Question

Write the irrational numbers between 2 and 7.

Answer 1

Hint:In order to this question, to find out the irrational numbers between 2 and 7, we will first explain the properties and the criteria about the irrational numbers and then we will write the set of irrational numbers on the basis of the criteria of irrational numbers between 2 and 7.

Complete step by step answer:
Irrational numbers are real numbers that cannot be represented as integer ratios. For example, $\sqrt 2 $ is an irrational number. Also, the decimal expansion of an irrational number is neither terminating nor recurrent. Irrational numbers are real numbers that cannot be expressed in $\dfrac{p}{q}$ form, where $p\,and\,q$ are integers and $q$ is not equal to zero.

The numbers $\sqrt 2 $ and $\sqrt 3 $ are examples of irrational numbers. Any number in the form $\dfrac{p}{q}$ , where $p\,and\,q$ are integers and $q$ is not zero, is considered a rational number. And, $pi(\pi )$ is an irrational number because it is non-terminating. The approximate value of pi is $\dfrac{{22}}{7}\,or\,3.141$ .

Now, we will find the irrational numbers between 2 and 7: as we know, the square root numbers except the perfect square root numbers are irrational numbers.
$2 = \sqrt 4 \\
\Rightarrow 7 = \sqrt {49} \\ $
So, the numbers lying between $\sqrt 4 \,and\,\sqrt {49} $ are irrational numbers, except the square root number which is the perfect square of any number.
$\sqrt 9 = 3 \\
\Rightarrow \sqrt {16} = 4 \\
\Rightarrow \sqrt {25} = 5 \\
\Rightarrow \sqrt {36} = 6 \\ $
These are not irrational numbers.

Therefore, the irrational numbers between 2 and 7 are \[\sqrt 5 ,\sqrt 6 ,\sqrt 7 ,\sqrt 8 ,\sqrt {10} ,\sqrt {11} ,\sqrt {12} ,\sqrt {13} ,\sqrt {14} ,\sqrt {15} ,\sqrt {17} \ldots \ldots \ldots \ldots \ldots \ldots .\sqrt {48} \].

Note:Since irrational numbers are the subsets of the real numbers, irrational numbers will obey all the properties of the real number system. The following are the properties of irrational numbers:
-When an irrational number and a rational number are added together, the result is an irrational number. Assume that $x$ is an irrational number and $y$ is a rational number, and that adding the two numbers $x + y$ yields the rational number $z$ .
-Any irrational number multiplied by any nonzero rational number yields an irrational number. Assume that if \[xy = z\] is rational, then $x = \dfrac{z}{y}$ is rational, which contradicts the irrationality of $x$ . As a result, the $xy$ product must be illogical.