Question

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

Answer 1

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.

Complete step by step answer:
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 \[R - Q\]