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 rational and irrational is always irrational.
  5. Division of rational and irrational is always irrational.
1. The first property says:
 
i) Irrational number \(+\) Irrational number \(=\) Irrational\(/\)Rational number.
Example:
Consider two irrational numbers \(2+\sqrt{3}\) and \(5+\sqrt{3}\).
Add the numbers.
 \(2+\sqrt{3}+5+\sqrt{3}=7+2\sqrt{3}\).
This is again the irrational number.
 
Now consider another two irrational numbers \(2+\sqrt{3}\) and \(2-\sqrt{3}\).
Add the numbers.
\(2+\sqrt{3}+2-\sqrt{3}=4\).
This is a rational number.
ii) Irrational number \(-\) Irrational number \(=\) Irrational\(/\)Rational number.
Example:
Consider two irrational numbers \(2+\sqrt{3}\) and \(5+\sqrt{3}\).
Substract the numbers.
\(2+\sqrt{3}+(5+\sqrt{3})=-3\).
It is a rational number.
 
Now consider another two irrational numbers \(2+\sqrt{3}\) and \(2-\sqrt{3}\).
Substract the numbers.
\(2+\sqrt{3}-2+\sqrt{3}=2\sqrt{3}\).
It is an irrational number.
iii) Irrational number \(×\) Irrational number \(=\) Irrational\(/\)Rational number.
Example:
Consider two irrational numbers \(2\sqrt{3}\) and \(\sqrt{3}\).
Multiply the numbers.
\(2\sqrt{3}\times \sqrt{3}=2\times 3=6\).
It is a rational number.
 
Now consider another two irrational numbers \(2\sqrt{3}\) and \(\sqrt{2}\).
Multiply the numbers.
\(2\sqrt{3}\times \sqrt{2} = 2\sqrt{6}\).
It is an irrational number.
iv) Irrational number ÷ Irrational number \(=\) Irrational\(/\)Rational number.
Example:
Consider two irrational numbers \(\sqrt{12}\) and \(\sqrt{3}\).
Divide the numbers.
\(\sqrt{12}\div \sqrt{3} = 2\sqrt{3}\div \sqrt{3} = 2\).
It is a rational number.
 
Now consider another two irrational numbers \(sqrt{15}\) and \(\sqrt{5}\).
Divide the numbers.
\(\sqrt{15}\div \sqrt{5} = \sqrt{3}\).
It is an irrational number.
2. Rational number \(+\) Irrational number \(=\) Irrational number.
Example:
Consider a rational number \(5\) and an irrational number \(e\).
Adding the rational and irrational numbers becomes \(5+e\).
 
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:
Consider a rational number \(-8\) and an irrational number \(\sqrt{19}\).
Substracting the rational and irrational numbers becomes \(-8-\sqrt{19}\).
 
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. Rational number \(×\) Irrational number \(=\) Irrational number.
Example:
Consider a rational number \(3\) and an irrational number \(\sqrt{7}\).
Multiplying the rational and irrational numbers becomes,
\(3\times\sqrt{7}=3\sqrt{7}\).
 
Multiplication of a 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. Rational number ÷ Irrational number \(=\) Irrational number.
Example:
Consider a rational number \(-\frac{3}{2}\) and an irrational number \(\pi\).
Dividing the rational and irrational numbers becomes,
\(-\frac{3}{2}\div \pi = -\frac{3}{2\pi}\)
 
Division of a rational number and irrational number will not affect the non-recurring and non-terminating nature of irrational. Thus, the resultant is an irrational number.