Searching a solution we often forget about the problem behind it. I'm familiar with the wrong focusing on 'how' instead on 'why', and what's more I've seen people doing the same mistakes as well. For example, let's just look at law regulations as synonym of complication paired with uncleanness. However, I'll drop that subject and introduce the similar problem occurring in maths. Do not expect any great proof, because I'll do that to explain it as simple as possible.
0.999... = 1
The proof for the equation above is the above derivation:
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1
If you are interested on it, you can find more about it on Wiki: 0.999... or at once read quite fine explanation by Cecil Adams. The fact is that he will not going to get Nobel price for that but in real numbers field his explanation is mostly accurate.
I wanted to explain the reason behind it rather than submitting next proof for it. For this purpose I'll introduce the proof that every person earns the same amount of money, of course using the same method as above.
m - best earnings on earth.
x - earnings some man.
y - earnings some other man.
We have then:
x >= 0
y >= 0
x <= m y <= m Let's assume: x = y We can divide by nonzero: x/2m = y/2m Let's bring it to integer numbers: using (x <= m) x/2m = 0 using (y <= m) y/2m = 0 So we have proved equality: 0 = 0
Thus, I proved that each of us earns the same amount of money, so it's proved the validity of the communist system.
Now, it's high time to end the torment and explain the point. Each of us earns the same amount, 'but' in accuracy to double amount of the best salary possible to be earned. The same thing happens in the case of
0.999 ... = 1. Except the fact that in the latter case the accuracy of the integers is 1, the precision of real numbers is (roughly) 1/∞. So we can safely confirm equality between 0.(9) and 1, 'BUT' with an accuracy of 1/∞, which shows that in the same time there is (roughly) the difference between these two numbers.
All proves (including those the Wiki) are accurate, but in the same time they do not carry too much sense.
This is how it is
Proper infinity algebra is not used with some fairly simple reasons (in general):
- Infinitely small values are 'relatively' meaningless, so for quite a while there was no need to include them in the model.
- Algebra with ∞ numbers begins to exhibit vector traits.
- The calculation results are interesting only if it is lifted upwards.
- Inclusion of infinity in the model is quite tiring.
- A few existing formulas would have to be corrected in order to properly fit infinity algebra.
If somebody wants to make some calculation, he/she may read about surreal numbers, that contain the infinite and infinitesimal numbers (greater than zero but less than any positive real numbers). The mathematical description you can find on wiki as well: 0.999...: Alternative number systems