Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.
There are infinite rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$
$\frac{3}{5}=\frac{3 \times 6}{5 \times 6}=\frac{18}{30}$
$\frac{4}{5}=\frac{4 \times 6}{5 \times 6}=\frac{24}{30}$
Therefore, $5$ rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ (i.e. $\frac{18}{30}$ $ \frac{24}{30})$ are
$\frac{19}{30}, \,\frac{20}{30},\, \frac{21}{30}, \,\frac{22}{30},\, \frac{23}{30}$
Show that $3.142678$ is a rational number. In other words, express $3.142678$ in the form $\frac {p }{q }$, where $p$ and $q$ are integers and $q \ne 0$.
You know that $\frac{1}{7}=0 . \overline{142857}$. Can you predict what the decimal expansions of $\frac{2 }{7},\, \frac{3}{7}$, $\frac{4}{7},\, \frac{5}{7}, \,\frac{6}{7}$ are, without actually doing the long division ? If so, how ?
Express $0.99999 \ldots$ in the form $\frac{p}{q}$. Are you surprised by your answer ? With your teacher and classmates discuss why the answer makes sense.
Classify the following numbers as rational or irrational :
$(i)$ $2-\sqrt{5}$
$(ii)$ $(3+\sqrt{23})-\sqrt{23}$
$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$
$(iv)$ $\frac{1}{\sqrt{2}}$
$(v)$ $2 \pi$
Rationalise the denominator of $\frac{1}{\sqrt{2}}$.