Question

Classify the following as rational and irrational numbers:
i) \[\sqrt{7}\]
ii) \[3+\sqrt{2}\]
iii) \[-7-\sqrt{3}\]
iv) \[\sqrt{4}\]
v) 5.7
vi) 0.73457456
viii) \[\dfrac{\pi }{2}\]
ix) 1.7324
x) 1.6244265

Answer 1

Hint: We have to solve this type of question with the concept of rational and irrational numbers. We have to first know the conditions of a rational and irrational number.
Then, using those conditions we can classify the given numbers as rational or irrational.

Complete step by step solution:
According to the question, we are asked to find whether the number is rational or irrational.
Let us first know the conditions of rational numbers.
1) The numbers in the form \[\dfrac{p}{q}\], where p and q are integers and \[q\ne 0\].
2) The decimal numbers where the number of digits after the decimal is finite and countable.
3) The decimal with infinite numbers after decimal but is repeating after an interval is also a rational number.
The conditions for irrational number are
1) The numbers in the form \[\dfrac{p}{q}\], where p and q are not integers and \[q\ne 0\].
2) The decimal number with an infinite number of digits after the decimal is an irrational number.
3) Square or cubic roots which are not perfect is also an irrational number.

Let us now classify the given numbers.
i) \[\sqrt{7}\]
\[\sqrt{7}\] can be written as \[\dfrac{\sqrt{7}}{1}\] which is of the form \[\dfrac{p}{q}\].
 \[\sqrt{7}\] is not an integer.
Therefore, \[\sqrt{7}\] is an irrational number.

ii) \[3+\sqrt{2}\]
Here, 3 is an integer and thus is a rational number.
And \[\sqrt{2}\] can be written as \[\dfrac{\sqrt{2}}{1}\] which is of the form \[\dfrac{p}{q}\].
\[\sqrt{2}\] is not an integer.
Addition of a rational number and an irrational number is always a rational number.
Therefore, \[3+\sqrt{2}\] is an irrational number.

iii) \[-7-\sqrt{3}\]
Here, -7 is an integer and thus is a rational number.
And \[-\sqrt{3}\] can be written as \[\dfrac{-\sqrt{3}}{1}\] which is of the form \[\dfrac{p}{q}\].
\[-\sqrt{3}\]is not an integer.
Addition of a rational number and an irrational number is always a rational number.
Therefore, \[-7-\sqrt{3}\] is an irrational number.

iv) \[\sqrt{4}\]
We know that \[\sqrt{4}=\pm 2\] and \[\pm 2\] is an integer.
All the integers are a rational number.
Therefore, \[\sqrt{4}\] is a rational number.

v) 5.7
Here, we have only one digit after the decimal.
Therefore, 5.7 is a rational number.

vi) 0.73457456
Here, we observe that there are a few numbers of digits after the decimal.
There is only a countable number of terms in the decimal.
Therefore, 0.73457456 is a rational number.

viii) \[\dfrac{\pi }{2}\]
We know that the value of pi is \[\pi =3.1459265358....\].
We find that the value of \[\pi \] has an infinite number of digits after the decimal.
Thus, \[\pi \] is an irrational number.
On dividing an irrational number by 2, we get an irrational number only.
Therefore, \[\dfrac{\pi }{2}\] is an irrational number.

ix) 1.7324
Here, we observe that there are a few numbers of digits after the decimal.
There is only a countable number of terms in the decimal.
Therefore, 1.7324 is a rational number.

x) 1.6244265
Here, we observe that there are a few numbers of digits after the decimal.
There is only a countable number of terms in the decimal.
Therefore, 1.6244265 is a rational number.

Note: We should not get confused with conditions of rational and irrational numbers. Also an irrational number added or subtracted with an integer is always an irrational number. A decimal number with infinite number of digits after the decimal point which is repeating after an interval is a rational number and not an irrational number.