Just how big is it?

[Qqmike]

Yes, the wiki needs editing, it's ridiculous. It's a good example of not believing everything you read, esp on the Internet (vs professional, peer-reviewed, published sources).
It is true that 1 equals the infinite summation represented by the shorthand 0.999..... But 1 is not defined to be that summation! There are infinitely many equations where the number 1 is on the left or the right side. Infinitely many of them can be rigged up to involve various infinite summations. So what! Are there infinitely many definitions of the number 1 then? Let's define 1 to be
1 + limit (as x --> infinity of 1/x)
That's one of infinitely many ways. In that case, we are coming down on 1 (from numbers greater than 1) as opposed to increasing up to 1 (as we are with the infinite summation represented by the shorthand notation 0.999....).
Quit quoting this stuff and think about it, and you'll get it. To be clear, 1 is not defined to be 0.999 ... Rather, 1 is the (unique) multiplicative identity of the real field, the unique real number that has the property that for every real number x, 1*x = x*1 = x. I implore you, quit quoting this stuff where either the author is mistaken, or the author is quoted out of context, or where you may not fully understand the intent or context of the author. Here, do this for yourself: go get a mathematician! Ask him or her to review this thread. Get a personal, second opinion. It's not that serious, of course, but it would settle this. It is a neat thing that the series limit (as n --> infinity) (9*(10^(-1) + 10^(-2) + 10^(-3) + ... 10^(-n))) equals the real number 1, but that is not how "1" is defined in the real field. (The term "field" is meant technically, http://en.wikipedia.org/wiki/Field_%28mathematics%29) I'm trying to be helpful here as I think you can get this if you'd try. But I agree, I don't think it's going to happen. Mods: not to worry on this one; I do now finally and formally cease and desist! ;-)

[jlittle]

I think what Qqmike is missing is that a real number is defined to be the limit of such sequences. Maths was stuck on "what is real number" till someone came up with that. 0.9999... equals 1 because it matches the definition of 1.

Regards, John Little

[Qqmike]

I just lost a post, just a second ...

[HalationEffect]

I guess you need to go and edit the Wikipedia page then - especially the bit that says:

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education

But in order for such an edit not to be immediately reverted, you'd also have to show that all the proofs listed on that page are wrong.

(Edit) Your mathematical knowledge is greater than mine, of that I have no doubt. You'll most assuredly be able to win any mathematical argument with me. However, there are those whose mathematical knowledge is greater than either of ours, and they aren't in agreement with you on this matter. I'm afraid that I'll be deferring to them rather than to you.

On one thing we do agree though: neither of us are interested in continuing this debate, as it will lead nowhere.

[Qqmike]

I know I said I had to go for now, but ...

HalationEffect, my friend, you are playing with words, with symbols, with shorthand notations, and citing other people playing with words. Stop. Think for yourself, quit quoting stuff like this. When you read something, you must take it in the context of which it was written, you can't just apply it to ANY context. Trust me, the number 1 is not defined to be that limit. It is true that the value of that limit happens to be the number 1. Hell, it could have been the number 4.276; but no, it is 1. The number 1 is altogether something else having NOTHING to do with that limit (i.e., with 0.99999.... and that expression IS nothing more than shorthand for an infinite summation which IS nothing more than shorthand for an infinite limit of finite summands). If you want to push this, OK, the number 1, in THIS context, is the multiplicative identity of the field of real numbers (every field, axiomatically must have a multiplicative identity; i.e., there exists an element, call it "1," such that for every element x of the real field, 1*x = x*1 = x). It is ridiculous to say that we define the multiplicative identity of the field of real numbers to be the limit represented by the shorthand symbol 0.9999.

That did it! :-)
Now I AM done and excuse myself from this nonsense!

[HalationEffect]

OK, how about this:

The formula for finding the sum of an infinite geometric sequence is S = a1 / (1 - r), where S is the sum, a1 is the first term in the sequence, and r is the common ratio.

