I thought the rule was simple: that if the abbreviation is formed by the elision of letters within the word, there is no ".": if it's formed by missing letters off the end, then you do use a "." (Just don'task what happens if you do both.)
jj

I have never before heard of such a rule. It appears to be more honored in the breach than in the
observance. Even on those occasions when I have seen "Mr" or "Mrs" without the "." it appeared to
be an error. I think the use of the "." is desirable after all abbreviations.

From my dictionary: note that abbreviation for "cotangent" does not require a "."
abbreviation
n.

1 a making shorter
2 the fact or state of being made shorter
3 a shortened form of a word or phrase, as N.Y. for New York, Mr. for Mister, lb. for pound, ctn for cotangent

more honored in the breach

Or in Britlands

Several transpondian references to deal with here:

1)
dr (debtor), Dr (Doctor), with or without a "."
The rule of which Dr Bill has not heard used to be quite firmly the case over here, but is not so any more. We use either form - with or without "." - and no-one comments; not even if one isn't consistent. However, words that have had bits lopped off to form the abbrev. should have the "."

B)
What do we call "."?
Full stop - whether it is at the end of a sentence or after an abbrev. There is no particular justification for this - it's just the way most of us do it. Some people have started to call it "period" but that ain't too common, yet.

Þ) comma, semi-colon, colon
comma, semi-colon, colon

But "!" is almost always called "exclamation mark"
and "?" is a "question mark".

I think that's all for now!

>Brits and some others

well, i was asking about it because poles are "some others" :P

I think the use of the "." is desirable after all abbreviations.

A lot of people seem to think that Ms is an abbreviation, and put a stop after it.

ctn for cotangent

Typo, dr bill? I've never seen anything but cot for cotangent.

I've seen it as both CTN and COT.


k

I honestly thought ctn was a mistake. Every math book I've ever seen uses cot not to mention pretty much every programming language. In fact I thought it might have reached the stage of not being language-specific. emanuela, are the usual trig function abbreviations in Italian sin, cos, tan, sec, csc, cot? Anyone else with math and another language background care to comment?

http://www.bartleby.com/61/87/C0788700.html[
(Interestingly, I found a site that says that the maple programming language uses ctn and another that says it uses cot - perhaps both are correct and there's been a change. One Mathematica page explicitly says it's cot AND NOT ctn in that language.)

And people are actually using the notation, too http://web.mit.edu/wwmath/calculus/differentiation/trig.html3

http://www.mansd.org/memorial/mainsite/bb/other/nhoule/subdir/math.html

What I'm not sure of is how this varies - regional, perhaps, or maybe subject area (perhaps engineers are more likely to use ctn over cot), or perhaps even temporally (in the process of changing).

Similar notational deviance in a program language would be the use of the caret for exponentiation in some languages (fortran, pascal, vb, even quickbasic) vs double asterisk (perl, and the older versions of basic). [C, for example, doesn't even HAVE an exponentiation operator! It has the exp function, but no operator!]

Another notational oddity is that for the natural logarithm. Commonly ln or LN denoted natural log, while log denoted "log base 10" unless otherwise stated. Nowadays, I've seen class notes and handouts, etc, in which log denotes "log base e" unless otherwise stated.

On a slight, er, 'tangent,' one of the most curious terminological impasses I've encountered was the definition of a "Natural Number" which has changed over the years. When a woman first indicated in a usenet group that what she learned was that the naturals were the same as the integers I thought sure she was mistaken.

The whole thread is at http://makeashorterlink.com/?H32121033

k

the naturals were the same as the integers

I thought it was non-negative integers.

the naturals were the same as the integers



I thought it was non-negative integers.



Most of us (me included) learn a thing and think "well, that's the way it is." And we have a strong inclination (especially me) to think that opposing views are necessarily wrong.


What *I* learned:
N = Natural Numbers > 0
W = Whole Numbers >= 0
J = Integers = any non-fractional number
-- yet another thing ... sometimes they use the symbol I to mean integers
while my daughter is in pre-algebra in 7th grade and they use
J(n) to represent integers.

