.9999999 repeating equals 1 depending on your tolerance, and if we are talking applied or theoretical math. If we divide 1 by 3 we get .333333 repeating. This is not a perfect representation of 1/3, as each repeating 3 gets you ever closer to 1/3. The question is when do mathematicians say "after this 3 let us just say it just equals 1 third." So yes .9999999 repeating can equal 1, but at some point you are rounding up, the amount you round up gets ever smaller as you reach more nines, but eventually you round up (i.e. with .999 you round up by .001, and with .99999 it is .00001)

Feel free to correct me if I am wrong, just thought this info was important to understanding the question.

This argument keeps surfacing on the forum and I keep trying to explain.  To save you having to find my earlier posts I'll say it again.

Numbers have no existence at all except as mathematical concepts, dreamed up to serve a purpose.  They get defined and that's what they mean.  If you don't want to use a number you don't have to. 

I use the common definition that ½ means one part in two but on a TI scientific calculator it looks like this:

They define it to mean the same thing and it's easy to see why.

When the builders were assigning positive integers to the houses in my street they decided to leave out the number between 12 and 14.  They thought it might be harder to sell a house with that label because some people think it is an unlucky number.  So that number doesn't exist in that context.  The lady who delivers the mail still manages to put the right letters in the right boxes so it doesn't seem to cause much of a problem. 

Some people have never heard of complex numbers, or quaternions but they still get used by those who have.

It is occasionally useful to define an expression as having a certain value because the formulas work better that way.  That's where the following definitions come from:

There are lots more like this.

Now it happens to suit some mathematicians to define:

You may not like it.  You may not want to use it.  That's Ok. You don't have to.  But it has been defined so it has as much existence as any other number. It's a waste of time debating that.

You might as well start arguing about whether the two symbols below  are the same.  They don't look the same but ........

Bob

john_gabriel wrote:

Muxdemux: Didn't you only prove (1)  for the natural numbers?

Of course. Are there any other numbers that you know of which I don't? Please don't tell me infinity is a number. It is a meaningless, non-real concept.

TheTick:  .9999.... NEVER equals 1 and it has nothing to do with tolerance.

At no point in the partial sums of 0.999... are you "rounding up". In other words, a carry is impossible. The problem with 0.999... is that it's not a number unless considered as an approximation. A non-terminating (repeating or non-repeating) decimal is an ill-defined concept.

I agree if we are talking theoretical math, however ask a manufacturer, engineer, architect, etc. if they measure one third to an infinite accuracy. As this can not be done realistically we "say" .999999... at some point equals 1, even though theoretically it does not. That is one I meant by .9999999... can equal 1.

Muxdemux: Of course I only proved (1) for natural numbers - this is all one can do with induction. The problem arises when you take the limit and the inequality results in an absurdity.

Bob Bundy: There are no personal attacks. Why the message? And of course I wrote the article. I never plagiarize the work of others and most people do not think or agree with me, which makes me feel pretty good. It's always nice to know that I am correct and the majority are in error. Either way, it does not really matter too much. People who read stuff on the internet (or anywhere else) should not take things personally. If your name is not in the perceived put-down or insult, then why take offence? Kind of silly, isn't it? The world does not revolve around any one individual. Please do not delete my post. I am sharing my unique knowledge with you on your forum. Whether you agree or not is irrelevant.

And one more thing: I don't like it that it says "novice" under my name. This is a major put-down for me. Please change it to something like Expert or Master. Thank you in advance!

Last edited by john_gabriel (2013-02-09 02:40:57)

Well, alright then. But I don't know how much longer I will be around. Sigh,...

10 posts can be a long, long time....

Yes, yes, I understand. Rules are rules.

Ha, ha. I hope you are right!

hi Agnishom,

If A intersect B = {} then {} is in A and {} is in B.

Bob

Expert or Master...

When a set is a subset of another set, it means that all elements of the first set are elements of the second set.

When a set is in another set, it means that the first set is an actual element of the second set.