Question

Find three different irrational numbers between the rational numbers $\frac{5}{9}$ and $\frac{5}{7}$.

• A. $\mathrm{0.6010010001...}$
• B. $\mathrm{0.6020020002...}$
• C. $\mathrm{0.6030030003...}$
• D. $0.6020020002$

Answer 1

Answer: (A), (B), (C)

$\mathrm{0.6020020002...}$
$\mathrm{0.6010010001...}$
$\mathrm{0.6030030003...}$

$\frac{5}{9}=\mathrm{0.555....}=0.\overline{5}$ and $\frac{5}{7}=\mathrm{0.7142857142857.....}=0.\overline{714285}$

Here, the two decimal expansions are non-terminating recurring.

Hence, $\frac{5}{9}$ and $\frac{5}{7}$ are two rational numbers.

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

An irrational number has non-terminating non-recurring decimal expansions.

Here, $\mathrm{0.6010010001...}$, $\mathrm{0.5020020002...}$ and $\mathrm{0.62030030003...}$ are non-terminating non-recurring,

while $0.6020020002$ is terminating.

Hence, the required three irrational numbers areHere, $\mathrm{0.6010010001...}$, $\mathrm{0.5020020002...}$ and $\mathrm{0.62030030003...}$.

Thus, options $A$, $B$ and $C$ are the correct answers.