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
Represent $ \sqrt{9.3}$ on the number line.
Represent $ \sqrt{9.3}$ on the number line.
Classroom activity (Constructing the 'square root spiral') : Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point $O$ and draw a line segment $OP_1$ of unit length. Draw a line segment $P_1P_2$ perpendicular to $OP_1$ of unit length (see Fig.). Now draw a line segment $P_2P_3$ perpendicular to $OP_2$. Then draw a line segment $P_3P_4 $ perpendicular to $OP_3$. Continuing in this manner, you can get the line segment $P_{n-1}P_n$ by drawing a line segment of unit length perpendicular to $OP_{n-1}$. In this manner, you will have created the points $P_2$, $P_3$,...., $P_n$,... ., and joined them to create a beautiful spiral depicting $\sqrt 2,\, \sqrt 3, \,\sqrt 4$, ..............
Classroom activity (Constructing the 'square root spiral') : Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point $O$ and draw a line segment $OP_1$ of unit length. Draw a line segment $P_1P_2$ perpendicular to $OP_1$ of unit length (see Fig.). Now draw a line segment $P_2P_3$ perpendicular to $OP_2$. Then draw a line segment $P_3P_4 $ perpendicular to $OP_3$. Continuing in this manner, you can get the line segment $P_{n-1}P_n$ by drawing a line segment of unit length perpendicular to $OP_{n-1}$. In this manner, you will have created the points $P_2$, $P_3$,...., $P_n$,... ., and joined them to create a beautiful spiral depicting $\sqrt 2,\, \sqrt 3, \,\sqrt 4$, ..............
Visualise $4. \overline{26}$ . on the number line, up to $4$ decimal places.
Visualise $4. \overline{26}$ . on the number line, up to $4$ decimal places.
Show how $\sqrt 5$ can be represented on the number line.
Show how $\sqrt 5$ can be represented on the number line.
State whether the following statements are true or false. Give reasons for your answers.
$(i)$ Every natural number is a whole number.
$(ii)$ Every integer is a whole number.
$(iii)$ Every rational number is a whole number
State whether the following statements are true or false. Give reasons for your answers.
$(i)$ Every natural number is a whole number.
$(ii)$ Every integer is a whole number.
$(iii)$ Every rational number is a whole number