Irrational Numbers Between 5 and 6?

In number system, there are two Types of Numbers– rational number and irrational number and both are part of Real Numbers in mathematics. There are certain properties which exist between Rational and Irrational Numbers -
Property 1: When we add two Rational Numbers, it produces rational number like 2 + 5 = 7.
Property 2: When we add an irrational number and a rational number, then it produces an irrational number like 2 + √5 = 2 + √5, which is an irrational number.
Now we discuss how we calculate an irrational numbers between 5 and 6:
We use following steps for evaluation of irrational numbers between 5 and 6 -
Step 1: First of all, we make a relation between given two rational numbers like we have two rational numbers r₁ = 5 and r₂ = 6, where r₂ is greater than r₁, so relationship between r₁ and r₂ is -
r₁ < r₂ or 5 < 6,
Now we assume a variable ‘z’, which exists between 5 and 6, then-
5 < z < 6 …..................equation (1),
Step 2: Now, after evaluation of certain Relations, we perform arithmetic operation on given equation (1)-
Like we have a relation 5 < z < 6.
Now we subtract √11 from both sides of rational number, then difference between the given two rational numbers is same, means ‘5 - √11’ and ‘6 - √11’ is same value,
So our generated equation is-
5 - √11 < z < 6 - √11 …........equation (2),
Step 3: Now we simplify the equation(2) -
5 - √11 < z < 6 - √11,
=> 5 < z + √11 < 6,
Now from property 2, we know that addition between irrational number and any rational number produces an irrational number. So, here ‘z + √11’ is an irrational number.