1 (one)
You can write the number 1 with an infinite number of decimals as
The three points indicate that there are infinitely many decimal places.
Calculation with a fraction
We start with the fraction
Here you see that the same number can be written in different ways.
Calculation with an equation
We start with
and multiply both sides with 10, so
At both sides we subtract x
and find
so that
From this follows that 1 = 0.999999…
Calculation with a geometric series
In a geometric series each successive element emerges from its predecessor by multiplication with a constant. If a is the first element of the series and r the constant, then the whole series is defined
For |r| < 1 you can write the sum of this series as
The number 0.999999… can be considred as a geometric series, because
where
en
so
From this follows 0.999999… = 1.
Using a calculator
On a calculator you get
but sometimes it gives
that is then caused by the rounding that is used.
Note
The number 0.999999… doesn't stop with a 9 but always continues forever.