Question

Show that $3.142678$ is a rational number. In other words, express $3.142678$ in the form $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$.

Answer 1

Hint: To express a decimal in fractional form, first of all we need to write the given decimal in numerator without decimal point and then count the digits after the decimal point. If there are 2 digits then write 100 in denominator, if 3 digits then 1000 and so on.

Complete step by step solution:
In this question, we are given a decimal number and we need to prove that it is a rational number. For proving a number to be a rational number, we need to express that decimal number in fractional form.
Given decimal number: 3.142678
So, we need to express 3.142678 in fractional form as $\dfrac{p}{q}$.
Steps to convert a decimal number into fractional number:
Step 1:
First of all, write the given decimal number in the numerator without any decimal point. Therefore, we get
\[ \Rightarrow \;3.142678 = \underline {3142678} \]
Step 2:
Now, count the number of digits after the decimal point in the given number. Here, in our given number, there are 6 digits after the decimal point. Hence, write 1000000 in denominator. If there were 3 digits after decimal point then we might have written 1000. Therefore, we get
\[ \Rightarrow \;3.142678 = \dfrac{{3142678}}{{1000000}}\]
Hence, 3.142678 is a rational number and it can be expressed as \[\dfrac{{3142678}}{{1000000}}\] in $\dfrac{p}{q}$ form.

Note:
Here, we can also express the obtained fraction in simplest form.
$ \Rightarrow $ Simplest form of \[\dfrac{{3142678}}{{1000000}} = \dfrac{{2 \times 1571339}}{{2 \times 500000}} = \dfrac{{1571339}}{{500000}}\].
Hence, the fractional form of 3.142678 is \[\dfrac{{3142678}}{{1000000}}\] and its simplified form is \[\dfrac{{1571339}}{{500000}}\].