What this woman learned was that N = J. I was sure she was mistaken, but after spending a few hours in our company library I found a real math book that used her definition. (Which I meantion as the last post in that long thread of usenet posts I pointed to in my last message.)

Yet another weirdness which probably most awad readers are aware of, but don't think about very much. In 8th grade algebra, I learned that sqrt(-1) = i, but later, in engineering school, they used j for that.

It doesn't really matter how the terms are defined, so long as the terms are established early. Makes it difficult to interpret something when you come into it with no point of reference. That's obviously why they try to standardize some things.

k

I guess the only real contenders are the positive numbers (the orthodox view) and the non-negative numbers (the heretical view). One site I looked at suggested dropping the term completely; there were enough heretics running around that it was best just to say positive or non-negative or whatever you meant.

>I've never seen anything but cot for cotangent

i have :P -ctg

The way I was taught when I was learning to type was that it depended on the form of punctuation you were using.

If you were using 'closed punctuation' (considered old-fashioned)you would put a full stop after every abbreviation and initial, so...
Dr. A. N. Other

And when writing you would not leave a clear line between paragraphs, but would instead indent the first line of the paragraph by five spaces (1 tab stop).

This produced a fairly dense, cluttered page of type.

In 'open punctuation' you omitted full stops for abbreviations and initials...
Dr A N Other

and simply left a clear line between paragraphs instead of indenting the first line.

This produced a much cleaner page of type that was easier on the eye.

Interesting this topic should come up. I was recently reading a letter written by my grandmother. She was an appallingly lazy writer (I always wondered where that came from) and she abbreviated damned near every second word which made the letter rather hard to read. Frustratingly, she reduced names to R. and P. and what have you. Can't phone her and ask her now.

But, getting to the point, she put a full stop after every single one of those contractions and abbreviations. Looks pretty weird to moi, and I can be pretty pedantic!



- Pfranz

The i vs. j thing for imaginary numbers is usually an engineering/physics divide. I was told it was because engineers have a silly habit of using i for current and since you use imaginary numbers in decribing current and voltage in circuits, two different uses for the same letter would be confusing.

When I took real analysis (a hardcore math course in which we started by proving 2 > 1 and ended the year eight months later proving how to do calculus) we had natural numbers (N), that is, the "counting numbers" 0, 1, 2, 3,..., integers (...,-3, -2, -1, 0, 1, 2, 3...) (symbolized by Z - why has no one else mentioned Z?), rational numbers (Q) (any number that can be written as the quotient of two integers), irrational numbers (I, I think) (any number that CAN'T be written as the quotient of two integers), real numbers (R) (rational and irrational together), and complex numbers (C) which included all the others.

Note that for all of those the symbol is not just the letter but the funny letter with an extra bar which is completely unreproducible outside of TeX. (Having said that, maybe I'll make a little TeX file and post it somewhere so you can see what I mean. Give me an hour or two...I'm supposed to be working!)

Anyway, the naming is probably not so important as knowing what differentiates one group from another, what properties make one group different from the other.

emanuela could probably give us some definitive definitions on what's current in the "real math" world (rather than the bizarre world of the inconsistent drivel they teach kids in school - all you have to do is change provinces and the natural numbers could take on a whole different meaning - they should get their act together!).

I think the use of the "." is desirable after all abbreviations.

I don't. (just weighing in with my two cents' worth)

Didn't I see it here - the article, I think it was by Steve Martin, that was carefully constructed so as to use only one full stop in the punctuation of the entire piece? the premise being that the full stop is becoming an endangered species and that publishers of all kinds of printed materials are rationing its use? Just as this post of mine uses only one full stop that isn't recycled, in case you hadn't noticed!

Save the full stop! (or period, some calls it)

carefully constructed so as to use only one full stop in the punctuation of the entire piece

He kind of cheated, as I remember, stringing three periods together and calling them an ellipsis.

