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### Theory:

There are many techniques to produce more and more irrational numbers between two numbers.

But sometimes it is easy to identify the provided number is irrational or not by checking the following properties.
Properties of irrational numbers:
1. Addition, subtraction, multiplication and division of two irrational number is may or may not be irrational.
2. Addition of rational and irrational number is always irrational.
3. Subtraction of rational and irrational number is always irrational.
4. Multiplication of a non-zeror rational and irrational is always irrational.
5. Division of a non-zero rational and irrational is always irrational.
1. The first property says :

i) Irrational number $$+$$ Irrational number $$=$$ Irrational$$/$$Rational number.
Example:
$\begin{array}{l}\mathit{Consider}\phantom{\rule{0.147em}{0ex}}\mathit{two}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}2+\sqrt{3}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}5+\sqrt{3}.\\ \mathit{Adding}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\mathit{will}\phantom{\rule{0.147em}{0ex}}\mathit{result}\phantom{\rule{0.147em}{0ex}}\mathit{in},\\ 2+\sqrt{3}+\phantom{\rule{0.147em}{0ex}}5+\sqrt{3}\phantom{\rule{0.147em}{0ex}}=7+2\sqrt{3}.\phantom{\rule{0.147em}{0ex}}\mathit{This}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}\mathit{agian}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{number}.\\ \\ \mathit{Now}\phantom{\rule{0.147em}{0ex}}\mathit{consider}\phantom{\rule{0.147em}{0ex}}\mathit{another}\phantom{\rule{0.147em}{0ex}}\mathit{two}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}2+\sqrt{3}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}2-\sqrt{3}.\\ \mathit{Adding}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\mathit{will}\phantom{\rule{0.147em}{0ex}}\mathit{result}\phantom{\rule{0.147em}{0ex}}\mathit{in},\\ 2+\overline{)\sqrt{3}}+\phantom{\rule{0.147em}{0ex}}2-\overline{)\sqrt{3}}=4.\phantom{\rule{0.147em}{0ex}}\mathit{This}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{number}.\end{array}$

ii) Irrational number $$-$$ Irrational number $$=$$ Irrational$$/$$Rational number.
Example:
$\begin{array}{l}\mathit{Consider}\phantom{\rule{0.147em}{0ex}}\mathit{two}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}2+\sqrt{3}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}5+\sqrt{3}\\ \mathit{Subtracting}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\mathit{will}\phantom{\rule{0.147em}{0ex}}\mathit{result}\phantom{\rule{0.147em}{0ex}}\mathit{in},\\ 2+\sqrt{3}-\left(\phantom{\rule{0.147em}{0ex}}5+\sqrt{3}\right)\phantom{\rule{0.147em}{0ex}}=-3.\phantom{\rule{0.147em}{0ex}}\mathit{It}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{number}.\\ \\ \mathit{Now}\phantom{\rule{0.147em}{0ex}}\mathit{consider}\phantom{\rule{0.147em}{0ex}}\mathit{another}\phantom{\rule{0.147em}{0ex}}\mathit{two}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}2+\sqrt{3}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}2-\sqrt{3}.\\ \mathit{Subtracting}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\mathit{will}\phantom{\rule{0.147em}{0ex}}\mathit{result}\phantom{\rule{0.147em}{0ex}}\mathit{in},\\ 2+\sqrt{3}-2+\sqrt{3}=2\sqrt{3}.\phantom{\rule{0.147em}{0ex}}\mathit{It}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}\mathit{an}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{number}.\end{array}$

