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 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 terminating decimal expansion or a non-terminating repeating decimal expansion. Also, find the numbers of places of decimals after which the decimal expansion terminates.
$\dfrac { 13 }{ 3125 } $

  1. $3$
  2. $4$
  3. $5$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The given value is $\dfrac{13}{3125}$ the denominator is 3125 which can be written as:


$3125=2^0 \times 5^5$ it is in the form of $2^m \times 5^n$

$max(m,n)=5$

$\therefore$ the expansion is terminating decimal it terminates after 

$max(m,n)=5$ places from the decimal [since  $ m=0,n=5$]

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

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :
(i) $\displaystyle \dfrac{7}{16}$ (ii) $\displaystyle \dfrac{23}{125}$
(iii) $\displaystyle \dfrac{9}{14}$ (iv) $\displaystyle \dfrac{32}{45}$
(v) $\displaystyle \dfrac{43}{50}$ (vi) $\displaystyle \dfrac{17}{40}$
(vii) $\displaystyle \dfrac{61}{75}$ (viii) $\displaystyle \dfrac{123}{250}$

  1. (i), (iii), (v), (vi) and (vii)

  2. (i), (ii), (v), (vi) and (viii)

  3. (i), (iii), (v), (vi) and (viii)

  4. (i), (ii), (v), (vi) and (vii)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

 The rational no having denominator $3, 7, 9, 11, 13, 17, 23, 27$.............. and multiple of these number will have non terminating decimal .
(1) $\dfrac{7}{16}$ the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(2) $\dfrac{23}{125}$ -- the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(3) $\dfrac{9}{14}$ --he denominator of this rational number is having these above number multiple of $7$, so this will have non terminating decimal.
(4)$\dfrac{32}{45}$--he denominator of this rational number is having these above number multiple of 9, so this will have non terminating decimal.
(5) $\dfrac{43}{50}$-- the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(6)$\dfrac{17}{40}$ -- the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(7)$\dfrac{61}{75}$-- he denominator of this rational number is having these above number multiple of 3, so this will have non terminating decimal.
(8)$\dfrac{123}{250}$--the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(i), (ii), (v), (vi) and (viii) will have  terminating decimal.

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

A rational number in its decimal expansion is $327.7081.$ What can you say about the prime factors of $q$, when this number is expressed in the form $\cfrac {p}{q}$?

  1. $q$ has prime factors $2$ or $5$ or both.
  2. $q$ has prime factors except $2$ and $5.$
  3. $q$ has no prime factors

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

We know that The rational no having denominator 3, 7, 9, 11, 13, 17, 23, 27.............. and multiple of these number will have non terminating decimal .
As  decimal expansion is 327.7081 which is terminating.
prime factors of q, when this number is expressed in the form p/q will not be above number, it will be 2 or 5 or both.

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

............... numbers have terminating and non- terminating repeating decimals.

  1. Integers

  2. Whole

  3. Rational

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

$\dfrac {1}{4} = 0.25$ is a terminating decimal.


$\dfrac {8}{3} = 2.666666666......$ is a non-terminating repeating decimal.

Both are rational numbers but it was non repeating then they are irrational numbers.
Therefore, $C$ is the correct answer.

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

A rational number can be expressed as a terminating decimal if the denominator has factors _________.

  1. $2$ or $5$
  2. $2$, $3$ or $5$
  3. $3$ or $5$
  4. Only $2$ and $3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Any rational number its denominator is in the form of ${ 2 }^{ m }\times { 5 }^{ n }$,where m,n are positive integer s are terminating decimals.

So correct answer will be option A

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

If $x =\dfrac{p}{q}$  be a rational number such that the prime factorization of $q$ is not of the form $2^n 5^m$, where $n, m$ are non-negative integers. Then $x$ has a decimal expansion which is terminating.

  1. True

  2. False

  3. Neither

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

If q is not of the form $2^n*5^m$ then definitely q can take any of the values 3,6,9,12,15,...etc.
As we know 4/3,20/6 all are non-terminating, thus required decimal expansion is non terminating.

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

The numbers 7.478478.... and 1.101001000100001.....are

  1. Rational and irrational respectively

  2. Both rationals

  3. Both irrationals

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

In 7.478478.......  we can see that after decimal the digits 478 are repeating itself again and again.

Hence it is a non-terminating and repeating decimal. Therefore it is a rational number.

In 1.101001000100001........  it is a non-terminating and non-repeating decimal.
Therefore it is an irrational number.
Hence option A is correct. 

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

Out of the rational numbers $\displaystyle\frac{-5} {11},\,\frac{-5}{12},\,\frac{-5}{17}$ which is greatest ?

  1. $\displaystyle\frac{-2}{11}$
  2. $\displaystyle\frac{5}{-12}$
  3. $\displaystyle\frac{-5}{17}$

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

$\displaystyle\frac{-5}{11},\,\frac{-5}{12},\,\frac{-5}{17}$

$\because$ All have same numerator. So the rational number having the least denominator is the greatest. But here all have negative sign. So the number having greatest denominator is greater.

Hence, $\displaystyle\frac{-5}{17}$ is greater.

Alter : Take any two given numbers. $\displaystyle\frac{-5}{11},\, \frac{-5}{12}$

$-5\,\times\,12, -5\,\times\, 11$

- 60, - 55 

$\because\, - 55\, >\, - 60$

So, $\displaystyle\frac{-5}{12}$ is greater.

