I don’t agree with that but I do agree that 3/3=.99…
4/3 <> 1.332…
4/3 = 1.333…
Likewise for 5/3, and 6/3. They are 1.666… and 1.999… respectively.
Darrell
I can’t agree with 6/3 = 1.998… but I can agree that 3/3 = 0.999…
You seem to have ignored what the ellipsis means. You stress your point rather than being totally accurate.
From a non-mathematically rigorous but somewhat intuitive view:
You have a rounding error. 4/3 <> 1.332…
You are adding 0.999… + 0.333… This is not 1.332…, because you fail to remember the 9 after the third decimal place, and round that into the sum. When you add the third decimal place, you have to carry in the 1 from the sum of the fourth decimal place (9 + 3 = 12). If you try your sum at the fourth decimal place, you must carry over the 1 from the fifth decimal place. This continues infinitely. Ergo, you must accept that 4/3 = 1.333…
Notice, in fact, the very way you had to write the number - 1.332…
By that notation, the 2 is repeating, not the three. But we all agree that 1.33222… <> 1.33333…
If you were trying to state the 2 is in the final decimal place, then you are misunderstanding the infinity. There is no final decimal place. Just because we only write the first couple of threes does not mean you only start adding at that decimal place. The whole point of the ellipses is to show the threes carry on infinitely. And so do the nines. Which means you can’t just truncate them in your addition like you do.
There really isn’t anything to argue over here. It is (as with most things) a matter of definition. If we say that 0.333~ means one-third, and thus that 0.999~ represents three-thirds, then 0.999~ = 1. But somewhere on that eternal number line, there is a point that can accurately be described as 0.999~ which is slightly smaller than 1.
So, if you come across a 0.999~ in the wild, make it show some ID.
“Everyone has won, and all must have prizes.” –Alice in Wonderland
I only hope you are joking, because if as a result of definitions (and as you say–most things are) we conclude that .99…=1, then it is illogical to simultaneously claim that this number is slightly smaller than 1. It’s either 1 or it isn’t. Come down on one side or the other, because right now you are strattling the fence and mathematics doesn’t allow for that.
If you can indeed “put your finger” so to speak on the point on the number line corresponding to .99…, then your finger isn’t moving. It’s at a very specific point. If this point so hapens to be “slightly smaller than 1” then I can put my finger on a point between yours and the point “1”. This process could theoretically continue indefinitely, IF .99… is indeed slightly less than 1.
The real numbers are a dense, connected set. This means if what you claim is true (the point corresponding to .999… is slightly less than the point corresponding to 1) then not only can I put my finger on a point between yours and 1, but infinitely many such points. IOW, my point is always closer to 0 than your point.
This should shed some light on the inadequacy of vague statements such as “slightly less than 1.” If it is indeed less than 1, then what’s the difference? IOW what is the result of 1-.99… ? If they are not equal, the result is nonzero.
Even in nonstandard analysis, where things like infinity and epsilon are not just vague notions but actual defined concepts, the expression .99…=1 is usually taken to be a true statement. The difference is not some nonzero “epsilon” because the meaning of the notation .99… remains that of the sum of the infinite series .0+.09+.009+… and this sum is 1.
Yes, by definition.
**
…And if there happens to be an “alias” line on that ID, it may very well say “1”.
No there isn’t.
No there isn’t.
No there isn’t.
[Repeat to infinity.]
Hmmm, I wonder if you could say, taking a different tack, that there could be infinitely many possible numbers between the two - pitting infinity against infinity. Since infinities are of undetermined size, you couldn’t say one is greater than the other and thus couldn’t say the first rules out the second. Obviously I’m just playing devil’s advocate here.
Note that I have coincidentally linked to your page in this discussion already!
Sorry that that typo. Of course, what I intended to write was my point is always closer to 1 than your point. The point is (pardon the pun) is that “slightly less than 1” does not specify any particular point on the number line. If .999… is indeed some real number, then it corresponds to exactly one point on the (real) number line. So if it’s less than 1 (even “slightly”) then you need to tell me how much less, specifically. IOW, you need to name me a real number (just one will do) that is between .99… and 1, noninclusive.
It’s not a “different” tact, it’s the same tact, since if indeed .99… and 1 are not the same number, not only can we say there are infinitely many other numbers between them, but due to established properties of the real numbers, that’s what we actually are saying.
