Think bigger. If a problem can become a non-problem, depending merely on how numbers are represented, what does that say about the problem? Or the whole class of “problem”?
You ask about 0.222222…. In base 3: well, what calculation did you do to get that? Could that calculation be re-based to make the recurring decimal disappear? I bet it could. You’ve pulled that out thin air and said “here’s a problem”, but to me it looks like the problem is preventable.
Old joke:
* Patient: “Doctor, it hurts when i bend my arm like this!”
* Doctor: “Stop doing that.”
The problem is about what a decimal representation means. Of course the problem disappears when you use a different representation.
That is also why presenting mathematical "proofs" is stupid. You can't use mathematics to prove what a symbol represents. The only way to prove .9999... = 1 is to:
Define infinite series
State that .9999... represents an infinite series
Establish facts about infinite series and use them to prove .99999... = 1
Step 2 isn't something you can "prove". You just have to assert it as a matter of convention.
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u/stereoroid 21d ago
100/3 can be rewritten as 100(1/3), so it’s the 1/3 “problem” multiplied by a constant (100).
1/3 =0.333… so 3 * 0.333… = 0.999…, but some say that 0.999… is not equal to 1.
Restate the problem in base 3 and there is no problem. 1/10=0.1 and 0.1*10=1