I have a middle grade math question for you. I would like to know why .9999... = 1 ? I can not use algebra to show this or the following:

We agree that 2 = 2 and that 2-2 = 0, so

0.000... = 0 therefore 0.9 = 1
----------OR--------------- 1/3 = 0.333333 and 3 X 1/3 = 1, so if 3 X 0.333... = 0.999... then 0.999... = 1

My teacher says that I can not use the above example to show why this is true, and that I must use a couple different examples. He says that there are several other ways. Do you know any? I could really use the help because I can't think of any other ways to show this is true. Thanks for any help you can give.


Hi Joan,

I really like the fact that your teacher is asking you to find more than one argument for the fact that 0.999... = 1. Sometime we think that questions in mathematics only have one answer.

There ia a note in the Quandaties and Queries database called Repeating decimals where Walter gives two arguments. His second argument is similar to your first argument but his first argument is quite different.

Here is a proof by contradiction.

If 1 and 0.999... are different numbers then  (1 + 0.999...)/2 = 1.999.../2 is yet another number, in fact it is half way between them. But  1.999.../2 = 0.999... and hence 1 and 0.999... cannot be different.

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