I keep being told that 0.999... is exactly 1, not approaching it. Every proof I have seen feels like a trick. The 10x one in particular looks like it assumes the answer, because you have to already believe the number behaves like an ordinary number before you can multiply it by ten.
Is there a version that does not feel like sleight of hand?
Accepted Answer
The honest answer is that it is true by definition, and the definition is the part nobody shows you.
0.999... is notation for the limit of the sequence 0.9, 0.99, 0.999 and so on. That is what an infinite decimal means. It is not a number that is still being built.
The limit of that sequence is 1, because the gap after n digits is 10 to the power of minus n, and you can make that smaller than any positive number you name.
So the question "is it close to 1 or equal to 1" has a hidden assumption, that 0.999... is something other than its limit. It is not. Once you accept the definition, there is nothing left to prove.
On the 10x proof, your suspicion is right and worth spelling out.
Writing x = 0.999..., then 10x = 9.999..., then subtracting to get 9x = 9, is valid. But every step relies on 0.999... already being a real number you can multiply and subtract. If someone doubts that infinite decimals denote genuine numbers, the proof is answering a question they did not ask.
It is a good argument for people who already accept the framework, and useless for people who do not. That is why it feels like a trick to you. You are the second group.
Try it in thirds. One third is 0.333..., and three thirds is 1. Multiply the decimal by three and you get 0.999... If you accept the first line you have already accepted the last.
1 more answer and the discussion on each one are open to members.
Join free to read the rest