**
I’m not sure I am clear on what two concepts of “infinity” you are referring to. One is obviously the “infinite” number of numbers between .99… and 1 that I alluded to (assuming they are not equal to begin with, which they are). The other infinity, I’m not sure what you have in mind. If by chance you are referring to something of the effect of .99… being infinitely close to 1, as some like to argue, then this is an infinite concept of the small kind (infinitely small), as opposed to an infinite concept of the large kind (infinitely large).
If so, then of these two infinite concepts, one is clearly “larger” than the other, so I don’t see your point.
If by chance you are somehow referring to two different infinites of the LARGE kind, and thinking about whether or not it makes any sense (in any abstract way even) to say that one is LARGER than another, then congratulations—you’re not alone. There are indeed different “magnitudes” of infinities, so to speak. If one is interested in such things, one should look a little into the areas of math dealing with cardinals and ordinals.
Without going into any specific details of cardinals and ordinals, here’s a quick example: Although there are infinitely many rational numbers, AND infinitely many irrational numbers, there are , in a very well defined sense, more irrational numbers than rational numbers. Chew on that for a while!
Interesting enough, is the same mathematics that tells us we have “more” irrationals than rationals, also tells us that if you take a line segment and cut in in half, both segments (both the original and the new, shorter one) contains the SAME number of points. In a sense, that is.
**
Indeed you have. I guess that thing actually does get read every now and then.
In response to Darrell:
Actually, I’m not joking.
Keep in mind that the exact number 0.333~ is not equal to one third. Close (oops, I said “close” again), but no cigar. Fractions like one-third just can’t be precisely expressed in the base ten system (cf. my first post in this string).
Just cuz we can’t put a finger on a specific point and say, “This is Mr. 0.999~” doesn’t mean he isn’t out there. Consider i [that’s italic lower-case i, meaning “the square root of -1”]. That dude is so far out there that it isn’t even close to the number line, but it does exist and, when plugged into the right formula, can even be made to do some useful work.
I know I’d explain this better if I muttered on at length, but a few minutes back the boss stormed into the building and I need to duck and cover.
I can rigorously and mathematically prove that it is exactly equal to one third. So can many of the other people on this thread. Can you give us a proof that it’s not exactly equal to one third?
And i is, indeed, close to the number line, if you consider “a distance of 1 away” to be close, and I can very easily put my finger on it. Sure, I have to extend the real line to the complex plane to do so, but that’s OK, because I know how to make that extension. Can you tell us how to extend the real numbers in such a way as to include the number .333~, which you claim is different from the number 1/3?
You’re joking.
How many times must we say it in this thread and all the others?
0.333~ is precisely equal to 1/3.
Not close or near or approaching or about. Equal.
We can express this perfectly well in base 10 or any other base. This is not the slightest question about this. There is nothing to argue about. There is not another side. Anybody who argues against it is purely and simply wrong.
Your posts show that you just don’t understand what is going on here and muttering on at length would only dig you deeper into the hole you’ve dug for yourself.
Gee RM, are you hunting me down for the kill? I’m just trying to have some fun here, lighten up.
Ah hah!!! Point well taken. Just playing devil’s advocate here, trying to find a crack in the reasoning 1 = 0.999…
OK then, you’re not joking. I wasn’t sure, to tell you the truth. The power of intuition is quite apparent, and has a strange way of making otherwise reasonable people disbelieve what has been proved to them. Will you accept the possibility that, perhaps, your intuition may be wrong? If so, I will attempt to address your concerns constructively, because at one time I felt the same way as you.
The first question I have of you is: If the exact number .33… is not 1/3, then which number is it? Before you answer that, however, let’s review the long division algorithm for this problem of 1 divided by 3:
.333... and so on...
________
3 |1.000...
9
_____
10
9
______
10
9
______
and so on...
(someone pls tell me how to include text verbatim, with a monospaced font)
Actually, in this context, the quotient .333… is just our way of denoting that we can never actually finish the algorithm in any finite number of steps. All we really know, at the time we were first introduced to this problem (usually long before any calculus is taken), is that there seems to be a pattern that so long as we continue the process, we keep tacking on 3’s to the quotient. When we wrote “.33…” at that time, this was just another way of saying that.
Then someone, somewhere, somehow, (again, before calculus class) mysteriously told us “this means 1/3 = .33…” and we blindly accepted this (at least all but the brightest did, anyway).
