Tag: real number

Questions Related to real number

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

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

  1. A rational number

  2. A natural number

  3. An irrational number

  4. An integer

Show answer
Correct Option: C
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

$2.13113111311113....$ is _____________.

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

  1. A rational number

  2. An integer

  3. An irrational number

  4. Not a real number

Show answer
Correct Option: C
Explanation:

$2.13113111311113....$ follows a definite pattern but the digits after decimal places are non-terminating and non-recurring. 

Hence, number is irrational.

State the following statement is True or False
$\dfrac{7}{9}$ has a value of non terminating decimal number

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

  1. True

  2. False

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Correct Option: A
Explanation:

$\dfrac{7}{9}=0.777...$, Non terminating decimal number.

The decimal expansion of $\sqrt{2}$ is :

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

  1. finite decimal

  2. 1.4121

  3. non - terminating recurring

  4. non - terminating non recurring

Show answer
Correct Option: D
Explanation:

Since, $\sqrt { 2 } $ is a irrational number. So, its decimal expansion is non- terminating non recurring. 

So, correct answer is option D.

Classify the decimal form of the given rational number into terminating non-terminating recurring type.

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

  1. $\dfrac { 13 }{ 5 }$

  2. $\dfrac { 2 }{ 11 }$

  3. $\dfrac { 29 }{ 16 }$

  4. $\dfrac { 17 }{ 125 }$

  5. $\dfrac { 11 }{ 6 }$

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Correct Option: A

State the following statement is True or False

$\dfrac {15}{1600}$ has a terminating decimal expansion .

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Given, $\displaystyle \frac {15}{1600}= \frac {15}{5^{2}2^{6}}$
As it is in the form of ${ 2 }^{ m }\times { 5 }^{ n }$ where ($n=6,m=2$).
So, the rational number $\displaystyle \frac {15}{1600}$ has a terminating decimal expansion

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.
$\dfrac {29}{343}$

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

  1. Terminating

  2. Non-terminating

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: B
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

If $9{x}^{2}+25{y}^{2}=181$ and $xy=-6$, find the value of $3x+5y$

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

  1. $\pm 3$

  2. $\pm 1$

  3. $\pm 2$

  4. None of the above

Show answer
Correct Option: B
Explanation:
Given 

$9{x}^{2}+25{y}^{2}=181$ and $xy=-6$,

formula,
$(a+b)^2=a^2+b^2+2ab$

$\therefore(3x+5y)^2=(3x)^2+(5y)^2+2(3x)(5y)$

$3x+5y=\sqrt{9x^2+25y^2+30xy}$

$=\sqrt{181+30(-6)}$

$=\pm 1$

which of the following represents the sum of numerator and denominator in the simplified expression of $\left(3.2\bar3 + 4.\bar {75}\right) $?

maths real number rounding decimals non-terminating recurring decimals in rational numbers rounding off decimals

  1. $798$

  2. $2967$

  3. $8901$

  4. $989$

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Correct Option: A


$20$ is written as the product of primes as :

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

  1. ${2\times 5 }$

  2. ${2\times 2\times 3\times 5}$

  3. ${2\times 2\times 5}$

  4. ${2\times 2\times 3}$

Show answer
Correct Option: C
Explanation:

To write a number as product of its primes, we divide it by various prime numbers $ 2, 3, 5, 7 $ etc one by one and check by which prime numbers it is divisible with and how many times.

Hence, $ 20 = 2 \times 10 = 2 \times 2 \times 5 $