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 operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

State True or False.

The five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are $ \displaystyle \frac{19}{30},\frac{20}{30},\frac{21}{30},\frac{22}{30},\frac{23}{30}$.

  1. True

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

Since we want five  numbers we write $\frac{3}{5}$ and $\frac{4}{5}$
So multiply in numerator and denominator by 5+1 = 6 we get
$\Rightarrow \frac{3}{5}=\frac{3\times 6}{5\times 6}=\frac{18}{30}$
$\Rightarrow \frac{4}{5}=\frac{4\times 6}{5\times 6}=\frac{24}{30}$
We know that $18<19<20<21<22<23<24$
$\Rightarrow \frac{18}{30}<\frac{19}{30}<\frac{20}{30}<\frac{21}{30}<\frac{22}{30}<\frac{23}{30}<\frac{24}{30}$
Hence 5 rational number between $\frac{3}{5} and  \frac{4}{5}$are
$\frac{19}{30},\frac{20}{30},\frac{21}{30},\frac{22}{30},\frac{23}{30},$

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Write any $10$ rational numbers between $0\;and\;2$.

  1. $\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{88}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$
  2. $\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{21}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$
  3. $\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{35}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$
  4. $\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let us write $0$ as $\displaystyle\frac{0}{10}\;and\;2$ as $\displaystyle\frac{20}{10}$.

The rational numbers between these are


$\displaystyle\frac{1}{10},\,\displaystyle\frac{2}{10},\,\displaystyle\frac{3}{10},\,\displaystyle\frac{4}{10},\,\displaystyle\frac{5}{10},\,\displaystyle\frac{6}{10},\,\displaystyle\frac{7}{10},\,\displaystyle\frac{8}{10},\,\displaystyle\frac{9}{10},\,\displaystyle\frac{10}{10},\,\displaystyle\frac{11}{10},\,\displaystyle\frac{12}{10},\,\displaystyle\frac{13}{10},\,\displaystyle\frac{14}{10},\,\displaystyle\frac{15}{10},\,\displaystyle\frac{16}{10},\,\displaystyle\frac{17}{10},\,\displaystyle\frac{18}{10},\,\displaystyle\frac{19}{10}$

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Choose the rational number which does not lie between rational numbers $ \displaystyle \frac{3}{5} $ and $ \displaystyle \frac{2}{3} $ :

  1. $ \displaystyle \frac{46}{75} $
  2. $ \displaystyle \frac{47}{75} $
  3. $ \displaystyle \frac{49}{75} $
  4. $ \displaystyle \frac{50}{75} $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For $\dfrac{3}{5}$ multiply numerator and denominator by $15$ to make denominator $75$ that comes into $\dfrac{45}{75}.$
Similarly doing for second then we have $\dfrac{50}{75}.$
Now question is asking about rational lying between them.

So, we need to check the numerator only that lies in between $45$ and $50$ or not.
Clearly $D$ is correct.

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Find five rational numbers between $1$ and $2.$

  1. $\dfrac {1}{10}, \dfrac {2}{10}, \dfrac {3}{10}, \dfrac {4}{10}, \dfrac {5}{10}$
  2. $\dfrac {1}{5}, \dfrac {2}{5}, \dfrac {3}{5}, \dfrac {4}{5}, \dfrac {5}{5}$
  3. $\dfrac {1}{2}, \dfrac {1}{3}, \dfrac {1}{4}, \dfrac {1}{5}, \dfrac {1}{6}$
  4. $\dfrac {8}{7}, \dfrac {9}{7}, \dfrac {10}{7}, \dfrac {11}{7}, \dfrac {12}{7}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The given rational numbers are $1$ and $2$.

Let us multiply both the numbers by $\dfrac {7}{7}$.
$1\times \dfrac {7}{7} = \dfrac {7}{7}$ and $2\times \dfrac {7}{7} = \dfrac {14}{7}$.
Thus, five rational numbers between $\dfrac {7}{7} = 1$ and $\dfrac {14}{7} = 2$ are $\dfrac {8}{7}, \dfrac {9}{7}, \dfrac {10}{7}, \dfrac {11}{7}, \dfrac {12}{7}$.

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Choose the rational number which does not lie between rational numbers $-\cfrac {2}{5}$ and $-\cfrac {1}{5}$

  1. $-\dfrac {1}{4}$
  2. $-\dfrac {3}{10}$
  3. $\dfrac {3}{10}$
  4. $-\dfrac {7}{20}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since the given rational numbers $-\dfrac {2}{5}$ and $-\dfrac {1}{5}$ are negative rational numbers, therefore, none of the positive rational number can lie between them.