The point is, at this juncture (at least according to the above demonstration) we are simply accepting on blind faith that 1/3 really does equal .33… when in fact, all the above really shows is that so long as we continue the steps, we get a number of the form .33…3 (that is, there is a final terminating 3 somewhere, depending on where we choose to stop with the procedure.)
Hence an understandable concern would be: Does this have anything to do with .33… since this is a recurring decimal (ie, the string of 3’s never stops)?
But, when you did stop somewhere, you had a certain remainder. What is the relevence of this remainder? It’s VERY relevent, since it’s not dividend divided by divisor that equals the quotient, it’s dividend divided by divisor equals quotient plus remainder.
The next question for you is: What is the remainder if we theoretically NEVER stop with this algorithm? Considering the fact that the notation .33… is just our little way of denoting the fact that we never (at least never in theory) are really 'stopping" with the algorithm since there is NO terminating step we can ever come across, what could the remainder possibly be?
Here’s the most important question to you I can think of presently: What MUST the remainder be if indeed .33… is NOT equal to 1/3? Something nonzero, right?
Problem is, the only way to have a nonzero remainder is to actually terminate the algorithm at some point, as suggested above. But the notation .33… denotes that we NEVER terminate the algorithm. Hence the “remainder” associated with the quotient .333… must be 0. IOW, we have:
dividend/divisor = quotient+remainder
1 / 3 = .33… + 0
1/3 = .33…
Then we take caluculus class at some juncture, and study more formal ways of showing, and in fact proving, that 1/3=.333… and also 1=.999…
**
If indeed we can’t “put our finger” on it, then on the contrary, he isn’t “out there” on the number line. Every real number corresponds to exactly one point on the number line and, conversely, every point on the number line corresponds to exactly one real number. If Mr. .99… exists at all as a real number, he necessarily is “out there” and we necessarily CAN “put our finger on it.” IOW, you can sketch a number line, plot a certain point, and say “There’s Mr. .99…”
**
i is not on any REAL number line, I assure you, so this is irrelevent to the discussion. The real number line is reserved parking for real numbers only. i is not a real number.
However, there is a common way of “graphing” complex numbers of the form x+iy where the x-axis is the “real” part and the y-axis is the “imaginary” part. The number 0+1i, the same “i” you are referring to, corresponds to the point (0,1) in this coordinate system. Note that in this coordinate system, all real numbers (which have form x+0i) correspond to precisely the same points on the x-axis as if this axis was actually a real number line. Again, this really has nothing to do with the discussion of .999… since it is well established to be a real number (all recurring decimals are not only real numbers, but RATIONAL even.)
Oops, forgot the emoticon. 
Can I just point out for those who fail to realise it, that if you are using the ~ notation for a repeating decimal,
0.3~ = 0.33~ = 0.333~ = 0.33333333333333333333333~ and so on to infinity.
There seems to be some belief that 0.333~ is less than 0.3333~ is less than 0.33333~ and so on, in the same way that 0.333 is less than 0.3333. Guess again.
BJMoose said:
I’m sorry, but you’re just wrong. I don’t know why you have that misconception. That is the whole point of the ~, or the …, or the bar over the repeating part, or whatever notation you wish to use. It means repeating infinitely. That allows us to write 1/3 in base 10. Note also how I just wrote 1/3 in base 10. Oops, did it again. 
You are correct that the ellipsis is a convention of notation, because otherwise we would find it difficult to write out the decimal form in base 10. But that’s the point of the ellipsis - we define it to mean “repeating forever”, and suddenly we can write the decimal form in base 10.
Reflecting on what reprisal just said, perhaps it’s the symbol that’s confusing you. Often ~ is used to mean approximately. In this case, though, it is being used instead of the ellipsis (…), to mean repeating.
0.333 ~= 1/3 is not what we’re writing.*
0.333~ = 1/3 is what we’re writing. This is instead supposed to mean
0.333… = 1/3
As with others, I don’t know why Cecil used the ~ format that we’ve been suffering in this thread.
- I cannot find a code for the tilde over the equals. I found it in charmap on Windows, but it requires Symbol font.
Yes, but you may not actually be one with Cecil, you may only be .999~ , and that seems to cause a lot of problems.
So what’s the square root of 0.999… ?