }{ 4 }  }$
$=2^{ \frac { 1 }{ 2 }  } \times 2^{ \frac { 1 }{ 2 }  }\times 3^{ \frac { 1 }{ 3 }  }$
$=2  \times3^{ \frac { 1 }{ 3 }  }$
$=2^{ \frac { 3 }{ 3 }  }\times3^{ \frac { 1 }{ 3 }  }  $
$=\sqrt [ 3 ]{ 2^{ 3 } }\times\sqrt[3]{3}$
$=\sqrt[3]{8\times3}$
$=\sqrt[3]{24}$