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 logarithm and its uses basic mathematical concepts physics

Value of $\displaystyle \log _{4}18 $ is:

  1. an irrational number

  2. a rational number

  3. natural number

  4. whole number
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\log _{4}{18}=\log _{2^{2}}{(2.3^{2})}=\dfrac{1}{2}\log _{2}(2.3^{2})$
$=\dfrac{1}{2}\left [ \log _{2}{2}+\log _{2}{3^{2}} \right ]=\dfrac{1}{2}\left [ 1+2.\log _{2}{3} \right ]$

$\log _{2}{3}$ is an irrational number.

Hence, $\dfrac{1}{2}\left [ 1+2.\log _{2}{3} \right ]$ is also an irrational number.
Multiple choice maths parts and whole multiplication of a fraction multiplication of a fractions multiplication of fraction finding the whole when a fraction is given

The product of two rational numbers $\displaystyle \frac{-9}{16}$. If one of the numbers is $\displaystyle \frac{-4}{3}$ then the other number is:

  1. $\displaystyle \frac{36}{48}$
  2. $\displaystyle \frac{25}{64}$
  3. $\displaystyle \frac{27}{49}$
  4. $\displaystyle \frac{27}{64}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let the required no. be $x$.
$\therefore x \times  {\cfrac{-4}{3} = \cfrac{-9}{16}}$
$\Rightarrow x =  {\cfrac{-9/16}{-4/3} = \cfrac{-9}{16} \times \cfrac{-3}{4} = \cfrac{27}{64}}$
Multiple choice maths parts and whole multiplication of a fraction multiplication of a fractions multiplication of fraction finding the whole when a fraction is given

Which of the following statements is true?

  1. Every point on the number line represents a rational number
  2. The product of a rational number and its reciprocal to $0$
  3. $(17\times 12)^{-1}=17^{-1}\times 12$
  4. Reciprocal of $\displaystyle\frac{1}{a}$, $a$ $\neq 0$ is $a$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

(1) Every point on the line doesn't represent a ration number, it represents the real number,  which includes both rational and irrational numbers.


(2) Let's say $a$ is a rational number,
Its reciprocal will be $\dfrac{1}{a}$
The product will be $ a \times  \dfrac{1}{a} =1$.

(3) $(17 \times  12)^{-1}$
$=$ $ \dfrac{1}{17\times 12}$
$=$ $ \dfrac{1}{17} \times  \dfrac{1}{12}$
$=$ ${17}^{-1} \times  {12}^{-1}$

(4) Reciprocal of $\dfrac{1}{a}$
$= \dfrac{1}{\dfrac{1}{a}}$
$=$ $a$
But $\dfrac{1}{a}$ is defined only if $ a \neq0$

Multiple choice maths parts and whole multiplication of a fraction multiplication of a fractions multiplication of fraction finding the whole when a fraction is given

Consider the following statements : 
A. The product of an integer and a rational number can never be a natural number 
B. The quotient of division of an integer by a rational number can never be an integer
Which of the statements given above is/are correct ?

  1. A only

  2. B only

  3. Both A and B

  4. Neither A nor B
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let integer = 4 and Rational number = $\displaystyle \frac{2}{1} $
Then product = $\displaystyle 4\times\frac{2}{1}=8 $ (a natural number)
and Quotient = $\displaystyle 4\div \frac{2}{1}=4\times \frac{1}{2}=2 $ (an integer)

Multiple choice maths mathematical reasoning implications principle of mathematical induction proofs in mathematics

Negation of the statement $p:\dfrac {1}{2}$ is rational and $\sqrt {3}$ is irrational is

  1. $\dfrac {1}{2}$ is rational or $\sqrt {3}$ is irrational
  2. $\dfrac {1}{2}$ is not rational or $\sqrt {3}$ is not irrational
  3. $\dfrac {1}{2}$ is not rational or $\sqrt {3}$ is irrational
  4. $\dfrac {1}{2}$ is rational and $\sqrt {3}$ is irrational
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The negation of (A AND B) is (NOT A OR NOT B). Negating '1/2 is rational' gives '1/2 is not rational', and negating 'sqrt(3) is irrational' gives 'sqrt(3) is rational'. However, the option provided uses the original statement parts in an OR format, which is a common simplification in logic tests.

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

If $\sqrt{a}$ is an irrational number, what is a? 

  1. Rational

  2. Irrational
  3. $0$

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

Consider the given irrational number$\sqrt{a}$ ,

Definition  of rational number- which number can be write in the form of $\dfrac{p}{q}$ but $q\ne 0$ is called rational number.

Hence, $a=\dfrac{a}{1}$

That why  $a$ is rational number

 

Hence, this is the answer.

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

Which of the following is irrational

  1. $\sqrt {\dfrac{4}{9}} $
  2. $\dfrac{4}{5}$
  3. $\sqrt 7 $
  4. $\sqrt {81} $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
A $=\sqrt{\dfrac{4}{9}}=\dfrac{2}{3}$         Rational

B $=\dfrac{4}{5}$                       Rational

C $=\sqrt7$                     Irrational

D $=\sqrt{81}=9$          Rational
Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

Which of the following rational number represents a terminating decimal expansion?

  1. $

    \dfrac { 77 } { 210 }

    $
  2. $

    \dfrac { 13 } { 125 }

    $
  3. $

    \dfrac { 2 } { 15 }

    $
  4. $

    \dfrac { 17 } { 18 }

    $
Reveal answer Fill a bubble to check yourself
B 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.

Solution is $B$ as $A$ is non terminating decimals.
$A =\dfrac{77}{210}= 0.366......$

$B =\dfrac{13}{125}= 0.104$

$C =\dfrac{2}{15}= 0.133.....$

$D =\dfrac{17}{18}=  0. 9444....$
Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

There can be a pair of irrational numbers whose sum is irrational 

Such as: $\displaystyle \sqrt{3}+2$ and $\displaystyle 5+\sqrt{2}$

  1. True

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

To get the sum as irrational, the numbers need to have an irrational part as well which are different from each other.

Example, the pair of numbers $ \sqrt{3} + 2 $ and $ 5 + \sqrt {2} $ have the sum $ \sqrt{3} + 2 + 5 + \sqrt {2} = 7 + \sqrt {2} + \sqrt {3} $ which is an irrational number too.

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

Which of the following is irrational?

  1. $\dfrac {22}{7}$
  2. $3.141592$
  3. $2.78181818$
  4. $0.123223222322223.......$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

An irrational number is any real number that cannot be expressed as a ratio of integers. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals.
Among all the options only $(D)$ $0.123223222322223$...... is non terminating and non repeating decimal.Therefore, it is a irrational number.

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

$\sqrt 7$ is

  1. A rational number

  2. An irrational number

  3. Not a real number

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

Rational numbers are those numbers which can be expressed in the form $ \dfrac {p}{q} $, where p and q are integers and $ q \neq 0 $
Numbers which are not rational numbers are called irrational numbers.
Since, $ \sqrt {7} $ cannot be written in $ \dfrac {p}{q} $, where $p$ and $q$ are integers and $ q \neq 0 $; it is an irrational number.