Question

Calculate the number of irrational numbers between 1 and 8.

Answer 1

Hint: Irrational numbers are the numbers that are expressed in decimal form and are non-terminating.
Here, we have to find the number of irrational numbers between two whole numbers 1 and 8. The definitions of rational and irrational numbers and how to find irrational numbers between 2 numbers is discussed below.

Complete step-by-step answer:
In this question, we are given two integers and we need to calculate how many irrational numbers lie between them.
Given numbers: 1 and 8.
Now, first of let us see what rational and irrational numbers are.
Rational numbers are numbers that can be expressed as a fraction. They can be positive, negative or zero. Rational numbers are of the form $\dfrac{p}{q}$, where $q \ne 0$.
For example: $\dfrac{1}{2}, - \dfrac{3}{4},\dfrac{0}{5}$, etc
Irrational numbers are written in decimal form. They are non-terminating decimals. The numbers that are not rational are called irrational numbers.
For example: $\dfrac{{22}}{7} = 3.143...$
Difference between Rational and Irrational numbers:

Rational Numbers	Irrational Numbers
They can be expressed in $\dfrac{p}{q}$ form.	They are expressed in decimal form and cannot be expressed in $\dfrac{p}{q}$ form.
Includes perfect squares	Includes surds.
Terminating.	Non – Terminating.

Now, there are infinite points between two integers. Hence, we can say that there are infinite irrational numbers between 1 and 8.
So, the correct answer is “infinite”.

Note: Here, we can find some irrational numbers between 1 and 8 by squaring method.
First of all, square 1 and 8.
Now, choose any integer between 1 and 8. Let us say we choose 3 and 5. Therefore, we get
$ \Rightarrow 1 < 3 < 64$ $ \Rightarrow 1 < 5 < 64$
Taking square root, we get
$
   \Rightarrow \sqrt 1 < \sqrt 3 < \sqrt {64} \\
   \Rightarrow 1 < \sqrt 3 < 8 \\
 $ $
   \Rightarrow \sqrt 1 < \sqrt 5 < \sqrt {64} \\
   \Rightarrow 1 < \sqrt 5 < 8 \;
 $
Hence, the two irrational numbers between 1 and 8 are $\sqrt 3 $ and $\sqrt 5 $.