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 $q \ne 0$.
We have $3.142678=\frac{3142678}{1000000}$, and hence is a rational number.
Now, let us consider the case when the decimal expansion is non-terminating recurring.
Visualise $3.765$ on the number line, using successive magnification.
Simplify
$(i)$ $2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}}$
$(ii)$ $\left(3^{\frac{1}{5}}\right)^{4}$
$(iii)$ $\frac{7^{\frac{1}{5}}}{7^{\frac{1}{3}}}$
$(iv)$ $13^{\frac{1}{5}} \cdot 17^{\frac{1}{5}}$
Show how $\sqrt 5$ can be represented on the number line.
Show that $0.3333... =$ $0 . \overline{3}$ can be expressed in the form $\frac {p }{q }$ , where $p$ and $q$ are integers and $q \ne 0$.
Find an irrational number between $\frac {1}{7}$ and $\frac {2}{7}$