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Nagy Vilmos1-Dec-15 23:44
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yiangos6-Apr-16 10:17
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Please take a look at the thread. There is a number of mathematical proofs that an infinite series of 9s in the decimal part is actually EXACTLY equal to 1. Heck, even I offered a proof.

There's no approximation there. 0.999... is EXACTLY equal to 1. Again, the dots play an important role. No dots, no equality.

EDIT No I didn't offer a proof.

Here's one.

0.999...= 9*(0.111...)
0.999...=9*[(1-1)+0.111...)
0.999...=9*[(1+0.111...)-1]
0.999...=9*[(1+0.1+0.01+0.001+...)-1]

Now 1+0.1+0.001+0.0001+... = 1/(1-0.1)=1/0.9

This is the closed form for the sum of an infinite number of terms of aa geometric series. This is NOT an approximation. This is the actual final value of summing infinite terms of a geometric series. You can see e.g. at Geometric series - Wikipedia, the free encyclopedia[^] or you can look it up in wolfram (too tired to look it up myself right now).

Therefore

0.999...=9*[(1/0.9)-1]
0.999...=(9/0.9) -9
0.999...=(90/9) -9
0.999...=10-9
0.999...=1
QED.

There's no pesky math there (such as dividing by zero, as the OP did).
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(Alas! We're devoured by lamb-guised wolves!)

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