Question

# Insert a rational number and an irrational number between the following:$$\dfrac {1}{3}$$ and $$\dfrac {1}{2}$$.

Solution

## $$\dfrac {1}{3} = \dfrac {1}{3} \times \dfrac {4}{4} = \dfrac {4}{12}$$ and $$\dfrac {1}{2} = \dfrac {1}{2}\times \dfrac {6}{6} = \dfrac {6}{12}$$Now, $$\dfrac {5}{12}$$ is a rational number between $$\dfrac {4}{12}$$ and $$\dfrac {6}{12}$$. So, $$\dfrac {5}{12}$$ is a rational number lying between $$\dfrac {1}{3}$$ and $$\dfrac {1}{2}$$.Again, $$\dfrac {1}{2} = 0.33333...$$ and $$\dfrac {1}{2} = 0.5$$.Now, $$0.414114111...$$ is a non-terminating and non-recurring decimal.Hence, $$0.414114111...$$ is an irrational number lying between $$\dfrac {1}{3}$$ and $$\dfrac {1}{2}$$.Maths

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