Tag: existence of irrational numbers

Questions Related to existence of irrational numbers

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

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

The value of  $\displaystyle \pi $ upto $50$ decimal places is $:\:314159265358979323846264338327950288419716939937510$
Which are the least occurring digits?
maths number systems existence of irrational numbers irrational numbers properties of irrational numbers

  1. $0$

  2. $1$

  3. $2$

  4. $3$

Show answer
Correct Option: A
Explanation:

We will be considering the digits only after the decimal point.


Digit after decimal point frequency
0 2
1 5
2 5
3 8
4 4
5 5
6 4
7 4
8 5
9 8
Total 50

The maximum occurring digits are 3 and 9. 
The least occurring digit is 0.

Which of the following is irrational?

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

  1. $\dfrac {22}{7}$

  2. $3.141592$

  3. $2.78181818$

  4. $0.123223222322223.......$

Show answer
Correct Option: D
Explanation:

An irrational number is any real number that cannot be expressed as a ratio of integers. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals.
Among all the options only $(D)$ $0.123223222322223$...... is non terminating and non repeating decimal.Therefore, it is a irrational number.