Mathematics · Quantitative Aptitude

Real Number System

261 Questions

The real number system encompasses rational and irrational numbers, forming the basis of arithmetic and number theory. It is a key area in the quantitative aptitude and mathematics sections of school level and competitive exams. Practice these questions to master the properties and identification of different types of numbers.

Identifying irrational numbersOrdering rational numbersProperties of square rootsPerfect number propertiesRational and irrational products

Real Number System Questions

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

Classify the following numbers as rational or irrational:  $\displaystyle \frac{\sqrt{12}}{\sqrt{75}}$

  1. Rational

  2. Irrational

  3. Can't be determined

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac { \sqrt { 12 }  }{ \sqrt { 75 }  } =\dfrac { 2\sqrt { 3 }  }{ 5\sqrt { 3 }  } =\dfrac { 2 }{ 3 } $ which is a rational number 

Hence, the correct answer will be option A

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

Which of the following are irrational numbers?
(i) $\sqrt{2+\sqrt{3}}$
(ii) $\sqrt{4+\sqrt{25}}$
(iii) $\sqrt[3]{5+\sqrt{7}}$
(iv) $\sqrt{8-\sqrt[3]{8}}$.

  1. (ii), (iii) and (iv)

  2. (i), (ii) and (iv)

  3. (i), (ii) and (iii)

  4. (i), (iii) and (iv)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Option (i)

$\sqrt3$ is irrational, so (i) is irrational.

Option (ii)
$\sqrt{25} = 5$, so we get $\sqrt{4+5} = \sqrt9 = 3$ which is rational.

Option (iii)
$\sqrt7$ is irrational, so (iii) is irrational.

Option (iv)
$\sqrt[3]{8} = 2$, so we get $\sqrt{8-2} = \sqrt6$ which is irrational.

$\therefore$ (i),(iii) and (iv) are irrational.

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

Which one of the following is an irrational number?

  1. $\pi$
  2. $\sqrt{9}$
  3. $\displaystyle\frac{1}{4}$
  4. $\displaystyle\frac{1}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A number having non-terminating and non-recurring decimal expansion is  a Irrational Number


A number having non-terminating and recurring decimal expansion is  a Rational Number

now looking at the options

$\pi$  is an irrational number 

$\pi = 3.1415926535897932384626433832............$


the number has non-terminating decimal expansion and non-recurring.

$\sqrt9 = 3$  is a rational number

$\dfrac14 = 0.25$ is a rational number

$\dfrac15 = 0.2$ is a rational number

So option $A $ is correct

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

Let $x$ be an irrational number then what can be said about ${x}^{2}$

  1. It is rational

  2. It can be irrational.

  3. It can be rational.
  4. Both $B$ and $C$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$x$ is any irrational number 
Let $x=\sqrt [ 4 ]{ 3 } $
$\Rightarrow { x }^{ 2 }=\sqrt { 3 } $
which is irrational so option $B$ is correct.
Now let $x=\sqrt 3$
$\Rightarrow {x}^{2}=3$
which is rational so option $C$ is correct.
So correct answer is $D$

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

The product of a non-zero rational number with an irrational number is always :

  1. Irrational number

  2. Rational number

  3. Whole number

  4. Natural number
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By definition, an irrational number in decimal form goes on forever without repeating (a non-repeating, non-terminating decimal). By definition, a rational number in decimal form either terminates or repeats. 

By multiplying a non repeating non terminating number to repeating or terminating/repeating number, the result will always be a non terminating non repeating number. 
So, option A is correct. 

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

Which is not an Irrational number?

