Locate $\sqrt 2$ on the number line.
Locate $\sqrt 2$ on the number line.
It is easy to see how the Greeks might have discovered $\sqrt 2$ . Consider a square $OABC$, with each side $1$ unit in length (see Fig. $1$). Then you can see by the Pythagoras theorem that $OB =\sqrt{1^{2}+1^{2}}=\sqrt{2}$. How do we represent $\sqrt 2$ on the number line ? This is easy. Transfer Fig $1$. onto the number line making sure that the vertex $O$ coincides with zero (see Fig.$2$)
We have just seen that $OB = \sqrt 2 $. Using a compass with centre $O$ and radius $OB$, draw an arc intersecting the number line at the point $P$. Then $P$ corresponds to $\sqrt 2$ on the number line.
Similar Questions
Find :
$(i)$ $9^{\frac{3}{2}}$
$(ii)$ $32^{\frac{2}{5}}$
$(iii)$ $16^{\frac{3}{4}}$
$(iv)$ $125^{\frac{-1}{3}}$
Find :
$(i)$ $9^{\frac{3}{2}}$
$(ii)$ $32^{\frac{2}{5}}$
$(iii)$ $16^{\frac{3}{4}}$
$(iv)$ $125^{\frac{-1}{3}}$
Rationalise the denominators of the following :
$(i)$ $\frac{1}{\sqrt{7}}$
$(ii)$ $\frac{1}{\sqrt{7}-\sqrt{6}}$
$(iii)$ $\frac{1}{\sqrt{5}+\sqrt{2}}$
$(iv)$ $\frac{1}{\sqrt{7}-2}$
Rationalise the denominators of the following :
$(i)$ $\frac{1}{\sqrt{7}}$
$(ii)$ $\frac{1}{\sqrt{7}-\sqrt{6}}$
$(iii)$ $\frac{1}{\sqrt{5}+\sqrt{2}}$
$(iv)$ $\frac{1}{\sqrt{7}-2}$
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 $\sqrt 2,\, \sqrt 3, \,\sqrt 4$, ..............
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.
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.
Locate $\sqrt 3$ on the number line.
Locate $\sqrt 3$ on the number line.