Tag: rational and irrational numbers

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Questions Related to rational and irrational numbers

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers
Say true or false:
$87, 54, 0, -13, -4.7, \sqrt{5}, 2{1}{7}, \sqrt{15}, -{8}{7}, 3\sqrt{2}, 4.807, 0.002, \sqrt{16}$ and $2+\sqrt{3}.$ are rational numbers
 

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sqrt { 5 } ,\quad \sqrt { 15 } ,\quad 3\sqrt { 2 } ,\quad 2\quad +\sqrt { 3 } $ are irrational numbers as they cannot be expressed as a ratio.

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

Say True or False
$3+2\sqrt 5$ is an irrational number

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let us assume, to the contrary, that $3+2\sqrt{5}$ is rational.


That is, we can find coprime integers $a$ and $b$ $(b0)$ such that $3+2\sqrt{5}=\dfrac{a}{b}$.

Therefore, $\dfrac{a}{b} - 3=2\sqrt{5}$

$\dfrac{a-3b}{b}=2\sqrt{5}$

$\dfrac{a-3b}{2b}=\sqrt{5}$

$\dfrac{a}{2b}-\frac{3}{2}=\sqrt{5}$

Since $a$ and $b$ are integers, we get $\dfrac{a}{2b}-\dfrac{3}{2}$ is rational, and so $\dfrac{a-3b}{2b}=\sqrt{5}$ is rational.

But this contradicts the fact that $\sqrt{5}$ is irrational.

This contradiction has arisen because of our incorrect assumption that $3+2\sqrt{5}$ is rational.
So, we conclude that  $3+2\sqrt{5}$ is irrational.

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

Say true or false:$0.120 1200 12000 120000 $....is a rational number

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given, $0.120 1200 12000 120000 ....$
Since, the decimal expansion is neither terminating nor non-terminating repeating, therefore, the given real number is not rational.
they are not rational, so we can't write of the form $\displaystyle \frac {p}{q}$.

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

The number $\displaystyle\frac{3-\sqrt{3}}{3+\sqrt{3}}$ is 

  1. Rational

  2. Irrational

  3. Both

  4. Can't say
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$Here,\quad we\quad will\quad carry\quad out\quad rationalization.\quad \ \frac { 3-\sqrt { 3 }  }{ 3+\sqrt { 3 }  } =\frac { 3-\sqrt { 3 }  }{ 3+\sqrt { 3 }  } x\frac { 3-\sqrt { 3 }  }{ 3-\sqrt { 3 }  } =\frac { { (3-\sqrt { 3 } ) }^{ 2 } }{ (3+\sqrt { 3) } (3-\sqrt { 3 } ) } =\frac { 9+3-6\sqrt { 3 }  }{ 9-3 } =\frac { 12-6\sqrt { 3 }  }{ 6 } =\frac { 2-\sqrt { 3 }  }{ 1 } \ Since\quad \sqrt { 3 } is\quad irrational\quad number\quad and\quad subtraction\quad of\quad rational\quad and\quad irrational\quad is\quad irrational.\ The\quad given\quad expression\quad is\quad irrational.\ \quad $

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

Give an example of two irrational numbers, whose sum is a rational number

  1. $4 +\sqrt{5},-\sqrt{5}$
  2. $4 +\sqrt{5},\sqrt{5}$
  3. $4 -\sqrt{5},-\sqrt{5}$
  4. $ 2+\sqrt{5},2+\sqrt{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $\sqrt{5}  and  -\sqrt{5}$
Sum of Number  $\left(\sqrt{5}\right) + \left(-\sqrt{5}\right)$
$\sqrt{5}-\sqrt{5} = 0$
Which is a rational number

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

Give an example of two irrational numbers, whose difference is an irrational number.

  1. $4\sqrt{3},2\sqrt{3}$
  2. $\sqrt{3},\sqrt{3}$
  3. $2\sqrt{3},2\sqrt{3}$
  4. $4\sqrt{3},4\sqrt{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $4\sqrt{3}  and  2\sqrt{3}$
Difference of Number  $4\sqrt{3} - 2\sqrt{3} = 2\sqrt{3}$
Which is a irrational number

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

Give an example of two irrational numbers, whose quotient is an irrational number.

  1. $\sqrt{15},\sqrt{5}$
  2. $\sqrt{45},\sqrt{5}$
  3. $\sqrt{20},\sqrt{5}$
  4. $\sqrt{80},\sqrt{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $\sqrt{15}  and  \sqrt{5}$
Quotient of Numbers  $\frac{\sqrt{15}}{\sqrt{5}} = \sqrt{\frac{15}{5}} = \sqrt{3} $
Which is a irrational number

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

Give an example of two irrational numbers, whose sum is an irrational number.

  1. $2\sqrt{5},3\sqrt{5}$
  2. $2\sqrt{5},-2\sqrt{5}$
  3. $2+\sqrt{5},2-\sqrt{5}$
  4. $2+\sqrt{5},3-\sqrt{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $2\sqrt{5}  and  3\sqrt{5}$
Sum of Number  $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$
Which is a irrational number

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

Give an example of two irrational numbers, whose quotient is a rational number.

  1. $\sqrt{5},\sqrt{2}$
  2. $\sqrt{8},\sqrt{2}$
  3. $\sqrt{3},\sqrt{2}$
  4. $\sqrt{7},\sqrt{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let be the Number are $\sqrt{8}  and  \sqrt{2}$
Quotient of Numbers  $\frac{\sqrt{8}}{\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2 $
Which is a rational number