He kind of cheated

true...but they WERE an ellipsis, so perhaps he can be forgiven...! After all, we mostly call them periods on their own - but if there are three of them together (or four), it's easier to call them "an ellipsis," rather than "three [or four] periods strung together" - yes?

An ellipsis and three periods are different creatures to a typesetter. Or so TeX would have you believe.

How can it be humanly possible that I have had to mention TeX twice in one day now? Strange.

An ellipsis and three periods are different creatures to a typesetter. Or so TeX would have you believe.

Yes they are two different things. The ellipsis is a lighter weight than the full stop and the spacing of the three dots is equal (from memory) to one em wide in the same font.

- Pfranz

Lessee:

...   three periods (full stops)
…   ellipsis
m   m (duh!)

Whatcha duhing about this time, Faldage?

I just knew that three dots was too much..

Duhing that I had to point out that m was m.

Three dots was too much for what?

So an ellipsis is three periods folowed by climacteric

Thanks Dr Bill…

sin, cos, tan, sec, csc, cot? Anyone else with math and another language background care to comment?

Russian + math: sin, cos, tg, sec, (?), ctg... I am not sure what is csc... cosecans? then cosec

Ridicolous to say, I am the last person apt to give a definitive answer. Mostly because I don't care about names, and simply try to understand from the context.
A rose without its name, wouldn't be a rose?
So I can ask the same question about numbers in "high" or "low" way:
example:
how many different decimal representation can a real number have?
Or
0,9999999999999999999....is or not equal to one?
To understand the second question implies to have understood the core of the first one.
(Anyway, Bean, you were perfect in your list of sets of numbers. Just a comment about irrational numbers . I have always seen them written as "R\Q" = real not rationa numbers)

I forgot to add the answer
Yes,
0,99999999...... is equal to 1.

0,9999999999999999999....is or not equal to one?

I've seen it argued that, since 1/3*3=1 and 0.33333333333…+0.33333333333…+0.33333333333…=0.99999999999… then 0.99999999999…=1. On the other hand, I would maintain that 0.99999999999… is in the open interval between 0 and 1, but 1 is not, therefore 0.99999999999… does not = 1

Dear Faldage: and I still think parallel lines can never meet.

"Infinity" was once *disproved to me by this explaination:

Take your thumb and your forefinger and spread them apart as far as possible. This represents a distance. Now make that distance one half as large. Continue to cut the distance between them in half.

Theoretically, your thumb and forefinger will never actually® touch each other.

0,99999999...... is equal to 1.

Does this mean that 0.9999999999… is not in the open interval from 0 to 1?

It's in both places.

in both places

1 is in the closed interval but not the open interval; 0.99999999999999999999999 (without the ellipsis) is in the open interval, no matter how many 9s you tack onto the end. Or so I thought.

said the Queen. "When I was younger, I always did it for half an hour a day. Why,
sometimes I've believed as many as six impossible things before breakfast."

no matter how many 9s you tack onto the end

I guess that's true only as long as the number of 9s you tack onto the end is finite.

I disagree that 0.9999999999... is equal to the integer 1. All integers are understood to be exactly 1 or exactly 2, etc, as if they had an infinite number of zeros after a decimel point. Of course, they are practically the same in any conceivable everyday context, but they differ by that tiny amount. For example, 1/(1-1) is not defined because one cannot divide by zero. 1/(1-0.9999999.....) approaches infinity asymptotically the more nines you tack on at the end.

This is basic calculus. You'd express the above as the limit of 1/(1-x) as x approaches 1 from zero.

I disagree that 0.9999999999... is equal to the integer 1

Why not, if you will accept the infinite number of zeros, also accept the infinite number of nines? Something to think about. But I do remember learning, and being upset by, and finally agreeing with, what emanuela said above.


Edit: To clarify: you have to differentiate between how we write down a number, and what it is. What emanuela is saying is that for the number which we commonly understand to be the integer 1, there are two ways to write it: 0.9999999999999999... or 1.00000000000...

