Mathematics · Quantitative Aptitude

Real Number System

277 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

$6+\sqrt{2}$ is a rational number.

  1. True

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

Let's assume that $6+\sqrt2$ is rational..... 

then 

$6+\sqrt2 = p/q $

$\sqrt2 =( p-6q)/(q) $ 

now take $p-6q$ to be P and $q$ to be Q........where P and Q are integers 

which means, $\sqrt2= P/Q$...... 

But this contradicts the fact that $\sqrt2$ is rational 

So our assumption is wrong and $6+\sqrt2$ is irrational.

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Let x and y be rational and irrational numbers, respectively, then x + y necessarily an irrational number.


State True or False.

  1. True

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

Yes.

Let x $= 21, y =\sqrt{2}$ be a rational number
Now $x+y=21 +\sqrt{2}=21+1.4142....=22.4142....$ , which is non-terminating and non-recurring. Hence x+y is irrational.

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Identify the irrational number(s) between $2\sqrt{3}$ and $3\sqrt{3}$

  1. $\sqrt{19}$
  2. $\sqrt{29}$
  3. $\cfrac { 4\sqrt { 3 } }{ \sqrt { 3 } } $
  4. $\sqrt{17}$
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation
$2\sqrt{3}=\sqrt{12}$
$3\sqrt{3}=\sqrt{27}$
$\therefore \sqrt{176}\sqrt{19}$ are irrational no between them $\sqrt{29}$ lie out of it.
As $\dfrac{4\sqrt{3}}{\sqrt{3}}=4$ (Rational)

Multiple choice comparison of irrational numbers surds and law of surds rational and irrational numbers exponents maths

Which one of the following is an irrational number?

  1. $\sqrt[3]{-27}$
  2. $\sqrt{2}(3\sqrt{2}+2\sqrt{8})$
  3. $\dfrac{3\sqrt{18}}{2\sqrt{6}}$
  4. $\sqrt{\dfrac{1}{2}}\cdot\sqrt{\dfrac{25}{2}}$
  5. $\dfrac{2\sqrt{5}}{\sqrt{45}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Option A: $\sqrt [ 3 ]{ -27 } ={ (-3) }^{ 3\times \frac { 1 }{ 3 }  }=-3$
Option B: $\sqrt { 2 } (3\sqrt { 2 } +2\sqrt { 8 } )=\sqrt { 2 } (3\sqrt { 2 } +4\sqrt { 2 } )=\sqrt { 2 } (7\sqrt { 2 } )=14$
Option C: $\dfrac { 3\sqrt { 18 }  }{ 2\sqrt { 6 }  } =\dfrac { 3\sqrt { 3 }  }{ 2 } $
Option D: $\sqrt { \dfrac { 1 }{ 2 }  } \sqrt { \dfrac { 25 }{ 2 }  } =\dfrac { 5 }{ 2 } $
Option E: $\dfrac { 2\sqrt { 5 }  }{ \sqrt { 45 }  } =\dfrac { 2\sqrt { 5 }  }{ 3\sqrt { 5 }  } =\dfrac { 2 }{ 3 } $
Therefore, all are rational except option $C$.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

The rational number is not lying between $\dfrac {5}{16}$ and $\dfrac {1}{2}$ is _________.

  1. $\dfrac {3}{8}$
  2. $\dfrac {7}{16}$
  3. $\dfrac {1}{4}$
  4. $\dfrac {13}{32}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

5/16 = 0.3125 and 1/2 = 0.5. Comparing options: 3/8 = 0.375 (in range), 7/16 = 0.4375 (in range), 1/4 = 0.25 (not in range), 13/32 = 0.40625 (in range).

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Find 9 rational numbers between  $2$ and $3$

  1. $2 < 2.1 < 2.2 < 3.3 < 2.4 < ... < 2.9 < 3$
  2. $2 < 4.1 < 2.2 < 2.3 < 2.4 < ... < 2.9 < 3$
  3. $2 < 2.1 < 2.2 < 2.3 < 2.4 < ... < 2.9 < 3$
  4. $2 < 2.1 < 2.2 < 2.3 < 2.4 < ... < 0.9 < 3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$2 < 2.1=(2+0.1) < 2.2=(2.1+0.1) < 2.3=(2.2+0.1) < 2.4=(2.3+0.1) < ... < 2.9=(2.8+0.1) < 3=(2.9+0.1)$


$2 < 2.1 < 2.2 < 2.3 < 2.4 < ... < 2.9 < 3$

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Write two rational numbers between $\displaystyle \sqrt{2}$ and $\displaystyle \sqrt{3}.$

  1. $1.5,\ 1.6$
  2. $1.4,\ 1.6$
  3. $1.5,\ 1.8$

  4. none of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We know that, $ \sqrt {2} = 1.414

$ and $ \sqrt {3} = 1.732 $

Hence two rational numbers between $ 1.414 $ and $ 1.732

$  can be $ 1.5 (= \frac {3}{2}) $ and $ 1.6 (= \frac {16}{10}= \frac {8}{5}) $

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Write three rational numbers between $\displaystyle \sqrt{3}$ and $\displaystyle \sqrt{5}$.

  1. 1.8,2 and 2.2

  2. 1.6,2 and 2.2

  3. 1.8,2.2 and 2.4

  4. none of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
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 \left(= \dfrac {18}{10}\right)  $ , $ 2 $ and $ 2.2 \left(= \dfrac {22}{10}\right) $.

Multiple choice maths fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Which one of the following is the rational number lying between $\displaystyle \frac{6}{7} \ and \ \frac{7}{8}?$

  1. $\displaystyle \frac{3}{4}$
  2. $\displaystyle \frac{99}{122}$
  3. $\displaystyle \frac{95}{112}$
  4. $\displaystyle \frac{97}{112}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Required rational number $\displaystyle =\frac{1}{2}\left ( \frac{6}{7}+\frac{7}{8} \right )=\frac{1}{2}\left ( \frac{48+49}{56} \right )=\frac{97}{112}$
Hence option (d) is correct