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
Represent $ \sqrt{9.3}$ on the number line.
Represent $ \sqrt{9.3}$ on the number line.
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.
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
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Show that $0.2353535 \ldots=0.2 \overline{35}$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$.
Show that $0.2353535 \ldots=0.2 \overline{35}$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$.