Classify the following numbers as rational or irrational: Class 9
Classify the following numbers as rational or irrational: Class 9
Question 1.
Classify the following numbers as rational or irrational: Class 9
i. $\sqrt{81}$
ii. $\sqrt{12}$
iii. 0.33333...
iv. 0.123451234512345...
v. 1.01001000100001... (Notice the pattern: Is it repeating a single block?)
vi. 23.560185612239874790120
Find the explicit fractions in case they are rational.
Solution:
i. $\sqrt{81}$ = 9 = $\frac{9}{1}$
∴ It is a rational number.
ii. $\sqrt{12}$ = 2$\sqrt{3}$
It cannot be expressed in the form $\frac{p}{q}$.
∴ It is an irrational number.
iii. 0.3333... is a repeating decimal.
Let x = 0.3333...
Since one digit repeats, multiply both sides by 10:
10x = 3.3333...
Subtracting the equations,
10x - x = 3.3333... -0.3333...
∴ 9x = 3
∴ x = $\frac{3}{9}=\frac{1}{3}$
∴ It is a rational number.
iv. 0.1234512345... is a repeating decimal.
Let x = 0.1234512345...
Since five digits repeat, multiply both sides by 105:
100000x = 12345.12345...
Subtracting the equations,
100000x - x = 12345.12345...- 0.1234512345...
∴ 99999x = 12345
∴ x = $\frac{12345}{99999}=\frac{4115}{33333}$
∴ It is a rational number.
v. 1.01001000100001... is non-terminating and non-repeating.
∴ It is an irrational number.
vi. 23.560185612239874790120 is a terminating decimal.
∴ It is a rational number.
Since there are 21 decimal places,
Fraction = $\frac{23560185612239874790120}{1000000000000000000000}$