4. Convert the decimal form of rational numbers to fractional form

1. Prove that $0.77777... = 0.$$\overline{7}$ is a rational number. That is, show that $0.$$\overline{7}$ can be expressed in $p/q$, where '$p$' and '$q$' are integers with $q$$\ne$$0$.

 

 

Let us take the provided number as '$x$'.

 

That is $x\ =\ 0.77777...$

 

Now observe the number '$x$'. The only repeated digit is $7$.

 

Here we have to make multiples of '$x$' in such a way that its decimal part will be the same as the given number.

 

Let us multiply '$x$' by $10$.

 

$10x\ =\ 7.77777...$

 

Now subtract '$x$' from $10x$,

 

$10x - x = 7.77777... - 0.777777...$

 

$9x = 7$

 

$x = 7/9$

 

Therefore, the fractional form of the rational number $0.$$\overline{7}$ is $7/9$.

 

 

2. Prove that $0 .2363636...\ =\ 0.2$$\overline{36}$ is a rational number. That is, show that $0.2$$\overline{36}$ can be expressed in $p/q$, where '$p$' and '$q$' are integers with $q$$\ne$$0$.

 

 

Let us take the provided number as '$x$'.

 

That is $x = 0.2363636...$

 

Let us multiply '$x$' by $10$ to get the repeated decimals separately.

 

$10x\ =\ 2.363636...$

 

Now multiply '$x$' by $1000$ to get the same decimal pars of $10x$,

 

$1000x\ =\ 236.363636...$

 

Subtract $10x$ from $1000x$.

 

$1000x - 10x = 236.363636... - 2.363636...$

 

$990x\ =\ 234$

 

$x\ =\ 234/990$

 

Therefore, the fractional form of the rational number $0.2$$\overline{36}$ is $234/990$.