Tag: properties of irrational numbers

Questions Related to properties of irrational numbers

State true or false:
There are numbers which cannot be written in the form $\frac{p}{q}$, where $q\neq 0$  and both p, q are integers.

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

  1. True

  2. False

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Correct Option: A
Explanation:

Any irrational number can not be written as $\dfrac{p}{q}$.

The product of a non-zero rational number with an irrational number is always :

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

  1. Irrational number

  2. Rational number

  3. Whole number

  4. Natural number

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Correct Option: A
Explanation:

By definition, an irrational number in decimal form goes on forever without repeating (a non-repeating, non-terminating decimal). By definition, a rational number in decimal form either terminates or repeats. 

By multiplying a non repeating non terminating number to repeating or terminating/repeating number, the result will always be a non terminating non repeating number. 
So, option A is correct. 

Which is not an Irrational number?

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

  1. $5-\sqrt{3}$

  2. $\sqrt{2}+\sqrt{5}$

  3. $4+\sqrt{2}$

  4. $6+\sqrt{9}$

Show answer
Correct Option: D
Explanation:

We know that sum of two irrational number or one rational and one irrational number will be irrational number. Option A, B , C stisfies this criteria but option D have two rational number i.e. $6 + \sqrt { 9 }$ = $6+ 3=9$

So correct answer is option D

$\left ( 2+\sqrt{5} \right )\left ( 2+\sqrt{5} \right )$ expression is :

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

  1. A rational number

  2. A whole number

  3. An irrational number

  4. A natural number

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Correct Option: C
Explanation:

${ (2+\sqrt { 5 } ) }^{ 2 }\ =4+5+4\sqrt { 5 } \ =9+4\sqrt { 5 } $

In the above equation $4\sqrt { 5 } $ is irrational number so $9+4\sqrt { 5 } $ will also be irrational number 
So correct answer is option C.

A pair of irrational numbers whose product is a rational number is:

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

  1. $\sqrt{16}, \sqrt{4}$

  2. $\sqrt{5}, \sqrt{2}$

  3. $\sqrt{3}, \sqrt{27}$

  4. $\sqrt{36}, \sqrt{2}$

Show answer
Correct Option: C
Explanation:

In the given options,

for option $(A)$ $\sqrt { 16 } \& \sqrt { 4 }$ are not irrational numbers i.e. their real values are 4 & 2 respectively. It cannot be correct answer. 

Now multiplying other options, we get

$(B):\sqrt { 5 } \times \sqrt { 2 } =\sqrt { 10 } $

$(C)\sqrt { 27 } \times \sqrt { 3 } =\sqrt { 81 } =9$

$(D) \sqrt { 36 } \times \sqrt { 2 } =\sqrt { 72 } $

So, correct answer is option C.

A number is an irrational if and only if its decimal representation is :

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

  1. non terminating

  2. non terminating and repeating

  3. non terminating and non repeating

  4. terminating

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Correct Option: C
Explanation:

According to definition of irrational number, If written in decimal notation, an irrational number would have an infinite number of digits to the right of the decimal point, without repetition.

So, correct answer is option C.

Which of the following is not an irrational number?

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

  1. $5-\sqrt{3}$

  2. $\sqrt{5}+\sqrt{3}$

  3. $4+\sqrt{2}$

  4. $5+\sqrt{9}$

Show answer
Correct Option: D
Explanation:

We know that if add  or subtract any number from irrational number then the result will be irrational number.
$\sqrt { 5 } $ , $\sqrt { 3 }$ , $\sqrt { 2 } $  are irrational number but $\sqrt { 9 } $ =3 is a rational number so option D is correct answer 

$\pi$ is _______

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

  1. a rational number

  2. an integer

  3. an irrational number

  4. a whole number

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Correct Option: C
Explanation:
Sometimes we use $π = 22/7$ which is a popular approximation

$π = 3.14159265358...$

$22/7 = 3.142857142857...$

But $π$ and $22/7$ are close but not accurate.

Rational Numbers - $P/Q$ when $Q$ is not equal to $0$.

Let $x = 33.33333…. $——-(1)

$10x = 33.333333….. $——-(2)

Equation $(2) - (1)$

$9x = 30$

$x = 30/9$ which is in form of $P/Q$ and $x = 33.3333…$

The digit $‘3′$ is repeating itself and that’s why it can be written as $100/3.$

When it’s π, the value is $3.14159265358...$ The order of digits will not repeat itself in it but in $22/7 = 3.142857142857….$ you can see that $142857…$ is repeating itself that’s why $22/7$ is rational but $π$ is irrational.

So, option C is correct.

Which of the following number is irrational ?

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

  1. $\sqrt{16}-4$

  2. $(3-\sqrt{3}) (3+\sqrt{3})$

  3. $\sqrt{5}+3$

  4. $-\sqrt{25}$

Show answer
Correct Option: C
Explanation:

In the given options $\sqrt { 16 }$ and $\sqrt { 25 } $ are irrational numbers. Their real values are 4 and 5 respectively. So, option A and C are incorrect.

Option B can solved and its real value becomes 6. So it is also a rational number.
In option C, $\sqrt { 5 }$ is a irrational number. So, option C is a irrational number. 
So, correct answer is option C. 

Which one of the following is an irrational number ?

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

  1. 0.14

  2. 0.1416

  3. 0.14169452

  4. 0.4014001400014.....

Show answer
Correct Option: D
Explanation:

In the given options, only option D is non terminating non recurring decimal. 

So correct answer is option D.