I would like to argue about .999

[Caramon]

Here's a proof that .999...=1.

.999... can be written as the infinite sum as follows:

$$.9 + .09 + .009 + .0009 + .00009 + ... = .9 \sum_{n=0}^\infty \left(\frac{1}{10}\right)^n$$

Now, evaluating the sum on the right, we use the fact (proven below) that...

$$\sum_{n=0}^\infty r^n = \frac{1}{1-r}$$

for all r with a magnitude less than 1. Using this fact, we find that...

$$.9 \sum_{n=0}^\infty \left(\frac{1}{10}\right)^n = .9 \frac{1}{1-.1} = \frac{.9}{.9}=1$$

Now, to prove that fact that we used, note the proof below:

$$S = \sum_{k=0}^{n-1} a r^k = a + a r + a r^2 + a r^3 + a r^4 + ... + a r^{n-1}$$

$$rS = a r + a r^2 + a r^3 + a r^4 + a r^5 + ... + a r^n$$

$$S - rS = a - a r^n = a (1 - r^n)$$

$$S(1-r) = a (1 - r^n)$$

$$S = \frac{a (1-r^n)}{1-r}$$

Now let n go to infinity. For r with a magnitude less than 1, r^n tends to 0 as n tends to infinity. Thus...

$$\lim_{n \rightarrow \infty} S = \lim_{n \rightarrow \infty} \frac{a (1-r^n)}{1-r} = \frac{a}{1-r}$$

Q E D

/thread

This thread is INFINITELY hilarious because people don't need to say anything other than, "oh wow this proof is great."

 

[Hurkyl]

Since the OP is no longer posting in the thread and these last few posts don't really look like mathematics, I think it's time to close this thread.

Calrid: if you want to post your ideas in the math subfora here, you're going to have to be clear and precise. For example, no making up an idea like one number equaling another number "at infinity" unless you first define what you mean (or at least make a reasonable attempt).

While I normally like philosophical discussions about math, they aren't very useful when they are overly vague/imprecise or one side appears to assert factually incorrect statements, even going so far as to explicitly refuses to acknowledge reality. (e.g. the reality that "beyond infinity" can and does make sense to some people. I assume from the context that you are referring to one infinite thing being larger than another -- but there are other cases where beyond infinity makes sense as written, such as the ideal points^* of hyperbolic geometry)

*: This might be the wrong name for them -- I'm having trouble finding a reference. For those who know hyperbolic geometry, I'm referring to the extension where any pair of distinct non-parallel lines meet in two points.

 

[HallsofIvy]

No, "ideal points" is correct. Although I would call them "points at infinity" rather than "beyond" infinity.

 

[Hurkyl]

No, "ideal points" is correct. Although I would call them "points at infinity" rather than "beyond" infinity.

Hrm. I would have expected the phrase "at infinity" to refer only to the boundary of the set of ordinary points, rather than including the points on the other side of the boundary. Then again, I honestly don't recall the last time I've seen "at infinity" used in a situation where the space of ordinary points has a nonempty exterior.

 

[HallsofIvy]

Okay, I mis-interpreted what you said. The "points at infinity" are the "ideal points" and the points "beyond" infinity are "ultra- ideal" points.