What is
0.777...
when written as a fraction?
Answer
We can write this number as
This is a geometric series with and . The sum of the infinite series is
If we put into a calculator, we get something like
0.77777777778.
Since the calculator can only display finitely many digits, it rounds the last seven up to an 8.
Write
0.787878...
as a rational number.
This is very similar to the previous problem. We use the repeating pattern to break up the decimal into a sum of fractions:
Since the repeating pattern is 2 digits long, we have to multiply each fraction by to get to the next fraction. We have a geometric series with and .
The sum of this series is
If we put the fraction into a calculator, we get something very close to 0.787878.... This is a good sanity check.
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and 
. The sum of the infinite series is
into a calculator, we get something like
to get to the next fraction. We have a geometric series with 
and 
.
into a calculator, we get something very close to 0.787878.... This is a good sanity check.