Question

State whether the following numbers are rational or irrational.
a) 1.5 b) \[\sqrt 7 \] c) \[0.\dot 7\]
d) \[0.7\ddot 3\] e) \[\sqrt {121} \] f) \[\pi \]

Answer 1

Hint: The rational numbers are numbers which can be expressed as a fraction and also as positive numbers, negative numbers and zero. It can be written as \[\dfrac{p}{q}\], where q is not equal to zero.
Irrational numbers could be written in decimals but not in the form of fractions which means it cannot be written as the ratio of two integers.
Irrational numbers have endless non-repeating digits after the decimal point.

Complete step-by-step answer:
a) The given number is \[1.5\]
This number can be represented as:
\[
   = \dfrac{{15}}{{10}} \\
   = \dfrac{3}{2} \\
 \]
Now since, it is expressed as fraction and can be written in the form of \[\dfrac{p}{q}\], where q is not equal to zero.
Therefore, it is a rational number.

b) The given number is \[\sqrt 7 \]
Now if we evaluate the value of $\sqrt 7 $ we get:-
$\sqrt 7 = 2.645751.....$
Now since the given number cannot be expressed as fraction and can be written in the form of \[\dfrac{p}{q}\]
Also, it has endless non repeating digits after decimal point
Therefore, it is an irrational number.

c) The given number is $0.\dot 7$
This number can be represented as:
$ = \dfrac{7}{{10}}$
Now since, it is expressed as fraction and can be written in the form of \[\dfrac{p}{q}\], where q is not equal to zero.
Therefore, it is a rational number.

d) The given number is \[0.7\ddot 3\]
This number can be represented as:
\[ = \dfrac{{73}}{{100}}\]
Now since, it is expressed as fraction and can be written in the form of \[\dfrac{p}{q}\], where q is not equal to zero.
Therefore, it is a rational number.

e) The given number is \[\sqrt {121} \]
The number 121 is the square of 11
Therefore this number can be represented as:
\[
   = \sqrt {{{\left( {11} \right)}^2}} \\
   = 11 \\
 \]
Now since the number 11 can be represented as:-
\[ = \dfrac{{11}}{1}\]
Hence, it is expressed as fraction and can be written in the form of \[\dfrac{p}{q}\], where q is not equal to zero.
Therefore, it is a rational number.

f) The given number is \[\pi \].
As we know that
\[\pi = 3.14159265358979..........\]
Now since the given number cannot be expressed as fraction and can be written in the form of \[\dfrac{p}{q}\]
Also, it has endless non repeating digits after decimal point
Therefore, it is an irrational number.

Note: Students should keep in mind that rational numbers are terminating or recurring numbers while irrational numbers are neither terminating nor recurring.
Also, in case of \[\pi \] it can be written in fraction as \[\dfrac{{22}}{7}\] but it is an approximate value and not the exact value
Hence \[\pi \] is always irrational.