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 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

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

Identify a rational number between $\sqrt{2}$ and $\sqrt{3}$.

  1. $\dfrac{\sqrt{2}.\sqrt{3}}{2}$
  2. $1.5$
  3. $1.8$
  4. $\dfrac{\sqrt{2}+\sqrt{3}}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

As $\sqrt 6$ is irrational therefore option A is wrong.

$1.5$ is rational and it lies between $\sqrt 2$ & $\sqrt 3$ hence Option B is correct.
$1.8$ is rational but it doesn't lies between $\sqrt {2}$ & $\sqrt 3$ implies option C is wrong.
As sum of $\sqrt 2$ & $\sqrt 3$ is irrational therefore option D is also wrong.

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 are three rational numbers between $-2$ and $-1$?

  1. $\dfrac { -1 }{ 2 } ,\dfrac { -1 }{ 3 } ,\dfrac { -1 }{ 5 } $
  2. $\dfrac { -3 }{ 2 } ,\dfrac { -7 }{ 4 } ,\dfrac { -5 }{ 4 } $
  3. $\dfrac { -12 }{ 5 } ,\dfrac { -22 }{ 5 } ,\dfrac { 12 }{ 5 } $
  4. $\dfrac { 3 }{ 2 } ,\dfrac { 7 }{ 4 } ,\dfrac { 5 }{ 4 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In option B,


$\dfrac{-3}{2} = -1.5$

$\dfrac{-7}{4} = -1.75$

$\dfrac{-5}{4} = -1.25$

All these numbers lie in between $(-2,-1)$

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 between the pair of number $\dfrac{1}{2}$ and $\sqrt 1$ is:

  1. $\dfrac{9}{4}$
  2. $\dfrac{3}{4}$
  3. $\dfrac{5}{4}$
  4. $\dfrac{7}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The rational number between $\dfrac12$ and $\sqrt1$ :

Since, $\sqrt1=1$
So. the rational number between $\dfrac12$ and $1=\dfrac12\times \left(\dfrac12+1\right)$
$=\dfrac12 \times \dfrac32$
$=\cfrac34$
So, $B$ is the correct option.

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 which is not lying between $\displaystyle\frac{5}{16}$ and $\displaystyle\frac{1}{2}$ is _________.

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

We know $\dfrac{5}{16} =0.3125$

and $\dfrac{1}{2}= 0.5$
Option A: $\dfrac{3}{8} =0.375$
lying between the gven numbers

Option B: $\dfrac{7}{16}= 0.4375$
lying between the given numbers.

Option C: $\dfrac{1}{4}=0.25$
NOT lying between the given numbers.

Option D: $\dfrac{13}{32}=0.40625$
lying between the given numbers.
So, option C is correct.

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

A rational number lie between $\displaystyle\frac{1}{4}$ and $\displaystyle\frac{1}{3}$ is _________.

  1. $\displaystyle\frac{7}{24}$
  2. $0.29$
  3. $\displaystyle\frac{13}{48}$

  4. All of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know $\dfrac{1}{4} = 0.25$

and $\dfrac{1}{3}=0.3333333$

Option A: $\dfrac{7}{24}$
$=0.291666$
lies between the given numbers

Option B: $ 0.29$
lies between the given numbers

Option C: $\dfrac{13}{48}$
$=0.27083333$
lies between the given numbers.

All the options are correct.

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

Number of rational numbers between $15$ and $18$ is:

  1. infinite

  2. finite

  3. zero

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

Between any two rational numbers we can find infinitely many rational numbers. 

Thus, number of rational numbers between $15$ and $18$ is infinite.
Hence, the answer is infinite.

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

A rational number -2/3 ______ .

  1. Lies to the left side of 0 on the number line.

  2. Lies to the right side of 0 on the number line.

  3. It is not possible to represent on the number line.

  4. Cannot be determined on which side the number lies.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

rational number always lies on the line.

 this rational number is $\dfrac{-2}{3}$ which is negative  hence it is always lies to left side of $0$ on the number line.
hence option $A$ is correct.