Question

Find five irrational numbers between 4 and 5.

Answer 1

Hint: Use the formula “irrational number between ‘a’ and ‘b’ = $\sqrt{ab}$” and find the irrational number in between given two numbers. Then use the same formula to find remaining irrational numbers in between.

Complete step by step answer:
TO solve the given question we will write the given numbers first and assume them as ‘a’ and ‘b’ respectively, therefore,
a = 4 and b = 5 …………………………………………………… (1)
As we have to insert five irrational numbers between given two numbers and for that we will insert one irrational number between ‘a’ and ‘b’ and to do that we should know the concept given below,
Concept: If two numbers ‘a’ and ‘b’ are such that their product ‘ab’ is not a perfect square then the irrational number between these two numbers in given by $\sqrt{ab}$
Therefore the irrational number between ‘a’ and ‘b’ = $\sqrt{ab}$ ……………………………………………. (2)
As we know that the value of $4\times 5$ is 20 which is not a perfect square therefore if we put the values of equation (1) in the above equation we will get,
Therefore irrational number between 4 and 5 = $\sqrt{4\times 5}$
As we know the square root of 4 is 2 therefore above equation will become,
Therefore irrational number between 4 and 5 = $2\sqrt{5}$ …………………………………………………… (3)
Now we have the numbers as 4, $2\sqrt{5}$, and 5. We have inserted one irrational number in the given two numbers now we will insert one irrational number in 4 and $2\sqrt{5}$.
As we know that the product of 4 and $2\sqrt{5}$ is $8\sqrt{5}$ which is not a perfect square therefore we can use the formula of equation (2) therefore we will get,
Therefore irrational number between 4 and \[2\sqrt{5}\] = \[\sqrt{4\times 2\sqrt{5}}\]
As we know that the square root of 4 is 2 therefore above equation will become,
Therefore irrational numbers between 4 and \[2\sqrt{5}\] = \[2\sqrt{2\sqrt{5}}\] …………………………………………. (4)
Now we will insert another irrational number in 4 and \[2\sqrt{2\sqrt{5}}\] therefore we will get by using equation (2) therefore we will get,
Therefore irrational number between 4 and \[2\sqrt{2\sqrt{5}}\] = \[\sqrt{4\times 2\sqrt{2\sqrt{5}}}\]
Further simplification in the above equation will give,
Therefore irrational number between 4 and \[2\sqrt{2\sqrt{5}}\] = \[2\sqrt{2\sqrt{2\sqrt{5}}}\] ………………………………… (5)
Now we will insert a irrational number between $2\sqrt{5}$ and 5, therefore by using equation (2) we will get,
Therefore irrational number between $2\sqrt{5}$ and 5 = \[\sqrt{2\sqrt{5}\times 5}\]
Therefore irrational number between $2\sqrt{5}$ and 5 = \[\sqrt{10\sqrt{5}}\] …………………………………………….. (6)
Also by using equation (2) the irrational number between \[\sqrt{10\sqrt{5}}\] and 5 is given by,
Therefore irrational number between \[\sqrt{10\sqrt{5}}\] and 5 = \[\sqrt{\sqrt{10\sqrt{5}}\times 5}\]
Therefore irrational number between \[\sqrt{10\sqrt{5}}\] and 5 = \[\sqrt{5\sqrt{10\sqrt{5}}}\] ………………………………………. (7)
From equation (3), equation (4), equation (5), equation (6) and equation (7) we can write,
The five irrational numbers between 4 and 5 are given by \[2\sqrt{2\sqrt{2\sqrt{5}}}\], \[2\sqrt{2\sqrt{5}}\], $2\sqrt{5}$,\[\sqrt{10\sqrt{5}}\] , \[\sqrt{5\sqrt{10\sqrt{5}}}\]. Therefore the numbers can be represented as,
4, \[2\sqrt{2\sqrt{2\sqrt{5}}}\], \[2\sqrt{2\sqrt{5}}\], $2\sqrt{5}$,\[\sqrt{10\sqrt{5}}\] , \[\sqrt{5\sqrt{10\sqrt{5}}}\], 5

Note: If you are solving it in a competitive exam then you can directly take the root of the product of two numbers between which you are considering the irrational number. While inserting the number always take one corner number to reduce the calculations.