< **1** 2 >

### 0.999...

In base 10 system 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 sequence 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 sequence as

The number 0.999999… can be considered as a geometric sequence, because

where

and

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.