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Why is .999... = 1 ?

Some have difficulty understanding why .999... = 1 .
The main reason seems to be related to the use of .999... by
mathematicians to represent 2 different ideas .
These 2 ideas are one finite and the other infinite , so the difference
is the difference between the finite and the infinite .
Sometimes .999... is used to mean a finite quantity close to 1 .
I.e. , .999... suggests writing , or equivalently adding , 9s to some
point , perhaps far to the right , and then stopping .
E.g. , perhaps .99999999999999999999 .
It is hard to conceive of any ordinary situation , apparatus or
construction where any practical difference between
.99999999999999999999 and 1 would arise .
Usually , mathematicians write .999... with a different meaning .
The construction for this meaning is the same as for the first meaning ,
except that the last step is skipped , and there is no stopping .
No matter how far to the right we look , we always find a 9 there .
There are an infinite number of 9s .
If you don't see the difference between these 2 constructions , then you
have not yet "bought" into the concept of the infinite , or infinity .
The product of
.99999999999999999999 and 10 is
9.9999999999999999999 .
The product of
.9999999999999999999 and 10 is
9.999999999999999999 .
Note that , just as
.99999999999999999999 is not =
.9999999999999999999 , neither is
9.9999999999999999999 =
9.999999999999999999 .
Mathematicians like to think very carefully about the exact meaning of
the following calculation , but you can see it's a powerful argument .
The product of .999... and 10 is 9.999... .
After multiplying by 10 , the infinite number of 9s is still there to
the right of the decimal point .
Every distant decimal place still has the same digit , namely , 9 .
The product of x and 10 is 10 * x = 10x = 9x + x .
Let x = .999... .
Then , 10x = 9.999... .
Then , 9x + .999... = 9.999... .
Then , 9x + .999... - .999... = 9.999... - .999... .
Then , 9x = 9 .
Then , x = 1 .
Thus , .999... = 1 .
Walter Nissen
2008-04-11