Question

Find an irrational number between \[3\] and \[4\] .

Answer 1

Hint: The given question is solved on the basis of the irrational number system. For that we know about the basic concept of irrational and rational numbers. On the basis of general facts we can solve the above problem irrational are those which are not in the form of \[\dfrac{p}{q}\] or we can say that it is not in the form of faction on that may we can conclude our result.

Complete step by step solution:
Any number which is not in the form of \[\dfrac{p}{q}\] is called the Irrational number. Like integers, decimals, roots….. so on.
Integers including numbers like $ 1,2,3,4, $ ………….. so on.
Decimals including numbers like $ 1.2,2.2,3.2,4.006, $ ……………. so on.
Numbers having roots like $ \sqrt {2,} \sqrt {30} ,\sqrt {20} ,\sqrt {15} , $ …………………. so on.
For solving the given question we are considered rational numbers.
Let $ a $ and $ b $ are two posture rational numbers.
 $ ab $ is not a perfect square of a rational number, which means it gives the value in $ \dfrac{{\sqrt {} }}{{root}} $ form as well.
We can say that $ \sqrt {ab} $ is an irrational number. It is lying between $ a $ and $ b $
From the given statement the irrational number is lying between \[3\] and \[4\] .
We are considering both the numbers to solve the given problem.
We have to take the root of both \[3\] and \[4\] as we can say that the irrational number is \[\sqrt {ab} \] .
 So, \[3\] and \[4\] can be written as..
 \[ = \sqrt {3 \times 4} \]
Separating the root of \[3\] and \[4\] in above term we get
 \[ = \sqrt 3 \times \sqrt 4 \]
Now solving the root of $ 4 $ i.e. \[\sqrt 4 \]
We get $ 2 $ , the equation becomes
 \[ = \sqrt 3 \times 2\]
 \[ = 2\sqrt 3 \]
So the irrational number lies between \[3\] and \[4\] is \[2\sqrt 3 \]
The given question is based on rational and irrational numbers.
So, the correct answer is “ \[2\sqrt 3 \] ”.

Note: There are so many numbers that can lie between \[3\] and \[4\] we can find a number \[2\sqrt 3 \] that lies between \[3\] and \[4\] . The practical application of irrational numbers is finding the circumference of a circle irrational numbers fill all the holes that exist in a set of rational numbers and make it possible to study continuity, derivatives …… so on.