Question

Question 3
Insert a rational number and an irrational number between the following
(i) 2 and 3
(ii) 0 and 0.1
(iii) $\frac{1}{3}$ and $\frac{1}{2}$
(iv) $\frac{-2}{5}$ and $\frac{1}{2}$
(v) 0.15 and 0.16
(vi) $\sqrt{2}$ and $\sqrt{3}$
(vii) 2.357 and 3.121
(viii) .0001 and .001
(ix) 3.623623 and 0.484848
(x) 3.375289 and 6.375738

Answer 1

We know that, there are infinitely many rational and irrational values between any two numbers.

(i) A rational number between 2 and 3 is 2.1
To find an irrational number between 2 and 3 . Find a number which is non-terminting non-recurring lying between them,
$\therefore$ Such number will be 2.040040004…….

(ii) A rational number between 0 and 0.1 is 0.05 and an irrational number between 0 and 0.1 is 0.007000700007……

(iii) A rational number between $\frac{1}{3}$ and $\frac{1}{2}$ i.e $\frac{4}{12}$ and $\frac{6}{12}$ is $\frac{5}{12}$. An irrational number between $\frac{1}{3}$ and $\frac{1}{2}$ i.e between $0.\overline{3}$ and 0.5 is 0.4141141114.....

(iv) A rational number between $\frac{-2}{5}$ and $\frac{1}{2}$ i.e between – 0.4 and 0.5 is 0.1 and an irrational between $\frac{-2}{5}$ and $\frac{1}{2}$ i.e between – 0.4 and 0.5 is 0.151151115……

(v) A rational number between 0.15 and 0.16 is 0.151. An irrational number between 0.15 and 0.16 is 0.1515515551……

(vi) A rational number between $\sqrt{2}$ and $\sqrt{3}$ i.e between 1.4142 …..and 1.7320…..is 1.5
An irrational number between $\sqrt{2}$ and $\sqrt{3}$ is 1.585585558……

(vii) A rational number between 2.357 and 3.121 is 3. An irrational number between 2.357 and 3.121 is 3.101101110…..

(viii) A rational number between 0.0001 and 0.001 is 0.00011. An irrational number between
0.0001 and 0.001 is 0.0001131331333……..

(ix) A rational number between 3.623623 and 0.484848 is 1. An irrational number between 3.623623 and 0.484848 is 1.909009000…..

(x) A rational number between 6.375289 and 6.375738 is 6.3753. An irrational number between 6.375289 and 6.375738 is 6.375414114111…..