0.99999999...=1 -- Infintesimal

This problem, while always leading to a wonderfully lively debate, does bring up the major problem of how you have valid proofs and still an idea that is fundamentally at odds with an intuitive understanding of how a created number system should work.

First off, it must be stated the proofs are (at least, as far as I can tell) unimpeachable--and any disputes stating the answer is wrong MUST also show how each of the proofs are fail to remain valid. I, by no means, am capable of achieving this; nor have I seen anyone (to date) that has posted such a rebuttal.

However, I would like to enter into evidence a concept referenced by an author of one of the formulations of this (current) identity (@Jake Lai)--the infinitesimal. This, for me, "solves" the problem; in fact, when pointed out, it is the perfect compliment to the infinite sum that underlies the more rigorous proofs.

The question, though, remains: What, exactly, does this mean for such an interpretation? Is not the infinitesimal used in the same way limits, imaginary numbers, and division by zero are?--i.e. as (temporary) work-arounds for shortcomings in the system as it stands?

This works, of course, as long as the destination actually DOES avoid (in the final accounting) the original paradox it was created to avoid. Of course, this isn't to say there aren't VALID interpretations where this is not the case--cf. Ramanujan Summation or Galois Theory--, but even here the applications require "interpretation" outside of the (current) number system. Would this application not also need to adhere to the same restriction of having either "net-error-avoidance", or at least "imprecise" application to the problem? [These are honest questions, as I am neither a mathematician, nor a student well-versed enough in number theory to seek verification on my own.]

So, in (forgive the pun) sum (hahah...no?...not even a little bit funny?):
1) Is the infinitesimal a valid "interpretation"?
2) [If 1 is true] How does the interpretation avoid the caveats above?
[Of course, the corollary question is--are the caveats above valid?]

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Jake Lai (See mention in the note)

Calvin Lin Staff - 4 years, 6 months ago

First of all, I'm really sorry, I've just read the title of this article....

First.- 0.999999... is not well- defined...

Second.- Every real number has one only decimal representation, for example, what is the decimal representation of 1?

a) 1.0

b) 1

c) 1.00

d) 1.000000000000000000000

Third.- it's usual on Brilliant, for instance, if you write 0.999999...., The ellipsis (suspension points) means that the number 9 is repeated an infinitum countable number of times after the last 9 written.

Fourth.- what is $\displaystyle \sum_{n = 1}^\infty \frac{9}{10^n} \space \text{?}$ 0.999999.....(This is a limit)? or 1?

P.S.- I hope this helps you. If someone wishes to reply me, he can do it, of course....

Guillermo Templado - 4 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$