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 number system representation of irrational numbers on number line existence of irrational numbers irrational numbers

Which of the following is/are correct?

  1. Every integer is a rational number.

  2. The sum of a rational number and an irrational number is an irrational number.

  3. Every real number is rational.

  4. Every point on the number line is associated with a real number
Reveal answer Fill a bubble to check yourself
A,B,D Correct answer
Explanation

Yes every integer can be represented in the form of $p/q$ where q is 1 for integers, so every integer is a rational number.
The sum of rational and irrational numbers is always irrational.
No, every real number is not rational. Real numbers are classified as rational numbers and irrational numbers.
Any number on the number line is a real number because that number can be either rational or irrational. Rational and irrational numbers together form real numbers.

Multiple choice maths number system representation of irrational numbers on number line existence of irrational numbers irrational numbers

To represent a rational number $\sqrt{2}$ on number line, take sides of right triangle as:

  1. $1$ and $1$
  2. $1$ and $2$
  3. $2$ and $0$
  4. $-1$ and $-1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Notice that $\sqrt{2}= \sqrt{(1^2 + 1^2)}$. So, we can form a length of $\sqrt{2}$ units using two mutually perpendicular sides of length $1$ unit each. (Since, $1,1,\sqrt{2}$ form sides of a right angled triangle by Pythagoras theorem).
Multiple choice maths number system representation of irrational numbers on number line existence of irrational numbers irrational numbers

Give an example of two irrational numbers whose difference is an irrational number.

  1. $\sqrt{3},-\sqrt{3}$
  2. $\sqrt{5,}-\sqrt{5}$
  3. $4\sqrt{3},-2\sqrt{3}$

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

$4\sqrt{3},2\sqrt{3}$ are the irrational numbers and thier difference,


$4\sqrt{3}-2\sqrt{3}=2\sqrt 3$ is also an irrational number.

Multiple choice maths number system representation of irrational numbers on number line existence of irrational numbers irrational numbers

Which is the wrong step that shows $\displaystyle 5-\sqrt{3}$ is irrational?
(I) Contradiction : Assume that $\displaystyle 5-\sqrt{3}$ is rational
(II) Find coprime a & b $\displaystyle \left ( b\neq 0 \right )$ such that $\displaystyle 5-\sqrt{3}=\frac{a}{b},\therefore 5-\frac{a}{b}=\sqrt{3}$
Rearranging above equation $\displaystyle \sqrt{3}=5-\frac{a}{b}=\frac{5b-a}{b}$
(III) Since a & b are integers we get $\displaystyle 5-\frac{a}{b}$ is irrational and so $\displaystyle \sqrt{3}$ is irrational
(IV) But this contradicts the fact that $\displaystyle \sqrt{3}$ is irrational Hence $\displaystyle 5-\sqrt{3}$ is irrational

  1. Both I and II

  2. Only III

  3. Only II

  4. Both II and III
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Step III is incorrect because it claims that since a and b are integers, 5 - a/b is irrational. In reality, 5 - a/b is rational if a and b are integers, which is the basis of the contradiction proof.

Multiple choice maths number system representation of irrational numbers on number line existence of irrational numbers irrational numbers

Which of the following irrational numbers lie between $4$ and $7$?

  1. $\sqrt{25}$
  2. $\sqrt{19}$
  3. $\sqrt{47}$
  4. $\sqrt{50}$
Reveal answer Fill a bubble to check yourself
B,C Correct answer
Explanation

$4^{2} = 16$

$5^{2} = 25$
$6^{2} = 36$
$7^{2} = 49$


$\Rightarrow \sqrt19$ and $\sqrt47$ are irrational numbers which lie between $4$ and $7$


$\sqrt25 = 5$ which is a rational number

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

The non terminating non-recurring decimal cannot be represented as

  1. irrational numbers

  2. rational numbers

  3. real numbers

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Write the decimal number as a fraction $0.2020020002...... $
It is Non- terminating decimal and cannot be represented as a quotient of two integers.
Therefore, B is the correct answer.

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

If a number has a non-terminating and non-recurring decimal expansion, then it is.

  1. A rational number

  2. A natural number

  3. An irrational number

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

A number having non-terminating and non-recurring decimal expansion is an Irrational Number


for example 

$\pi$  is an irrational number 

$\pi = 3.1415926535897932384626433832............$


the number has non-terminating decimal expansion and non-recurring.

So option $C $ is correct

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 numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
$\dfrac {29}{343}$

  1. Terminating

  2. Non-terminating

  3. Ambiguous

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

Given, $\displaystyle \frac {29}{343}= \frac {29}{7^{3}}$
As it is not in the form of ${ 2 }^{ m }\times { 5 }^{ n }$.
So, the rational number $\displaystyle \frac {29}{343}$ has a non terminating decimal expansion

Multiple choice maths real number fundemental theorem of arithmetic real numbers on number line fundamental theorem of arithmetic

Fundamental theorem of arithmetic is basically used for ________

  1. proving the irrationality of numbers.

  2. to explore when exactly the decimal expansion of a rational number is terminating or non-terminating repeating.

  3. prime numbers.

  4. both A and B.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Fundamental theorem of arithmetic is basically used for firstly proving the irrationality of numbers and secondly to explore when exactly the decimal expansion of a rational number is terminating and when it is non - terminating repeating.
Therefore, $ B$ is the correct answer.