"Math is not an exact science."
-ron obvious

In reply to:

---

We would still say we had one couch.

---

...and that is why I say that they are the same in any conceivable, everyday aspect. Of course if you are talking about real couches, 0.99999999999999 of a couch is macroscopically indistinguishable from 1.00000000 of a couch.

What I am saying is that strictly speaking, in the abstract world of mathematics, there is a difference between the following numbers:

2.99999
3
3.00001

To prove my point, here is a function which will return a different value depending on which of the three above numbers you plug in: f(x)=3/(x-3)

Now if those three numbers were truly the same, then you should get the same result from the function above, but you don't.

3/(2.99999-3) = 3/(-0.00001) = (-300,000)

3/(3 -3) = 3/(0) = not defined/infinity/or, you go to jail for dividing by zero and your mother is very disappointed

3/(3.00001-3) = 3/(0.00001) = 300,000

f(x)=3/(x-3)

Which is certainly true if we have a finite number of 9s after the decimal point. If we have an infinite number of 9s, there is an infinite number of 0s before we get to that pesky 1 in the .000000…0001 meaning that we'll never get there.

Yeah, that's the difference here, which maybe wasn't clear. The ellipsis after a number in math means "going on to infinity" so 0.99999999... means nines going on to infinity, that is, you never run out of digits. That is emphatically NOT the same as 0.99999999. Which means any kind of numerical example given with a finite number of nines isn't talking about the same thing.

Anyway, I looked it up in my analysis book last night, and you can prove pretty straightfowardly (if you've read the first three chapters of the book) that both decimal representations (that is, 0.999999.... and 1.000000...) are, indeed, EQUAL to the number 1.

Another thing I thought of, Alex, is

If you don't believe that 0.9999999... is equal to 1, then please tell me how far from 1 it is. That is, what is 1 - 0.9999999... = ???? If you are right, the difference must be equal to some probably quite small but finite number.

0.999... = 1

however

0.99999 <> 1

The repeating decimal is identical to 1, whereas the fixed decimal is not.

let x = 0.999... (where the ellipsis denotes infinite repetition of '9')

so 10x = 9.999...

Clearly, 10x - x = 9x

10x = 9.999...
- x = - 0.999...
------ ----------
9x = 9

Dividing both sides by 9 gives x=1.

Therefore 0.999... is identically equal to 1, mathematically, theoretically, and, in fact. (Well, that's what I think anyway.)

k

I have seen in various posts both
0,9999 and
0,9999...(meaning that we are imaginig infinite 9's)

Well, here is the question.
What do you mean by saying that a given number is given by an infinity of digits? In other words, which is the meaning of the decimal representation of a number, when it is not finite?
The number is the limit of the partial ways of writing. In our example, to say that 0,9999999........ = 1
means exactly that 1 is the limit of the succession
0,9
0,99
0,999
0,9999
........and so on

that puzzled me a lot, indeed. But the explanation is so easy to seem obvious: roughly speaking, I would explain this way:
Suppose that you are working in the plane - the usual plane, the plane you can imagine , the plane of the blackboard prolonged to the infinity - the so called "affine plane".
Consider there two parallel lines: don't worry, they don't meet.

But,if you consider the so called "projective plane" , obtained from the above plane by adding more "points", each one representing a "direction", the two lines will have the same "point",or, in other words, they have the same direction.

In reply to:

---

If you don't believe that 0.9999999... is equal to 1, then please tell me how far from 1 it is. That is, what is 1 - 0.9999999... = ???? If you are right, the difference must be equal to some probably quite small but finite number

---

That is exactly what I am saying. 1- (0.999999...) is an infinitesimally small value not quite equal to zero. When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one.

The example I offered suffered from a lack of ellipses, but it holds true in differentiating 3.000...0001 from 2.9999999.... In fact, the more zeros or nines in the decimal point, the greater the difference between the two defined outcomes.

That is exactly what I am saying. 1- (0.999999...) is an infinitesimally small value not quite equal to zero. When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one.



