Look at several examples of rational numbers in the form $\frac{p}{q}(q \neq 0),$ where $p$ and $q$ are integers with no common factors other than $1$ and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy ?
Look at several examples of rational numbers in the form $\frac{p}{q}(q \neq 0),$ where $p$ and $q$ are integers with no common factors other than $1$ and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy ?
Let us look at decimal expansion of the following terminating rational numbers :
$\frac{3}{2}=\frac{3 \times 5}{2 \times 5}=\frac{15}{10}=1.5$ $\left[\right.$ Denominator $\left.=2=2^{1}\right]$
$\frac{1}{5}=\frac{1 \times 2}{5 \times 2}=\frac{2}{10}=0.2$ $\left[\right.$ Denominator $\left.=5=5^{1}\right]$
$\frac{7}{8}=\frac{7 \times 125}{8 \times 125}=\frac{875}{1000}=0.875$ $\left[\right.$ Denominator $\left.=8=2^{3}\right]$
$\frac{8}{125}=\frac{8 \times 8}{125 \times 8}=\frac{64}{1000}=0.064$ $\left[\right.$ Denominator $\left.=125=5^{3}\right]$
$\frac{13}{20}=\frac{13 \times 5}{20 \times 5}=\frac{65}{100}=0.65$ $\left[\right.$ Denominator $\left.=20=2^{2} \times 5^{1}\right]$
$\frac{17}{16}=\frac{17 \times 625}{16 \times 625}=\frac{10625}{10,000}=1.0625$ $\left[\right.$ Denominator $\left.=16=2^{4}\right]$
We observe that the prime factorisation of $q$ (i.e. denominator) has only powers of $2$ or powers of $5$ or powers of both.
Similar Questions
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 ?
Check whether $7 \sqrt{5}, \,\frac{7}{\sqrt{5}}, \,\sqrt{2}+21, \,\pi-2$ are irrational numbers or not.
Check whether $7 \sqrt{5}, \,\frac{7}{\sqrt{5}}, \,\sqrt{2}+21, \,\pi-2$ are irrational numbers or not.
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) $ $125^{\frac{1}{3}}$
Is zero a rational number ? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$ ?
Is zero a rational number ? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$ ?
Find :
$(i)$ $9^{\frac{3}{2}}$
$(ii)$ $32^{\frac{2}{5}}$
$(iii)$ $16^{\frac{3}{4}}$
$(iv)$ $125^{\frac{-1}{3}}$
Find :
$(i)$ $9^{\frac{3}{2}}$
$(ii)$ $32^{\frac{2}{5}}$
$(iii)$ $16^{\frac{3}{4}}$
$(iv)$ $125^{\frac{-1}{3}}$