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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.

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 \[R - Q\]