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 fractions, decimals and rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

 ________ are rational numbers between $\displaystyle -\dfrac{3}{4}$ and $\displaystyle \dfrac{1}{2}.$

  1. $\dfrac{-7}{16}, \dfrac{-1}{8}, \dfrac{9}{16}$
  2. $\dfrac{-15}{16}, \dfrac{-1}{8}, \dfrac{3}{16}$
  3. $\dfrac{-7}{16}, \dfrac{-1}{8}, \dfrac{3}{16}$

  4. none of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
A rational number between two numbers $ a $ and $ b = \dfrac {(a + b)}{2} $ 

So,
a rational number between $ - \dfrac {3}{4} $ and $ \dfrac {1}{2} $
$= \dfrac {-\dfrac {3}{4} + \dfrac {1}{2}}{2} = - \dfrac {1}{8} $

Now,
another rational number between $ - \dfrac {3}{4} $ and $ - \dfrac {1}{8} $
$=\dfrac {- \dfrac {3}{4} - \dfrac {1}{8}}{2} = - \dfrac {7}{16} $ 

Another rational number between $ - \dfrac {1}{8} $ and $ \dfrac {1}{2} =$

$\dfrac { - \dfrac {1}{8} + \dfrac {1}{2}} {2} =  \dfrac {3}{16} $ 

Hence, required three rational numbers between $ - \dfrac {3}{4} $ and $  \dfrac {1}{2} $ are $ - \dfrac {3}{4}, - \dfrac {7}{16},  - \dfrac {1}{8}, \dfrac {3}{16}, \dfrac {1}{2} $
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 lying between the numbers $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{3}{4}$ are

  1. $\displaystyle \frac{97}{300}$,$\displaystyle \frac{299}{500}$
  2. $\displaystyle \frac{99}{300}$,$\displaystyle \frac{301}{400}$
  3. $\displaystyle \frac{95}{300}$,$\displaystyle \frac{301}{400}$
  4. $\displaystyle \frac{117}{300}$,$\displaystyle \frac{287}{400}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To insert rational numbers between $2$ numbers, we will arrange the options and check if they are in ascending order.
$\dfrac { 1 }{ 3 } { ? }\dfrac { 117 }{ 300 } \ \Longrightarrow 300<351\ \qquad \dfrac { 3 }{ 4 } { ? }\dfrac { 287 }{ 400 } \ \Longrightarrow 1200>1148$
They are in ascending order, i.e., $\dfrac { 1 }{ 3 } ,\dfrac { 117 }{ 300 } ,\dfrac { 287 }{ 400 } ,\dfrac { 3 }{ 4 } $
From the given options only option $D$ satisfies this condition. Hence, $D$ is the answer.

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

Let a, b, c be positive integers such that $\frac {a\sqrt 2+b}{b\sqrt 2+c}$ is a rational number, then which of the following is always an integers?

  1. $\frac {2a^2+b^2}{2b^2+c^2}$
  2. $\frac {a^2+b^2-c^2}{a+b-c}$
  3. $\frac {a^2 _2b^2}{b^2+2c^2}$
  4. $\frac {a^2+b^2+c^2}{a+c-b}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For (a*sqrt(2)+b)/(b*sqrt(2)+c) to be rational, the irrational parts must cancel. This implies a/b = b/c, so b^2 = ac. Testing option D: (a^2+b^2+c^2)/(a+c-b) is a standard algebraic identity related to this condition.

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 of these is true?
$(I)$ $5\sqrt {3}$ is not a rational number
$(II)$ $1$ is not the cube of a rational number
$(III)$ If a is rational and $n$ is an integer greater than $1$, then ${a}^{n}$ is rational.

  1. $I$ and $II$
  2. $II$ and $III$
  3. $III$ and $I$

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

(I) In $5\sqrt{3}$

5 ia s rational number and $\sqrt{3}$ is an Irrational number
As we know, The product of a rational and irrational number is an irrational number.
So, $5\sqrt{3}$ is not a rational number.
Hence, the option (I) is true
(II) 1 is a rational number 
and cube of 1 is 1 only, which is a rational number
Hence the option (II) is False
(III) We know that product of two rational number is always a rational number
Hence if a is a rational number and n is greater than one 
Then,
a2 = a x a is a rational number.

a3 = a2 x a is a rational number,

a4 = a3x a is a rational number,

......

......

∴ an = an-1 x a is a rational number.

So, the option (III) is true

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 of the following represents a rational number between $-6$ and $-7$?

  1. $\dfrac {-6 - 7}{2}$
  2. $\dfrac {-6 + 7}{2}$
  3. $\dfrac {6 + 7}{2}$
  4. $-6 - 7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Mean $= \dfrac {(-6) + (-7)}{2} = \dfrac {-6 -7}{2}$.

Mean of two numbers always lies between the two numbers.
So, answer is option $A.$

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 between $\dfrac {-9}{10}$ and $\dfrac {4}{5}$ is:

  1. $\left (\dfrac {-9}{10} + \dfrac {4}{5}\right ) \times \dfrac {1}{2}$
  2. $\left (\dfrac {-9}{10} - \dfrac {4}{5}\right ) + \dfrac {1}{2}$
  3. $\left (\dfrac {-9}{10} + \dfrac {4}{5}\right ) \times 2$

  4. All above are correct
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Mean of two numbers always lies between the two numbers.