Hence, the rational number $\dfrac {3}{10}$ does not lie between the rational numbers $-\dfrac {2}{5}$ and $-\dfrac {1}{5}$ 

Multiple choice maths operations on rational numbers rational numbers on number line rational numbers and their decimal expansions rational numbers between two rational numbers

Identity the rational number that does not lie between  $ \cfrac{3}{5}$ and $ \cfrac{2}{3}$.

  1. $ \cfrac{46}{75}$
  2. $ \cfrac{47}{75}$
  3. $ \cfrac{49}{75}$
  4. $ \cfrac{50}{75}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Changing the fractions to denominator $=75$


$\dfrac{3}{5} = \dfrac{45}{75}$

$\dfrac{2}{3} = \dfrac{50}{75}$

$\therefore \dfrac{50}{75}$ doesn't lie in between $\dfrac{3}{5}$ and $\dfrac{2}{3}$

Multiple choice physics logarithms introduction to logarithm logarithmic notation exponential and logarithms

$\log _{ 4 }{ 18 } $ is

  1. A rational number

  2. An irrational number

  3. A prime number

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$log _418=\dfrac{log18}{log4}$
$=\dfrac{log2+log9}{2log2}$
$=\dfrac{log2+2log3}{2log2}$
$=\dfrac{log2}{2log2}+\dfrac{2log3}{2log2}$
$=\dfrac{1}{2}+log _23$
Since $log _23$ is irrational
Hence $=\dfrac{1}{2}+log _23$ is irrational.
Therefore
$log _418$ is irrational number
Multiple choice physics logarithms introduction to logarithm logarithmic notation exponential and logarithms

Which of the following is true for $\log _25$?

  1. An integer

  2. A rational number

  3. An irrational number

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

Let us assume that $\log _{ 2 }{ 5 } =\frac { p }{ q } $ , where $p,q$ are integers

We have $5=2^{\frac{p}{q}}$
$\Rightarrow 5^{q}=2^{p}$
This suggest that $5$ and $2$ are not mutually prime , But $2$ and $5$ are mutually prime
Therefore $\log _{ 2 }{ 5 } $ is an irrational number

Multiple choice maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

The rational number which can be expressed as a terminating decimal is

  1. $\displaystyle \frac{1}{6}$
  2. $\displaystyle \frac{1}{12}$
  3. $\displaystyle \frac{1}{15}$
  4. $\displaystyle \frac{1}{20}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits. 
$\displaystyle \frac {1}{20}= 0.05$ 

In the other options, the decimal does not end with a finite number of digits.

Multiple choice maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or non -terminating decimal expansion

$\displaystyle \frac{15}{1600}$

  1. Terminating decimal expansion

  2. Non-terminating decimal expansion

  3. Cannot be determined

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

Factorize the denominator we get $1600=2 \times 2 \times 2 \times2 \times 2 \times 2 \times 5 \times 5  = 2^{6} \times 5^{2}$
so denominator is in form of $2^n \times 5^m$
Hence $\frac{15}{1600}$  is  terminating.

Multiple choice maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or non -terminating decimal expansion

$\displaystyle \frac{7}{210}$

  1. Terminating decimal expansion

  2. Non -terminating decimal expansion

  3. Cannot be determined

  4. None
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Simplify it by dividing nominator and denominator both by 7 we get $\frac{1}{30}$
Factorize the denominator we get  $30=2 \times 3 \times 5$
Denominator has 3 also in denominator
So denominator is not in form of $2^{n} \times 5^{n}$
 Hence it is non terminating.

Multiple choice maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

A number having non-terminating and recurring decimal expansion is.

  1. An integer

  2. A rational number

  3. An irrational number

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

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


for example 

$\dfrac{2}{7}$  is a rational number 

$\dfrac{2}{7} =0.285714285714285714.......... $

or   $\dfrac{2}{7} =\overline{0.285714}.......... $

the number has non-terminating decimal expansion but recurring after every 6 digits after decimal

So option $B $ is correct

Multiple choice maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

Let $x=\dfrac { p }{ q } $ be a rational number, such that the prime factorization of $q$ is of the form $2^n 5^m$, where $n, m$ are non-negative integers. Then $x$ has a decimal expansion which terminates.

  1. True

  2. False

  3. Neither

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

The form of q is $2^n*5^m$
q can be $1,2,5,10,20,40....$
Any integer divided by these numbers will always give a terminating decimal number.