Find the decimal expansions of $\frac{10}{3},\, \frac{7}{8}$ and $\frac{1}{7}$.
Find the decimal expansions of $\frac{10}{3},\, \frac{7}{8}$ and $\frac{1}{7}$.
$\frac {10}{3}= 3.3333$ ............
Remainders : $1$, $1$, $1$, $1$, $1$ ............
Divisor : $3$
$\frac {7}{8}= 0.875$ ............
Remainders : $6$, $4$, $0$, ............
Divisor : $8$
$\frac {1}{7}= 0.142857$ ............
Remainders : $3$, $2$, $6$, $4$, $5$, $1$, $3$, $2$, $6$, $4$, $5$, $1$ ............
Divisor : $7$
What have you noticed ? You should have noticed at least three things:
$(i)$ The remainders either become $0$ after a certain stage, or start repeating themselves.
$(ii)$ The number of entries in the repeating string of remainders is less than the divisor (in $\frac {10 }{3}$ one number repeats itself and the divisor is $3$, in $\frac {1 }{7}$ there are six entries $326451$ in the repeating string of remainders and $7$ is the divisor).
$(iii)$ If the remainders repeat, then we get a repeating block of digits in the quotient (for $\frac {10} {3}$ , $3$ repeats in the quotient and for $\frac {1} {7}$ , we get the repeating block $142857$ in the quotient).
Similar Questions
Are the following statements true or false ? Give reasons for your answers.
$(i)$ Every whole number is a natural number.
$(ii)$ Every integer is a rational number.
$(iii)$ Every rational number is an integer.
Are the following statements true or false ? Give reasons for your answers.
$(i)$ Every whole number is a natural number.
$(ii)$ Every integer is a rational number.
$(iii)$ Every rational number is an integer.
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 ?
Find an irrational number between $\frac {1}{7}$ and $\frac {2}{7}$
Find an irrational number between $\frac {1}{7}$ and $\frac {2}{7}$
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 ?
Show that $3.142678$ is a rational number. In other words, express $3.142678$ in the form $\frac {p }{q }$, where $p$ and $q$ are integers and $q \ne 0$.
Show that $3.142678$ is a rational number. In other words, express $3.142678$ in the form $\frac {p }{q }$, where $p$ and $q$ are integers and $q \ne 0$.