Show that $0.3333... =$ $0 . \overline{3}$ can be expressed in the form $\frac {p }{q }$ , where $p$ and $q$ are integers and $q \ne 0$.
Show that $0.3333... =$ $0 . \overline{3}$ can be expressed in the form $\frac {p }{q }$ , where $p$ and $q$ are integers and $q \ne 0$.
since we do not know what $0 . \overline{3}$ is , let us call it $'x'$ and so
$x=0.3333 \ldots$
Now here is where the trick comes in. Look at
$10 x=10 \times(0.333 \ldots)=3.333 \ldots$
Now, $3.3333 \ldots=3+x,$ since $x=0.3333 \ldots$
Therefore, $10 x=3+x$
Solving for $x,$ we get
$9 x=3,$ i.e., $x=\frac{1}{3}$
Similar Questions
Find :
$(i)$ $64^{\frac{1}{2}}$
$(ii)$ $32^{\frac{1}{5}}$
$(iii) $ $125^{\frac{1}{3}}$
Find :
$(i)$ $64^{\frac{1}{2}}$
$(ii)$ $32^{\frac{1}{5}}$
$(iii) $ $125^{\frac{1}{3}}$
Are the following statements true or false ? Give reasons for your answers.
$(i)$ Every whole number is a natural number.
$(ii)$ Every integer is a rational number.
$(iii)$ Every rational number is an integer.
Are the following statements true or false ? Give reasons for your answers.
$(i)$ Every whole number is a natural number.
$(ii)$ Every integer is a rational number.
$(iii)$ Every rational number is an integer.
Find an irrational number between $\frac {1}{7}$ and $\frac {2}{7}$
Find an irrational number between $\frac {1}{7}$ and $\frac {2}{7}$
Find six rational numbers between $3$ and $4$.
Find six rational numbers between $3$ and $4$.
Classify the following numbers as rational or irrational :
$(i)$ $2-\sqrt{5}$
$(ii)$ $(3+\sqrt{23})-\sqrt{23}$
$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$
$(iv)$ $\frac{1}{\sqrt{2}}$
$(v)$ $2 \pi$
Classify the following numbers as rational or irrational :
$(i)$ $2-\sqrt{5}$
$(ii)$ $(3+\sqrt{23})-\sqrt{23}$
$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$
$(iv)$ $\frac{1}{\sqrt{2}}$
$(v)$ $2 \pi$