I don't think so. The limit is the limit. That is, the limit is taken as exactly equal to one - well, that's the way I learned (but as Bean pointed out, we learn a lot of incorrect stuff).


lim x = 1 (identically)
x->1


For example, we don't say that integrals solved exactly are "approximately equal to blah blah."

The integral of cos(x) is sin(x) + C, not approximately sin(x) + C (and what is an integral, but a limit?).


OTOH, we don't say that differentials are identically equal to zero. It's a bit confusing. Back in HS, before actually taking calc, I read this book by a guy named Boyer called "The History of Calculus and its Conceptual Development." From what I recall (of more than 20 years ago), these concepts of differentials and limits that we take for granted today were a really big deal at one time - very controversial.

I'm a bit envious of Bean for actually having taken Real Analysis. I'm sure she's got a much better handle on the particulars because of it. (I think maybe Real Analysis is - ahem, approximately - to calculus as number theory is to arithmetic.)

k

I think my problem with this concept is that saying 0.999999999…=1 seems inelegant to me.

1- (0.999999...) is an infinitesimally small value not quite equal to zero

OK, if it's not zero, then what is it? If you're so sure that it's a nonzero value you must be able to tell me what it is! And the leaving-out or writing-in of ellipses makes all the difference here! No ellipsis means that the number ends just where you leave off.

When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one.

No, we are talking about the decimal representation of a number, that is, how do we write it? We all have a concept of what the integer "one" means. We are discussing whether it is more valid to write 1.00000.... or 0.999999... to express that number. And the point here is that either way is equally acceptable, both decimals (yes! including the 1.0000000...) must be extended to infinity to correctly represent the integer we all call 1.

The proof that shows that writing 0.99999... means the exact same thing as writing 1.00000... shows how to write it as an infinite series (because that's what decimal really is), that is, an infinite sum of numbers. The value of the series approaches 1 as the number of the terms in the sum becomes infinite. For any small difference from 1 that you choose to inquire about, I can always add enough terms that I am closer to 1 than the difference you've specified. (This is more or less the epsilon-delta definition of a limit: you prove that for ANY small difference you are interested in, no matter show small, you can make your function/number/decimal representation get closer to its limit than that specified difference.)

0.999999999…=1 seems inelegant to me.

So does 0.33333... = 1/3 also seem inelegant? The problem is these silly, finite sheets of paper on which we are supposed to write!

I think the problem some of us (non-mathematicians) have with this argument is that we view 1 as a 'pure' number, being unity. we don' need no stinking infinite string of zeros after the dot. it's just one. and to say that .9999999999999999999999999999999999999999999999999999999999999999999999999999... is equal to 1 is just yentzing with perfection.

but then, math isn't an exact science.
-ron o.

No one anywhere said it was a better or more convenient way to write it. Certainly the most convenient way to write it is just 1. But the point is that the other way is equally valid. That's all. Even a mathematician probably might not find a good USE for it. But it's important to know it's there!

does 0.33333... = 1/3 also seem inelegant?

No it doesn't. The reason that 0.33333333…=1/3 doesn't seem inelegant is that there is no other way to give the decimal representation of 1/3. 1.00000000… is vastly superior to 0.99999999… as a way of representing 1.

There is also the minor problem that there seems to be a conception that adding 9s to 0.99999, even an infinite number of 9s doesn't get to 1. The difference when you get the infinite number of 9s on the 0.9999999… is 0.0000000…0001.

You run into another communication problem using the term integer with computer types who may not have the math background. In computer parlance, 0.99999999… cannot be the representation of an integer. Neither can 1.0000000… There is no provision for representing a fractional part of an integer since there can be no fractional part.

Teach me to take a half hour to finely craft a response

OK, twenty minutes

The difference when you get the infinite number of 9s on the 0.9999999… is 0.0000000…0001.

No, it can't be. If you know where to put that last "1" then you don't have an infinite number of digits anymore. That is the difference between the infinite and finite number of digits.