  1. $5-\sqrt{3}$
  2. $\sqrt{2}+\sqrt{5}$
  3. $4+\sqrt{2}$
  4. $6+\sqrt{9}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know that sum of two irrational number or one rational and one irrational number will be irrational number. Option A, B , C stisfies this criteria but option D have two rational number i.e. $6 + \sqrt { 9 }$ = $6+ 3=9$

So correct answer is option D

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

A pair of irrational numbers whose product is a rational number is:

  1. $\sqrt{16}, \sqrt{4}$
  2. $\sqrt{5}, \sqrt{2}$
  3. $\sqrt{3}, \sqrt{27}$
  4. $\sqrt{36}, \sqrt{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In the given options,

for option $(A)$ $\sqrt { 16 } \& \sqrt { 4 }$ are not irrational numbers i.e. their real values are 4 & 2 respectively. It cannot be correct answer. 

Now multiplying other options, we get

$(B):\sqrt { 5 } \times \sqrt { 2 } =\sqrt { 10 } $

$(C)\sqrt { 27 } \times \sqrt { 3 } =\sqrt { 81 } =9$

$(D) \sqrt { 36 } \times \sqrt { 2 } =\sqrt { 72 } $

So, correct answer is option C.

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

A number is an irrational if and only if its decimal representation is :

  1. non terminating

  2. non terminating and repeating

  3. non terminating and non repeating

  4. terminating
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to definition of irrational number, If written in decimal notation, an irrational number would have an infinite number of digits to the right of the decimal point, without repetition.

So, correct answer is option C.

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

Which of the following is not an irrational number?

  1. $5-\sqrt{3}$
  2. $\sqrt{5}+\sqrt{3}$
  3. $4+\sqrt{2}$
  4. $5+\sqrt{9}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know that if add  or subtract any number from irrational number then the result will be irrational number.
$\sqrt { 5 } $ , $\sqrt { 3 }$ , $\sqrt { 2 } $  are irrational number but $\sqrt { 9 } $ =3 is a rational number so option D is correct answer 

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

$\pi$ is _______

  1. a rational number

  2. an integer

  3. an irrational number

  4. a whole number
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Sometimes we use $π = 22/7$ which is a popular approximation

$π = 3.14159265358...$

$22/7 = 3.142857142857...$

But $π$ and $22/7$ are close but not accurate.

Rational Numbers - $P/Q$ when $Q$ is not equal to $0$.

Let $x = 33.33333…. $——-(1)

$10x = 33.333333….. $——-(2)

Equation $(2) - (1)$

$9x = 30$

$x = 30/9$ which is in form of $P/Q$ and $x = 33.3333…$

The digit $‘3′$ is repeating itself and that’s why it can be written as $100/3.$

When it’s π, the value is $3.14159265358...$ The order of digits will not repeat itself in it but in $22/7 = 3.142857142857….$ you can see that $142857…$ is repeating itself that’s why $22/7$ is rational but $π$ is irrational.

So, option C is correct.
Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

Which of the following number is irrational ?

  1. $\sqrt{16}-4$
  2. $(3-\sqrt{3}) (3+\sqrt{3})$
  3. $\sqrt{5}+3$
  4. $-\sqrt{25}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In the given options $\sqrt { 16 }$ and $\sqrt { 25 } $ are irrational numbers. Their real values are 4 and 5 respectively. So, option A and C are incorrect.

Option B can solved and its real value becomes 6. So it is also a rational number.
In option C, $\sqrt { 5 }$ is a irrational number. So, option C is a irrational number. 
So, correct answer is option C. 

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

A number is an irrational if and only if its decimal representation is :

  1. non $-$ terminating
  2. non $-$ terminating and repeating
  3. non $-$ terminating and non $-$ repeating

  4. terminating
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Irrational numbers have decimal expansions that neither terminate nor repeating

So the correct answer is option C.

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

Which of the following is an irrational number ?

  1. $\sqrt{23}$
  2. $\sqrt{225}$
  3. $0.3796$
  4. $7.478$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In the given options, 

$\sqrt { 225 }$ = 15. So, it is not a irrational number,
Option C and D are terminating decimals. So, they are also rational numbers.
$\sqrt{23}$ is a irrational number. 
So, option A is correct answer.