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 ?
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 ?
We are given that $\frac{1}{7}=0 . \overline{142857}$
$\frac{2}{7}=2 \times \frac{1}{7}=2 \times(0 . \overline{142857})=0 . \overline{285714}$
$\frac{3}{7}=3 \times \frac{1}{7}=3 \times(0 . \overline{142857})=0.4 \overline{28571}$
$\frac{4}{7}=4 \times \frac{1}{7}=4 \times(0 . \overline{142857})=0 . \overline{571428}$
$\frac{5}{7}=5 \times \frac{1}{7}=5 \times(0 . \overline{142857})=0 . \overline{714285}$
$\frac{6}{7}=6 \times \frac{1}{7}=6 \times(0 . \overline{142857})=0 . \overline{857142}$
Thus, without actually doing the long division we can predict the decimal expansions of the above given rational numbers.
Similar Questions
Classify the following numbers as rational or irrational :
$(i)$ $\sqrt{23}$
$(ii)$ $\sqrt{225}$
$(iii)$ $0.3796$
$(iv)$ $7.478478 \ldots$
$(v)$ $1.101001000100001 \ldots$
Classify the following numbers as rational or irrational :
$(i)$ $\sqrt{23}$
$(ii)$ $\sqrt{225}$
$(iii)$ $0.3796$
$(iv)$ $7.478478 \ldots$
$(v)$ $1.101001000100001 \ldots$
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 five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.
Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.
Simplify the following expressions :
$(i)$ $(5+\sqrt{7})(2+\sqrt{5})$
$(ii)$ $(5+\sqrt{5})(5-\sqrt{5})$
$(iii)$ $(\sqrt{3}+\sqrt{7})^{2}$
$(iv)$ $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$
Simplify the following expressions :
$(i)$ $(5+\sqrt{7})(2+\sqrt{5})$
$(ii)$ $(5+\sqrt{5})(5-\sqrt{5})$
$(iii)$ $(\sqrt{3}+\sqrt{7})^{2}$
$(iv)$ $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$
State whether the following statements are true or false. Justify your answers.
$(i)$ Every irrational number is a real number.
$(ii)$ Every point on the number line is of the form $\sqrt m$ , where $m$ is a natural number.
$(iii)$ Every real number is an irrational number.
State whether the following statements are true or false. Justify your answers.
$(i)$ Every irrational number is a real number.
$(ii)$ Every point on the number line is of the form $\sqrt m$ , where $m$ is a natural number.
$(iii)$ Every real number is an irrational number.