Mean 

$= \dfrac{\left (\dfrac {-9}{10} + \dfrac {4}{5}\right )}{2}$

$= \left (\dfrac {-9}{10} + \dfrac {4}{5}\right )\times \dfrac {1}{2}$.

So, answer is option $A.$
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 of the following rational numbers lies between $\dfrac {3}{4}$ and $\dfrac {13}{8}$?

  1. $\dfrac {11}{16}$
  2. $\dfrac {12}{16}$
  3. $\dfrac {19}{16}$
  4. $\dfrac {26}{16}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$Mean = \dfrac{\left (\dfrac {3}{4} + \dfrac {13}{8}\right )}{2} = \dfrac {6 + 13}{8} \times \dfrac {1}{2} = \dfrac {19}{16}$.

Mean of two numbers always lies between the two numbers.
So, answer is 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

Which of the following rational number lies between $\dfrac {4}{9}$ and $\dfrac {4}{5}$?

  1. $-1$
  2. $\dfrac {28}{45}$
  3. $0$
  4. $1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$Mean = \dfrac{\left (\dfrac {4}{9} + \dfrac {4}{5}\right )}{2} = \left (\dfrac {20 + 36}{45}\right ) \times \dfrac {1}{2} = \dfrac {56}{45}\times \dfrac {1}{2}$
$= \dfrac {28}{45}$

Mean of two numbers always lies between the two numbers.
So, answer is option $B.$ 

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

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

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

The given rational numbers $-\dfrac { 2 }{ 3 }$ and $-\dfrac { 1 }{ 5 }$ are negative rational numbers because the numerator and denominator of both the rational numbers are of opposite signs that is the numerator of both the integers is negative while the denominators are positive.


Therefore, none of the positive rational number can lie between the given negative rational numbers $-\dfrac { 2 }{ 3 }$ and $-\dfrac { 1 }{ 5 }$.


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

Multiple choice maths operations on rational numbers representation of rational numbers on number line rational numbers on the number line rational numbers between two rational numbers

Rational number between $\dfrac{3}{8}$ and $\dfrac{7}{12}$ are 

  1. $\dfrac{3}{8},\dfrac{41}{96},\dfrac{23}{48},\dfrac{7}{12}$
  2. $\dfrac{3}{8},\dfrac{41}{196},\dfrac{23}{48},\dfrac{7}{12}$
  3. $\dfrac{3}{8},\dfrac{41}{96},\dfrac{23}{148},\dfrac{7}{12}$

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

3/8 = 0.375 and 7/12 = 0.5833. Option A: 41/96 = 0.427 and 23/48 = 0.479. Both are between 0.375 and 0.5833.

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

The rational form of $-25.6875$ is

  1. $\displaystyle-\frac{411}{16}$
  2. $\displaystyle-\frac{421}{16}$
  3. $\displaystyle-\frac{431}{16}$
  4. $\displaystyle-\frac{441}{16}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Consider the given decimal number

$ \Rightarrow -25.6875 $

$ \Rightarrow -\dfrac{256875}{10000} $

$ \Rightarrow -\dfrac{51375}{2000} $

$ \Rightarrow -\dfrac{10275}{400} $

$ \Rightarrow -\dfrac{2055}{80} $

$ \Rightarrow -\dfrac{411}{16} $


Hence, this is the answer.

Multiple choice

What are the three main types of numbers?

  1. Natural numbers, integers, and rational numbers

  2. Natural numbers, integers, and real numbers

  3. Natural numbers, rational numbers, and irrational numbers

  4. Integers, rational numbers, and irrational numbers
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The three main types of numbers are natural numbers, integers, and rational numbers. Natural numbers are the numbers that we use to count things, such as 1, 2, 3, and so on. Integers are the numbers that include the natural numbers and their negatives, such as -1, -2, -3, and so on. Rational numbers are the numbers that can be expressed as a fraction of two integers, such as 1/2, 3/4, and so on.

Multiple choice

What is the difference between a rational number and an irrational number?

  1. A rational number can be expressed as a fraction of two integers, while an irrational number cannot

  2. A rational number is a number that can be written as a decimal that terminates or repeats, while an irrational number is a number that cannot be written as a decimal that terminates or repeats

  3. A rational number is a number that is less than 1, while an irrational number is a number that is greater than 1

  4. A rational number is a number that is a whole number, while an irrational number is a number that is a fraction
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The difference between a rational number and an irrational number is that a rational number can be expressed as a fraction of two integers, while an irrational number cannot. For example, the number 1/2 is a rational number because it can be expressed as a fraction of the integers 1 and 2. The number (\sqrt{2}) is an irrational number because it cannot be expressed as a fraction of two integers.

Multiple choice

The set of all rational numbers is:

  1. Open.

  2. Closed.

  3. Both open and closed.

  4. Neither open nor closed.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The set of all rational numbers is neither open nor closed because it contains some of its boundary points but not all of them.