2/5 - sqrt(3)/5 = 2 - 7/sqrt(3). Same way that 3 - 2 = 1 - 0 (what’s the difference between that and 1?). Same way that .999… = 1.
Let me give you another example: 1/2 = 2/4, even though they are distinct numbers. What’s the difference between that and .999… = 1?
Well, this sounds like semantics to me. So you’re saying that the phrase “three minus two equals one” is incorrect, but the phrase “subtracting two from three results in one” is correct?
Would it help if I used the phrase, “When the quantities 9 divided by (10 to the power if i) are added together for all i from one to infinity, the resulting sum is one”? Would you believe that?
Those who are getting impatient with the length of this thread are overlooking the fact that at least some of the issues raised have been, historically, significant to the discipline of mathematics; and some of these controversies still continue. There are disagreements between what a number “really is,” for example, between the Platonists, Intuitivists, and Constructivist schools. (Forgive me if the names aren’t quite correct…)
The question of just how to interpret the notion of the “limit of a series” is not utterly moot. If a series is said to converge upon a certain number N as a limit, is N a member of that series? Is it the first number not a member of the series? More significantly, how sure are we that we grasp the distinction between these two descriptions, which on the face of them appear to be disjunct?
The question of whether there are, or are not, “completed” infinities (infinite series) was still alive in the 19th Century. One answer became fashionable and accepted: but that is not to say the other was shown to be self-contradictory. Mathematics, like any science, will sometimes choose one road over another for the purpose of convenience.
.9rep is obviously quantitatively indistinguishable from 1. The difference between them is “infinitely slight,” which entails, rigorously, that it is undetectible in any and all circumstances. They are two symbolizations of the same quantity; are they then the very same entity? Perhaps, after Frege, one might say that they are two senses with the same referent. Any takers?
When you perform the operation 1/3, what is the result? Is it 0.3…?
okay, how about someone explain this to me – 1 has a descrete value, 1. 1/3 has a descrete value, 1/3. Does 0.999 have a descrite value you can put your finger on? how about 0.333… – If there is no descrete value you can put your finger on (IE – its here on the numberline <point>) then it cannot be the descrete value 1, or 1/3
.333… does have a discrete value, which is exactly equal to 1/3. Or would you like to argue against the mapping from decimal representations of real numbers to real numbers?
I think I see some of your confusion. There’s a difference between talking about actually completing an infinite operation, and talking about the result of that operation. Of course, we know that no one can ever add up all the 9’s that go into .999…; it would take forever. But if you could, it would very well be equal to 1. I can appreciate how this might sound like trickery, but you can do this stuff and still have a consistent theory (I mean “consistent” in it’s formal sense here (i.e., there are statements that have no proof)). Mathematicians, especially those involved in studying the foundations of math, do this all the time.
If we have a consistent theory that gives useful results about the real world and beautiful results about the abstract world, why would you reject it?
I’d agree with this in one sense, but not in another. The disagreement is between formalists and platonists. Formalists more or less believe that we’re just manipulating symbols on paper, while platonists believe that our abstract notions have some basis in reality (it’s the old “invention vs. discovery” question). The thing is, this is one of those questions where the evidence comes down pretty strongly in favor of whichever side you already believe. Since the distinction makes no difference to the working mathematician, and we’re not discussing a metamathematical concept here, the distinction isn’t particularly important.
Now here’s the sense in which I won’t agree with you. Both schools will agree that natural numbers can be defined as successor sets, that rationals can be defined as an equivalence relation on N[sup]2[/sup], that reals can be defined as the set of Dedekind cuts on the rationals, and so on and so forth. Whether you believe that numbers have abstract existences or they’re just handy concepts, you probably won’t disagree with the definitions.
I’m not aware of any school of philosophy that includes the limit of a series in that series, unless it already occurs in there (i.e., all terms after a certain point are zero). If you have a link to such a thesis, I’d appreciate it.
(btw, I’m not sure that “the first number not a member of the series” really makes sense, unless you’re using a set-based definition of series. The one analysis book I have doesn’t, so if you could direct me to something that does, I’d appreciate it.)
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This is true. However, as you point out, modern mathematics does complete its infinities. Since you were the first person to mention this, I find it unlikely that others were arguing from this standpoint.
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Ah. Now we seem to agree. .999… and 1 are definitely two different representations of the same number. To be more precise, .999… and 1 point to elements p and q of R such that =[sub]R[/sub](p, q) is true.