1.00000000… is vastly superior to 0.99999999… as a way of representing 1.

Is it? That's an opinion. When you're talking math, you're not really dealing with opinions. Therefore, as I said to tsuwm above, it may be ugly to use 0.999999..., but that doesn't make it untrue.

Bean's arguments are compelling and elegant but nobody seems to have an answer as to why, if (2.9999.....), the integer 3, and (3.00000...0001) [that is, an infinite number of zeros before the final one]) are the same and truly equal, they produce different answers when plugged into certain functions.

>(3.00000...0001) [that is, an infinite number of zeros before the final one])

you lose me with this. how can you have a *final one after an *infinite number of zeros??

I think there has been some confusion here. (Which explains why I feel like I am not being understood.) 3.000....00001 is not the same as 3. I meant that 3.000...000...000...000000... is the same as 3, the integer. (no final nonzero digit, just zeros to hell and back and then some more) Just as 2.9999999... is another way to write 3.

I think the crucial thing here is the infinite digits; you simply can't allow yourself to imagine the digits having an ending. Which is super-difficult because we as humans are on the whole pretty bad at imagining infinity. I mean really, really trying to imagine infinity.

I'm not very good at imagining infinity but I can work with it fairly well. Just as, say, a doctor can't really sit down and visualize everything that happens in the human body, all at once in real-time like some kind of bizzarre movie, but they will believe in it anyway and then work with the part(s) that needs working on.

a *final one after an *infinite number of zeros?

It ain't easy, but when has Aint Easy stopped a determined mathematician?

In reply to:

---

>(3.00000...0001) [that is, an infinite number of zeros before the final one])

you lose me with this. how can you have a *final one after an *infinite number of zeros??

---

Well it seems as easy to visualize (4 minus 0.99999...) as it does to visualize (2.99999999...). They are each numbers that are very close to three, but on opposite sides of the integer three. I was expressing (4-0.99999...) as 3.000...0001. Maybe it would have been more precise to have expressed it as simply (4-0.99999...).

blergh.

ya, I'm stoopit, and the ekwazhuns are real purty, but I jus don't see how .999...… anything could be equal to 1
seems to de-feat the purpose of the dessimal point. but then, maybe it ain't no dessimal point neither. could be some other ar-cane cymbal.

I think it's all hocus-pocus.

In reply to:

---

a *final one after an *infinite number of zeros?

---

Oh yeah I forgot to mention. You just add them on at that middle, right behind the decimal. More seriously, you could do it this way:

1/10^n where n is infinity = 0.0000000000000000...0001

1/10^n where n is infinity

Is that equal to zero?

I was expressing (4-0.99999...) as 3.000...0001
It is not correct to use "usual method" for computation in the infinite case.
Before computation, you have to give a precise meaning to symbols!
Infinity has been very controversial in the history of maths.
The fact is that a real number (that can be thought as a point in the line, measured respect a given unit segment" )has just one decimal representation, or two, but it can happen just in the cases similar to that one, for example
2,34 = 2,33999999999999.......

The only time I have seen this fact used was to prove that
"real numbers" are "more" than "integers numbers"
I know that it is hard to imagine that there are several steps of infinities... (infinite,indeed) and in fact it has been understood at the end of XIX century . The meaning - more or less - is that it is possible to imagine the integers numbers as a subset of the real numbers, but not viceversa.

I suppose that this post will start another discussion:
we would say that I am throwing a stone in the pond ( gettare un sasso nello stagno), in order to excite something quiet.

Thank you but I am not having trouble with the concept of infinity or infinities, and I understand that the integers are a subset of the real numbers. In fact, the relationship of integers to real numbers can be used to illustrate that there is more than one infinity. We all agree that there are an infinite number of integers, right? And yet for any two consecutive integers such 2 and 3 (or any two integers for that matter), there is an infinite number of real numbers that can be represented in decimal form between them, such as 2.9, 2.99, 2.998, 2.999... So the infinite number of real numbers must be greater than the infinite number of integers.