iii) Irrational number $$×$$ Irrational number $$=$$ Irrational$$/$$Rational number.
Example:
$\begin{array}{l}\mathit{Consider}\phantom{\rule{0.147em}{0ex}}\mathit{two}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}2\sqrt{3}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\sqrt{3}\\ \mathit{Multiplying}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\mathit{will}\phantom{\rule{0.147em}{0ex}}\mathit{result}\phantom{\rule{0.147em}{0ex}}\mathit{in},\\ 2\sqrt{3}×\left(\phantom{\rule{0.147em}{0ex}}\sqrt{3}\right)\phantom{\rule{0.147em}{0ex}}=2×3=6.\phantom{\rule{0.147em}{0ex}}\mathit{It}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{number}.\\ \\ \mathit{Now}\phantom{\rule{0.147em}{0ex}}\mathit{consider}\phantom{\rule{0.147em}{0ex}}\mathit{another}\phantom{\rule{0.147em}{0ex}}\mathit{two}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}2\sqrt{3}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\sqrt{2}.\\ \mathit{Multiplying}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\mathit{will}\phantom{\rule{0.147em}{0ex}}\mathit{result}\phantom{\rule{0.147em}{0ex}}\mathit{in},\\ 2\sqrt{3}×\left(\sqrt{2}\right)=2\sqrt{6}.\phantom{\rule{0.147em}{0ex}}\mathit{It}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}\mathit{an}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{number}.\end{array}$

iv) Irrational number $÷$ Irrational number $$=$$ Irrational$$/$$Rational number.
Example:
$\begin{array}{l}\mathit{Consider}\phantom{\rule{0.147em}{0ex}}\mathit{two}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\sqrt{12}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\sqrt{3}\\ \mathit{Dividing}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\mathit{will}\phantom{\rule{0.147em}{0ex}}\mathit{result}\phantom{\rule{0.147em}{0ex}}\mathit{in},\\ \sqrt{12}÷\left(\phantom{\rule{0.147em}{0ex}}\sqrt{3}\right)\phantom{\rule{0.147em}{0ex}}=2\sqrt{3}÷\sqrt{3}=2.\phantom{\rule{0.147em}{0ex}}\mathit{It}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{number}.\\ \\ \mathit{Now}\phantom{\rule{0.147em}{0ex}}\mathit{consider}\phantom{\rule{0.147em}{0ex}}\mathit{another}\phantom{\rule{0.147em}{0ex}}\mathit{two}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\sqrt{15}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\sqrt{5}.\\ \mathit{Dividing}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\mathit{will}\phantom{\rule{0.147em}{0ex}}\mathit{result}\phantom{\rule{0.147em}{0ex}}\mathit{in},\\ 15÷\left(\sqrt{5}\right)=\sqrt{3}.\phantom{\rule{0.147em}{0ex}}\mathit{It}\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}\mathit{an}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{number}.\end{array}$

2. Rational number $$+$$ Irrational number $$=$$ Irrational number.
Example:
$\begin{array}{l}\mathit{Consider}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}5\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{an}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\mathrm{e}.\\ \mathit{Adding}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\mathit{becomes},\\ 5+\mathrm{e}.\phantom{\rule{0.147em}{0ex}}\end{array}$

Adding a rational number with an irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.
3. Rational number $$-$$ Irrational number $$=$$ Irrational number.
Example:
$\begin{array}{l}\mathit{Consider}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}-8\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{an}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\sqrt{19}.\\ \mathit{Subtracting}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\mathit{becomes},\\ -8-\sqrt{19}.\end{array}$

Subtracting a rational number from the irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.
4. Non-zero rational number $$×$$ Irrational number $$=$$ Irrational number.
Example:
$\begin{array}{l}\mathit{Consider}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{non}-\mathit{zero}\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}3\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{an}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\sqrt{7}.\\ \mathit{Multiplying}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\mathit{becomes},\\ 3×\sqrt{7}=3\sqrt{7}.\end{array}$

Multiplication of a non-zero rational number and irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.
5. Non-zero rational number $÷$ Irrational number $$=$$ Irrational number.
Example:
$\begin{array}{l}\mathit{Consider}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{non}-\mathit{zero}\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}-3.2\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{an}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{number}\phantom{\rule{0.147em}{0ex}}\mathrm{\pi }.\\ \mathit{Dividing}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{non}-\mathit{zero}\phantom{\rule{0.147em}{0ex}}\mathit{rational}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{irrational}\phantom{\rule{0.147em}{0ex}}\mathit{numbers}\phantom{\rule{0.147em}{0ex}}\mathit{becomes},\\ -3.2÷\mathrm{\pi }=-3/2\mathrm{\pi }.\end{array}$

Division of a non-zero rational number and irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.