Question

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

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Hint: Irrational numbers are a real number that, when expressed as a decimal, 90 on after (forever) after the decimal and never repeat.
Irrational are the real numbers that cannot be represented as a simple fraction It cannot be expressed in the form of a ratio such as $\dfrac{p}{q}$ where p & q are
Integers, $q \ne o$ it is a contradiction of rational numbers
For example, $\sqrt 5 ,\sqrt {11} ,\sqrt {21}$etc are irrational numbers
Properties of irrational number
1) The addition of an irrational number and rational number gives an irrational number for example $x =$ irrational$y =$ rational
$\Rightarrow x + y =$ irrational number
2) Multiplication of any irrational number with non-zero rational number results is an irrational number example $x =$ irrational $y =$ rational
$\Rightarrow x + y =$irrational
The LCM of two irrational number may or may NOT exists.
The addition or multiplication of two irrational numbers may be rational for example $\sqrt 2 .\sqrt 2 = 2$ Here $\sqrt 2$ is irrational 2 is rational.

Let us take
$(\sqrt 3 ,\sqrt 3 )$ are irrational numbers.
Difference b/w $(\sqrt 3 ,\& - \sqrt 3$
$= \sqrt 3 , - \sqrt 3$
$= \sqrt 3 + \sqrt 3$
$= \sqrt[2]{3}$(irrational number)

Difference between $\sqrt 5 , - \sqrt 5$ (irrational number)
$\Rightarrow \sqrt 3 - ( - \sqrt 5 )$
$= \sqrt 5 + \sqrt 5$
$= \sqrt[2]{5}$ (irrational number)

$\sqrt[4]{3}, - \sqrt[2]{3}$
difference between $\sqrt[4]{3} - ( - \sqrt[2]{3}) = \sqrt[6]{3}$ irrational number
Above all options A, B, C are the example of irrational numbers whose difference is an irrational number.

Note:
The decimal expansion of an irrational number is neither terminating nor recurring
Pi$(\pi )$ is an irrational number because it is non-terminating the approximate value of pi is
The set of an irrational number is NOT closed under the multiplication process unlike the set of rational numbers.
Integers are a rational number but Not irrational.
$\pi = 3.14159265358$
$e = 2.718281845$ are irrational numbers.
An irrational number is represented by using the set difference of the real minus rational numbers in a way $R - Q$