[split] 0.999... equals 1

[Violet]

An infinitely large number cannot have any number larger than it. An infinitely long number cannot have any number longer than it

I said something wrong too I was trying to note that as a number gets longer (ie 9 < 99 < 999, just as .9 < .99 < .999) it also gets larger, and as a number gets larger it often gets longer (IE 1 < 10 < 100.) (Neither of these is always true, but I thought it might be an interesting correlation to draw between the two.)

I understand that .9^ - .9^ = 0... And as I am sure I showed in a recent post (the long one with hide tags I think, though it might have been the one before or after that one), in doing so we have cancelled out an unsolvable. As with (1 + x)(1 + x)/(1 + x), you can cancel out difficult (and in some cases impossible), and make a finite number of infinite or otherwise unsolvable numbers, as in: (1 + x)/1.

As i've said before, 1/3 is perfect as a fraction... but not as a decimal. Fractions really are much more perfect than decimals... and calculate otherwise impossible to calculate things (See pi) accurately. Just as we could not hope to call Pi's decimal representation perfectly accurate, we cannot call .3^ or .142857^^ perfectly accurate. Infinitely long numbers (representing fractions) cannot be represented perfectly in decimal form... as they will always require an infinitesimally small round up. This is negligible for all practical applications... but an inequality is an inequality, no matter how small

And actually, why wouldn't infinity - infinity would cancel itself out into 0... just as much any other infinite concept? Unless the two infinites are of different length of course, which would make one of them not infinite.

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Sae - you raise some good points, but the clincher is the proof - Adrian is right when you look at the mathematics

1 = 9/9 = 9 x 1/9 = 9 x 0.111...=0.999

This is a non intuitive fact, the idea that two decimals can represent the same number, but it works in the proof. As non intuitive as this is when you think about it there is no getting away from the proof.

So Sae - are you going to stick to your predefined notions on the nature of infantecimals, or believe the proof when it is inconsistent with your existing understanding? You'd be doing something those other people always do.......

What's inconsistent about it, if you don't mind my asking? I don't doubt that the mathematicians probably have had the right idea... I simply don't see it yet Which other people specifically?

As far as I can see... the problem in understanding this comes not from the mathematics themselves... but from the decimal representation of them (base ten vs fractions). This is a lot like reading different languages... where in this language, I say A, B, C... and in the other language I say A, B, D&E. Translating between these two languages is not perfect, because there is no C in one, and the other lacks D&E. Some languages just lack C, and have no new letters.

From fraction to decimal is no different... and translation isn't perfect. Now, with both fraction and decimal cited (EG: 1/3x3=1 = .3^x3=1) [as many of the proofs are written]: It makes sense because of the conversion apparent between the two. However, decimal alone lacks the words we need to turn .9^ into 1... and leaves an infinitesimal gap between the two... just as .989^ leaves an infinitesimal gap between .989^ and .99. The gap is there with infinite decimals (note: or else there would be no use for them)... and that is just part of the 'grammar' of the language. That gap needs to be recognized, and condoned. It is negligible to the point that it doesn't matter in the slightest... but it remains in existence.