In the case of 0.999... we get S = 0.9 / (1 - 1/10) = 0.9 / 0.9 = 1

Source: http://www.bungie.net/it-it/Forum/Post?id=61117593

Also (from the same source) let's say that x = 0.999...
10x = 9.999...
10x - x = 9x = 9.000...

If 9x = 9.000... then x must equal 1.000..., even though we stipulated at the beginning that x = 0.999... therefore 0.999... and 1.000... must be the same.

From what I've been reading, amongst top mathematicians the notion that 0.999... equals 1 isn't even controversial; it's broadly accepted. I know that's the logical fallacy of 'argument from authority', but it'll do for me.

[Qqmike]

S-R, Post #29: Yes, I think ;-)

I've never been good at practical, word, math problems, and was surrounded by practical engineers and scientists in undergrad college who were (Rose Polytechnic Inst of Technology, now called Rose-Hulman Inst of Tech). As I say, a pure math guy here.

But here, yes, miles per hour --> miles/hour => distance divided by time. Your formula looks right. 30 mph takes 1/30 of an hour to cover 1 mile (speed1*time1); ditto for speed2*time2.
Not being good at this sort of thing, I can only get at it naively, x being the unknown speed to solve for:

mph must be 60

miles/hour must be 60

(1+1)/((1/30) + (1/x)) = 60
which reduces to 1/30 + 1/x = 1/30, and so implies x --> infinity (so 1/x --> 0).

That equation--and the thinking behind it--comes down what you are saying and your equation, Steve, right?

Sorry about my lack of typesetting skills for math symbols.

(I'll check back here later or in the morning, got some family urgency/emergency brewing long distance this evening.)

[SteveRiley]

I car goes around a circular one mile track, on the first lap the driver averages 30 MPH. How fast will he have to drive the second lap to average 60 mph for both laps?

I've seen this one before, including all the various debates. Here's how it was explained to me...

People who arrive at 90 mph for the second lap are led to this mistake in two ways: they don't realize that time, not distance, matters, and they're blind to the fact that this is really a weighted average problem because the "weight" here is a distance of 1, making the divide-by-2 seem intuitive. The incorrect formula that people are shortcutting is:

           speed[SUB]1[/SUB] * distance[SUB]1[/SUB] + speed[SUB]2[/SUB] + distance[SUB]2[/SUB]
speed[SUB]avg[/SUB] = ---------------------------------------
                    distance[SUB]1[/SUB] + distance[SUB]2[/SUB]

Since what matters in this problem is the time, not the distance, the formula should instead be:

           speed[SUB]1[/SUB] * time[SUB]1[/SUB] + speed[SUB]2[/SUB] + time[SUB]2[/SUB]
speed[SUB]avg[/SUB] = -------------------------------
                    time[SUB]1[/SUB] + time[SUB]2[/SUB]

How to know that the weight is time? Look at the phrasing of the question: specifically, which denominator is used. Here it's hour. Thus, time.

http://gmatclub.com/forum/a-certain-...-a-127918.html

Qqmike: seems reasonable, would you agree?

[Qqmike]

Yes, HalationEffect, me, too, I'm 64 and haven't done this work since about 1987 or so; although every now and then, a neighbor or a relative drags me back into it. Yes, I think we are on the same page. My bent is * pure * math, and so that sort of makes me, a priori, kind of an a*hole in how I communicate sometimes as I insist upon clinically clean rigor! ;-)

[Qqmike]

"No matter how small you want that difference to be, I can find a term where the difference is even smaller."

That's my epsilon in Post #21.

I think what you guys are wrestling with--or addressing--is how we humans can understand the notion of mathematical limit, and that's why I linked in post #21 to
http://en.wikipedia.org/wiki/Limit_%28mathematics%29

To beginning students of math, the notion of limit is confusing, or difficult to grasp, it's the first step in understanding differential calculus. It helps to actually write down an arbitrarily small positive real number epsilon (like 1/1000000000000) and then as an exercise actually produce the partial, finite summation that is within that epsilon of the number 1; and then to pick an even small epsilon and do it again, if necessary, until one becomes convinced.

