Question

How to convert decimal numbers to \[\dfrac{p}{q}\] from?

Answer 1

Hint: A decimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point. Any decimal number which are terminating or recurring decimal numbers which are rational numbers however large can be converted in the form of \[\dfrac{p}{q}\]. Where p and q integers and \[q \ne 0\]. \[p\] And \[q\] should be co-primes that is whose Highest common factor is \[1\].

Complete step by step answer:
To convert a decimal number into \[\dfrac{p}{q}\] from what we have to do according to the type of decimal number it is.
If the decimal number is a terminating decimal number then count the decimal places in the decimal number.
Take \[1\] annexed with as many as zeroes as the number of decimal places as in the given decimal number or in other words put zeroes after \[1\] as many as the number of decimal places as given in the decimal number. After multiplying and dividing with this multiple of \[10\] the given rational number will be converted in \[\dfrac{p}{q}\] form with ease.
For example \[6.233\] will be written \[\dfrac{{6233}}{{1000}}\] as \[\dfrac{p}{q}\] form
If the given decimal number is non-terminating recurring decimal numbers then let the number be \[x\] first count the total numbers after decimal which are recurring. Let \[n\] numbers are recurring after decimal then multiply the number with \[{10^n}\] then subtract the given decimal number from the number obtained after multiplying \[{10^n}\] then on simplification for \[x\] we get the number\[x\]in the form of \[\dfrac{p}{q}\].
For example \[1.626262...\]
It could be written as
\[
  x = 1.6262 \\
  x = 1.\overline {62} \\
 \]
As it have \[2\] recurring digits so on multiplying it by \[{10^2}\] we get
\[100x = 100 \times 1.\overline {62} \]
On subtracting \[x\] from \[100x\] we have
\[100x - x = 162.\overline {62} - 1.\overline {62} \]
\[99x = 161\]
\[x = \dfrac{{161}}{{99}}\]
Which is in the form of \[\dfrac{p}{q}\]
And if the given decimal number is non-terminating non-recurring then it cannot be converted into \[\dfrac{p}{q}\] form as it is an irrational number.

Note:
The decimal numbers are classified in three types: terminating decimal numbers, Non-terminating recurring decimal numbers and non-terminating and non-recurring decimal numbers. The first two kinds are rational numbers whereas the non-terminating non-recurring decimal numbers are irrational numbers. Any irrational cannot be represented in the form of \[\dfrac{p}{q}\].