Ok.
I think I figured it out. At least for me.
The “trick” to the so-called proofs is that they obfuscate the importance of the decimal place. By having an “infinite” repetition of nines, the 10x – x = 9.999… -0.999…
appears to yield 9x = 9
. But it does not. Or it does, if you’re willing to ignore the decimal.
Here’s why. Try this without the repetitive nines.
x = 0.99
10x = 9.9
<== here’s your first clue.
10x – x = 9.9 – 0.99
9x = 8.91
<=== see that, how by respecting the decimal place earlier, you get a non-magical value?
x = 0.99
x ≠ 1
Now I can imagine some responses.
1) But look at all the other proofs!—I don’t care about other proofs. I need only disprove it once.
2) But this is about repetitive nines!—Yah, I know. But as I mentioned earlier with pi, there are many cases where a decimal number can only approximate a value, never define it completely.
3) But really, this is about repetitive nines!—If I put you on the 10 yard line, and with each play you get half-way to the goal line, how many plays will it take for you to get a touchdown? Infinity!
4) But, why are you hung up on the decimal place?—Look at your paycheck. Move the decimal one place to the left. Tell me you don’t care.
5) Last time, this is about the repetitive nines, for cryin’ out loud!—Yes, it is. Because of the supposed infinite list of nines, it allows us to be sloppy. That’s the point of the proof. But when you must account for the decimal places by using a finite, easy to comprehend number of digits, then you can understand that the same must be done for the repetitive nines, too. In the wiki proof, that’s not being done.
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Don’t hate me because I’m beautiful.