Now compare this with $\displaystyle\frac{-5}{17}$ 

$\displaystyle\frac{-5}{12},\, \frac{-5}{17}$

$-5\,\times\,17, \, -5\,\times\, 12$

- 85, - 60

$\because\,- 60\, >\,- 85$

So, $\displaystyle\frac{-5}{17}$ is greater. 

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

Out of the rational numbers $\displaystyle {\frac{-5}{11},\, \frac{-5}{12},\, \frac{-5}{17}}$, which is greater ?

  1. $\displaystyle \frac{-5}{11}$
  2. $\displaystyle \frac{5}{-12}$
  3. $\displaystyle \frac{-5}{17}$

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

$\displaystyle {\frac{-5}{11},\, \frac{-5}{12},\, \frac{-5}{17}}$
$\because$ All have same numerator. Sothe rational number having theleast denominator is the greatest.But here all have negative sign.So, the number having greatestdenominator is greater. Hence, $\displaystyle \frac{-5}{17}$ is greater

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

The given rational numbers are $\displaystyle \frac{1}{2},\, \displaystyle \frac{4}{-5},\, \displaystyle \frac{- 7}{8}$. If these numbers are arranged in the ascending order or descending order, then the middle number is

  1. $\displaystyle \frac{1}{2}$
  2. $\displaystyle \frac{- 7}{8}$
  3. $\displaystyle \frac{4}{- 5}$

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

Let given numbers arranged in the descending order.
$\displaystyle \frac{1}{2},\, \displaystyle \frac{-4}{5},\, \displaystyle \frac{- 7}{8}$
$- 4\, \times\, 8,\, 5\, \times\, - 7$
$- 32,\, - 35$
$\displaystyle \frac{-4}{5}\, >\, \displaystyle \frac{-7}{8}$
The descending order is $\displaystyle \frac{1}{2}\, >\, \displaystyle \frac{-4}{5}\, >\, \displaystyle \frac{- 7}{8}$
So middle number is $\displaystyle \frac{- 4}{5}$.

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

The given rational numbers are $\displaystyle {\frac{1}{2},\, \frac{4}{-5},\, \frac{-7}{8}}.$ If these numbers are arranged in the ascending order or descending order, then the middle number is:

  1. $\displaystyle \frac{1}{2}$
  2. $\displaystyle \frac{-7}{8}$
  3. $\displaystyle \frac{4}{-5}$

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

The rational number $\dfrac {4}{-5}$ is same as $\dfrac {-4}{5}$.


Now consider the given rational numbers $\dfrac {1}{2},\dfrac {-4}{5}$ and $\dfrac {-7}{8}$ and make their denominator same by taking the LCM of the denominators as follows:

LCM$(2,5,8)=40$

The given fractions now with denominator $40$ can be written as:

$\dfrac { 1\times 20 }{ 2\times 20 } =\dfrac { 20 }{ 40 } \ \dfrac { -4\times 8 }{ 5\times 8 } =\dfrac { -32 }{ 40 } \ \dfrac { -7\times 5 }{ 8\times 5 } =\dfrac { -35 }{ 40 }$ 

The ascending order of the rational numbers is:

$\dfrac { -35 }{ 40 } ,\dfrac { -32 }{ 40 } ,\dfrac { 20 }{ 40 } \ \Rightarrow \dfrac { -7 }{ 8 } ,\dfrac { 4 }{ -5 } ,\dfrac { 1 }{ 2 } .......(1)$

The descending order of the rational numbers is:

$\dfrac { 20 }{ 40 } ,\dfrac { -32 }{ 40 } ,\dfrac { -35 }{ 40 } \ \Rightarrow \dfrac { 1 }{ 2 } ,\dfrac { 4 }{ -5 } ,\dfrac { -7 }{ 8 } .......(2)$ 

From equations 1 and 2, we conclude that the middle rational number in both ascending and descending order is same that is $\dfrac {4}{-5}$.

Hence, the middle number is $\dfrac {4}{-5}$.

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

Out of the rational numbers $\displaystyle\frac{7}{-13},\,\frac{-5}{13},\,\frac{-11}{13}$ which is smaller ?

  1. $\displaystyle \frac{7}{13}$
  2. $\displaystyle \frac{- 5}{13}$
  3. $\displaystyle \frac{- 11}{13}$

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

Take any two given rational numbers
$\displaystyle \frac{-7}{13},\, \displaystyle \frac{-5}{13}$
$- 7\, \times\, 13,\, -5\, \times\, 13$
$- 91,\, -65$
$\because\, - 65\, >\, - 91$
So $\displaystyle \frac{-7}{13}$ is smaller.
Now compare this with $\displaystyle \frac{- 11}{13}$
$\displaystyle \frac{- 7}{13},\, \displaystyle \frac{- 11}{13}$
$-7\, \times\, 13,\, - 11\, \times\, 13$
$- 91\, ,\, - 143$
$\because\, - 91\, >\, - 143$
So $\displaystyle \frac{- 11}{13}$ is smaller.

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

Which of the following is an irrational number? 

  1. $0.14$
  2. $0.14 \overline{16}$
  3. $1.1 {416}$
  4. $0.4014001400014....$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
The decimal expansion of a rational number is either terminating or non-terminating repeating.

(A) $0.14$ is terminating, so it is a rational number

(B) $0.14\bar{16}=0.141616....$ 

is also rational ( non-terminating repeating ), where digits $16$ are repeating.

(C) $0.1416$ is terminating, so it is a rational number

(D) $0.4014001400014.....$

is an irrational number because it is neither terminating nor repeating.