0.9999 does not actually equal 1. ONLY in the sense that 0.999 is shorthand notation for an infinite series (of finite partial summations) that converges in the real number field to the real number 1.

I really mean no offense here, but I think I've said it about as well as most mathematicians could (is there another mathematician here who could pitch in?), and so I'll step back for now and let you mull this over. I can't tell you how many calculus classes I have taught at the master's level.

[HalationEffect]

For a good explanation (better than Wikipedia's IMO) of why limits and arbitrary closeness don't disprove that 0.999... = 1, see: http://www.purplemath.com/modules/howcan1.htm

Especially the last section "Argument from semantics", which, as far as I can tell, outlines the objection you're raising.

A common objection is that, while 0.999... "gets arbitrarily close" to 1, it is never actually equal to 1. But what is meant by "gets arbitrarily close"? It's not like the number is moving at all; it is what it is, and it just sits there, looking at you. It doesn't "come" or "go" or "move" or "get close" to anything.

On the other hand, the terms of the associated sequence, 0.9, 0.99, 0.999, 0.9999, ..., do "get arbitrarily close" to 1, in the sense that, for each term in the progression, the difference between that term and 1 gets smaller and smaller. No matter how small you want that difference to be, I can find a term where the difference is even smaller.

This "getting arbitrarily close" process refers to something called "limits". You'll learn about limits later, probably in calculus. And, according to limit theory, "getting arbitrarily close" means that they're equal: 0.999... does indeed equal 1.

Emphasis mine.

Yeah, we probably do mean the same thing, and are just expressing it differently. It's been almost 30 years since I had to study any maths beyond high school level, and I've forgotten most of it.

[Qqmike]

"One ninth (1/9) expressed as a decimal is 0.111...
0.111... x 9 = 0.999..."

Only in the sense of convergence of infinite series (convergence of the series of finite summations as the number of summands approaches infinity). See my post #22 above. I think we mean the same things here, I'm just making it mathematically precise. It is not an axiom that 9 times one-ninth equals 1. Strictly speaking, for the number 9 in the real field, by the axioms of a field (the real field in this case), there * exists * a multiplicative inverse, and we agree to denote that inverse by, say, 1/9 or 9^(-1), but we could agree to denote it by anything, 9^*, egg9, z9, 9z, whatever.

[bsniadajewski]

Yes, I have seen folks ponder on this question for hours without coming up with the correct answer when it is a simple Distance, Time, Rate problem. He must complete the two laps in two minutes to average 60 mph. He already used the two minutes on the first lap.

Yeah, tyhat problem had me thinking about it for a bit. At first I thought "90 MPH" due to averages ((30+90)/2 = 60), but then I read this then I said "Ahhh, I see".

[HalationEffect]

Another way of looking at it - The set of real numbers is subject to the Archimedean property (that there are no non-zero infinitesimals; i.e. all infinitesimals are equal to zero), therefore saying that the difference between 0.999... and 1 is infinitesimal is the same thing as saying the difference between them is zero.

But for me, the best proof is the simplest, and the Wikipedia page has it:

One ninth (1/9) expressed as a decimal is 0.111...
0.111... x 9 = 0.999...
But axiomatically, one ninth multiplied by nine *must* equal one, right?

[Qqmike]

This might help.
0.9999 means
9*(10^(-1) + 10^(-2) + 10^(-3) + ... )
and that infinite summation equals (in the sense that the partial, finite sums converges to) "1" (as the number of summands approaches infinity). None of this makes any sense unless you do have a field, in this case the real field (with multiplicative identity and inverses and so on). All of this is true for any base (here, the base is 10).