Multiple choice maths concepts of seven and eight digit numbers comparison of numbers comparing numbers operations on rational numbers indian place value chart largest and smallest numbers writing and expanding numbers

Arrange the following rational number in ascending order $\displaystyle \frac{3}{7},\frac{4}{5},\frac{7}{9},\frac{1}{2}$

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

The LCM of 2, 5, 7 and 9 is 630.
$\frac{3}{7} = \frac{270}{630}$
$\frac{4}{5} = \frac{504}{630}$
$\frac{7}{9}=\frac{490}{630}$
$\frac{1}{2}=\frac{315}{630}$
The ascending order will be $\frac{3}{7},\frac{1}{2},\frac{7}{9},\frac{4}{5}$

Multiple choice maths concepts of seven and eight digit numbers comparison of numbers comparing numbers operations on rational numbers indian place value chart largest and smallest numbers writing and expanding numbers

Write the following rational numbers in ascending order:

$\dfrac{3}{4},\dfrac{7}{12}, \dfrac{15}{11}, \dfrac{22}{19}, \dfrac{101}{100}, \dfrac{-4}{5}, \dfrac{-102}{81}, \dfrac{-13}{7}$.

  1. $\dfrac{-13}{7},\dfrac{-102}{81}, \dfrac{-4}{5}, \dfrac{22}{19}, \dfrac{101}{100}, \dfrac{15}{11}, \dfrac{7}{12}, \dfrac{3}{4}$.
  2. $\dfrac{-13}{7},\dfrac{-102}{81}, \dfrac{-4}{5}, \dfrac{7}{12}, \dfrac{3}{4}, \dfrac{22}{19}, \dfrac{101}{100}, \dfrac{15}{11}$.
  3. $\dfrac{-13}{7},\dfrac{-102}{81}, \dfrac{-4}{5}, \dfrac{7}{12}, \dfrac{3}{4}, \dfrac{101}{100}, \dfrac{22}{19}, \dfrac{15}{11}$.
  4. $\dfrac{3}{4},\dfrac{7}{12}, \dfrac{15}{11}, \dfrac{22}{19}, \dfrac{101}{100}, \dfrac{-4}{5}, \dfrac{-102}{81}, \dfrac{-13}{7}$.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Convert to decimals: -13/7 approx -1.85, -102/81 approx -1.26, -4/5 = -0.8, 7/12 approx 0.58, 3/4 = 0.75, 101/100 = 1.01, 22/19 approx 1.16, 15/11 approx 1.36. The ascending order matches option C.

Multiple choice maths concepts of seven and eight digit numbers comparison of numbers comparing numbers operations on rational numbers indian place value chart largest and smallest numbers writing and expanding numbers

Find the rational numbers between the following numbers. 

$-0.2$ and $-0.22$.

  1. $-0.210 > -0.211 > -0.312 > -0.213 > -0.314 > 0.220$
  2. $-0.210 > -0.211 > -0.212 > -0.213 > -0.314 > 0.220$
  3. $-0.210 > -0.211 > -0.312 > -0.213 > -0.214 > 0.220$
  4. $-0.210 > -0.211 > -0.212 > -0.213 > -0.214 > 0.220$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Rational number between two numbers $ a $ and $ b = \dfrac {(a +

b)}{2} $

So, a rational number between $ -0.2 $ and $ - 0.22 = \dfrac {(-0.2 - 0.22)}{2} = -0.21 $

Now, another rational number between $ -0.21 $ and $ - 0.22 = \dfrac {(-0.21 - 0.22)}{2} = -0.215 $

rational number between $ -0.215 $ and $ - 0.21 = \dfrac {(-0.215 - 0.21)}{2} = -0.212 $ 

rational number between $ -0.215 $ and $ - 0.212 = \dfrac {(-0.215 - 0.212)}{2} = -0.213 $ 

rational number between $ -0.215 $ and $ - 0.213 = \dfrac {(-0.215 - 0.213)}{2} = -0.214 $ 

Similarly,
rational numbers between $ -0.2 $ and $ - 0.22 $ are $-0.210, -0.211 , -0.212, -0.213 , -0.214$ etc

Multiple choice maths concepts of seven and eight digit numbers comparison of numbers comparing numbers operations on rational numbers indian place value chart largest and smallest numbers writing and expanding numbers

Find the five rational numbers between $-5$ and $-6$

  1. $-5.1 ,-5.2 , -3.3 , -5.4 , -5.5 $
  2. $-5.1 ,-5.2 , -5.3 , -5.4 , -5.5 $
  3. $-5.1 , -6.2 , -5.3 , -5.4 , -5.5 $
  4. $-6.1 , -5.2 , -5.3 , -5.4 , -5.5 $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$−5>(−5−0.1)=−5.1>−5.2=(−5.1−0.1)>−5.3=(−5.2−0.1)>−5.4\\=(−5.3−0.1)>−5.5=(−5.4−0.1)>...>−6$

$-5>−5.1>−5.2>−5.3>−5.4>−5.5...>−6$

The five rational numbers between $−5$ and $−6$
$-5.1 ,-5.2 , -5.3 , -5.4 , -5.5 $