All I am arguing is that there are subtle mathematical differences between two numbers if one of them a tiny bit larger than 3 and the other is an equally tiny bit smaller than three. I can easily imagine 2.99999... I can just as easily imagine a number which is greater than three by the same degree. They are all three points on a number line, with the integer 3 in the middle and the two repeating-decimal numbers flanking it, equidistant on either side. The fact that the difference between them is essentially unmeasurably small doesn't change the fact that they are different.

To put it another way, take the limit of x as x approaches 3 from a number greater than three, and then take the limit of x as x approaches 3 from a value less than three. You plug those two values into f(x)=3/(x-3) and your answers will converge on to two different values. In fact, the answers converge on infinity, the first one with a positive sign and the other with a negative sign. That's because in the first case the denominator remains positive, because you are subtracting a number less than 3 from 3. As x converges on its limit of 3 from a number greater than three, (x-3) converges on zero, but remains greater than zero. The casual observer can see that for a number converging on 3 from say, 2, (x-3) remains a negative number that converges on zero "from below."

So my argument is that if 2.9999... is the same as 3, then (4-0.999999999...) is the same as three. But they are not as f(x)=3/(x-3) shows. What I am arguing more or less is that 2.9999... converges on 3 but is not quite the same as 3. Now maybe there is some flaw in my reasoning, but nobody has pointed it out yet to my satisfaction.

I think that by putting some of your above statements together, we've got all the bits we need to prove that 2.999999... = 3. The key one was where you said that you were writing 0.0000....000001 as 1/(10^n) (or 10^(-n)) with n approaching infinity. And as Faldage correctly stated, that's equal to zero. So now put it all together:

2.99999999... = lim (n->inf) (3 - 10^(-n)) - this is more or less what was stated above, right? That it differs from 3 by a small amount, namely 10^(-n).

But using the rules of limits, which have been implicitly used above, so we must be in agreement on them, re-write the RHS, so now we have

2.99999999... = lim (n->inf) (3) - lim (n->inf) (10^(-n))

Again using the rules of limits, the limit of a constant (the first term) is a constant itself, and the limit of the second term is, as Faldage said, zero, so we have

2.99999999... = 3 - 0
2.99999999... = 3

Making up an entirely different function from the one being discussed that shows that the limit OF THE PARTICULAR FUNCTION is different when approaching from above or below has no relevance to this argument, which is "how do we write the number 3?", not "can we think of a function that has a limit that is undefined at 3?". Certainly f(x) = 3/(x-3) is undefined at x = 3 but that doesn't tell you a lot about the point x = 3 on the number line.

In emanuela's post above, about the real numbers being of a higher order of infinity than the integers, it should be noted that the numbers represented by, e.g., 2.999999999... do not exhaustively represent the real numbers. A rational number is one that can be represented as the ratio of two integers. Thus, 0.3333333... can be represented as the ratio of 1 to 3. A feature of the rational numbers is that they can be represented as a repeating decimal, i.e., one in which a pattern of some finite number of digits repeats indefinitely at the end. 1/11=0.090909..., for example and 1/13=0.076923076923076923.... In some cases the repeating digit(s) will be 0, e.g., 1/8=0.12500000... In this case there is a non-repeating string of digits before we get to the repeating part. The repeating part of one of this type doesn't have to be zero; 2/15=0.13333333...

There is another type of real number, the irrational. Examples are pi, the square root of two, and e, the base of the natural logs. These cannot be represented as repeating decimals.

The the order of infinity of the rational numbers is the same as the order of infinity of the integers. Bean or emanuela could prove this for us. Add in the irrational numbers and *that's when you get the higher order of infinity.

This notwithstanding the fact that, as Alex pointed out, between any two rational numbers you can put more rational numbers, even an infinite number of rational numbers; there are no adjacent pairs of rational numbers.

From M-W:...

