## Show that the infinite decimal 0.9999 • • • is equal to 1. How Can 0.999… = 1? | .99999 = 1 proof

### Solution

## The number “0.9999…” can be Written as: 0.9999… = 0.9 + 0.09 + 0.009 + 0.0009 + …

### This may also be written as:

0.999...=\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\frac{9}{10000}+...

this is a infinite geometric series with first term a = 9/10 and common ratio r = 1/10. Since the value of the common ratio r is less than 1, we can use the infinite-sum formula to find the value:

0.999...=\left ( \frac{9}{10} \right )\left ( \frac{1}{1-\frac{1}{10}} \right )

=\left ( \frac{9}{10} \right )\left ( \frac{1}{\frac{9}{10}} \right )

=\left ( \frac{9}{10} \right )\left ( \frac{10}{9} \right )=1

So the formula proves that 0.9999… = 1.