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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.

 


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