In reply to:

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LIMIT: 6 a : a number whose numerical difference from a mathematical function is arbitrarily small for all values of the independent variables that are sufficiently close to but not equal to given prescribed numbers or that are sufficiently large positively or negatively b : a number that for an infinite sequence of numbers is such that ultimately each of the remaining terms of the sequence differs from this number by less than any given positive amount

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The above definition explains why I think that 2.9999... illustrates the concept of a limit of 3. The limit is 3, but as x approaches the limit it becomes infinitely close to 3 without quite becoming three. Sounds like 2.99999... to me.

Bean wrote:

In reply to:

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Making up an entirely different function from the one being discussed that shows that the limit OF THE PARTICULAR FUNCTION is different when approaching from above or below has no relevance to this argument, which is "how do we write the number 3?", not "can we think of a function that has a limit that is undefined at 3?". Certainly f(x) = 3/(x-3) is undefined at x = 3 but that doesn't tell you a lot about the point x = 3 on the number line..

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Yes but my point has to do with those two values that are not quite three. The function serves to illustrate the difference between 2.999999... and (4-0.99999...). The point at which x=3 itself has no bearing whatsoever on my argument.

Of course the big problem here is that I seem to be picking one nit and you seem to be picking another.

My question is : what do you say to be a number?
You cannot treat numbers before knowing what they are, and giving a meaning to such infinite way of writing.

My answer would be " a point in the real line, in which I fixed a point 0, and a point 1, in order to have the unit to measure segments with"
In this context, which number is 2,99......?
BY DEFINITION, it is the limit of the points given by the succession
2
2,9
2,99 ...
so that your arguments prove indeed that 2.99999...is 1.


Interesting to me, the discussion here is similar to that historically happened about infinity "in fieri" = latin,for "becoming" and "in acto" (not sure) = "already existing".

your arguments prove indeed that 2.99999...is 1.

This is, I suppose, a corollary of the proof that 2=1

http://www.anvari.org/fun/Misc/Two_Equals_One_Proof.html

I feel like emanuela and I are arguing semantics at this point rather than mathematics, and it looks like the kind of thing that can just go around and around in circles (thus leading to a discussion of pi, perhaps).

I may be wrong or I may be right; I don't know. I don't claim to be a professional mathematician, and it has been 8 years since my bachelor's degree in mathematics, so the odds are that I'm wrong and emanuela is right. But my intellectual curiosity is not really satisfied. Maybe someone else can set me straight on this. There must be some demonstrable flaw in my logic.

I think the point here, Alex, is that all values of the independent variables that are sufficiently close to but not equal to given prescribed numbers holds as long as you have a finite number of 9s trailing after your 2. You can add 9s arbitrarily and you will get closer and closer to the limit of 3 as long as you have a finite number of 9s. Once you reach an infinite number of 9s (not possible in the real world, but what the notation 2.99999... means) you are outside the realm of the defintion you quoted. You are no longer approaching the limit, you have reached it.

One problem repeatedy appearing in this thread is that the intuitive meaning of some of our words is not the same as the mathematical one. In particular (not that I would ever try to get us into a political discussion) it all comes down to precisely what the meaning of "is" is. Or "equals," more specifically.

In mathematics, "equality" means that no matter how small a difference you specify, the difference between the two objects under consideration is less than that. For the matter at hand, it translates as: no matter how close you want to require, the series 0.999999... is closer still to 1.000000... . And there is no "exactly 1"; it's just a convenient shorthand for the endless decimal 1.000000..., which is the same as 0.999999..., just written in a different way.

Mathematics has some very precise meanings for its words, if for no other reason than to resolve this kind of quandary. True, if you reject the mathematical definition of the word, the discrepancy won't go away. There is an underlying assumption of "under the rules of mathematics..." that some are invoking and others are ignoring.

"Infinite" and "infinity" are two other words that cause confusion, because the mathematical and the common meanings aren't quite the same.

See - it all comes back to words, after all!

I agree more or less with everything in your post, wof.

But, about infinity, there are several questions and not correct intuitions in this thread, so that I am planning - as soon as I feel strong enough - to start a specific thread.