THat’s fine for you, Ultralifter, but I think what kinoons was really asking is for you to not change the representing form of .3333… and point it out on the number line. That means no 1/3, one third, or any other way of saying it EXCEPT .33333…
The people that retaliate with the style you are choosing are comparable to the people who, when asked the question “Why did that man die?”, and reply with “Because it’s God’s Will.” It’s a refusal to look behind the curtain attitude.
It’s also important to point out that the common reply to the proposal of “show me the value of .333…/.666…/999… in a finite form” is “the finite form is 1/3 or 2/3 or 1.” This is a reciprocal response, and proves nothing, it only restates what is already being argued about.
If someone wants you to reveal your magic trick after you’ve shown it to them several times, the only way to show them is by uncovering the wires/trapdoors/mirrors etc… not by showing them the trick again.
this would be a reasonable point if, when dividing 3 into 1, you were able to finish the problem. But since you could never finish it, there is never a “result”. And 1/3 has a “result”, but it’s for use with finite numbers, or pies, or splitting the dinner bill.
1/3 and .333… are COMPLETELY different ideas.
It has nothing to do with ‘like’ or ‘dislike’ my finely filtered friend.
And all an infinite number of 9’s get you is an infinite number of fractions. And because the fraction represents the difference between .999… and 1, and that fraction is as infinite as the 9’s, .999… can never truly equal 1.
Here;
http://www.c-parr.freeserve.co.uk/hcp/infinity.htm
Hector C. Parr discusses infinity in mathematics. In part;
What that means, I think, is that .999… is the same as 1/9, 1/99, 1/999, 1/9999… .Right?
You’ll always have that annoying fraction.
But I do agree that, for mathematicians, .999… = 1. It has to.
Peace,
mangeorge
sorry for posting three times in a row, but when you gotta say something, sometimes it’s hard to wait till you’ve read it all…
The anti “.999… = 1” people here don’t reject the notion if you’re only saying it in order to make life easier, and to move on to other things. But NO ONE should ACCEPT it as true, and swear by it like a fanatic. EVERYONE should know that it’s a “wink wink” situation, just so we can sleep at night.
It’s the same as saying that we all know how utterly pointless life is, how we’ll never make a point in this universe, how each of us will just die someday and it will all seem like it was for nothing. But we don’t let it ruin our lives. Sometimes we can still see the worth in eating a banana split sundae.
But we all still know that we’re doomed.
But it’s not a “wink wink” situation, it’s an absolute proven fact.
I’m consistently impressed at your ability to find teeth-clenching metaphysical significance that keeps you up at night in trivial mathematical constructs.
No, there’s a difference between arbitrarily long finite sequence of 9’s, and an infinitely long one. Parr’s point is the same; for any finite value of n, 1/n is not equal to zero, but the limit of the sequence is 0. And before you claim that .999… just converges to 1, let me remind you that any series is equal to the limit of its sequence of partial sums (cf. Binmore’s Mathematical Analysis).
The wires, trapdoors, and mirrors are all found in the unusual properties of series (in a formal context, every series is an infinite sum, so saying “infinite series” is redundant).
If you want to argue that .333… is not equal to 1/3, I’d love to hear what you have to say. Might I add that neither is what we actually mean by 1/3? I don’t think you would like Dedekind cuts, so I won’t introduce them just yet.
You’re right; they are different ideas. They also both happen to represent the same number. This is one of those cases where we can talk about the results we’d get from finishing an infinite process in a manner consistent with what we’ve already said.
P.S.: All the numbers we’re talking about are finite, and every decimal representation is of infinite length. But when all the digits after a certain point are 0, we leave them off, and if the digits after a certain point repeat (like they do with any rational number), we use the ellipsis (…) or a bar over the repeating part to indicate that. Translation? 1 is infinite in exactly the same way as .999… is.
how can you explain that 1/3 and 0.333… are the same value? one is a specific value, exactly 1/3. No infinity or repeating necessary…
if you want to say that 0.333… is an exact value, and therefor is equal to 1/3, than 0.999… must be an exact value and therefor -not- equal to one.
Except that the 1 and the 3 are both followed by an infinite number of zeroes. We just omit them to speed up our typing.
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Are you even listening to what you’re saying? Given that 1/3 = .333…, it follows that 3 * 1/3 = 3 * .333… 3 * 1/3 is obviously one, and 3 * .333 is obviously .999… If you can explain why I’m wrong, I’d love to hear it. (Specifically, I’m arguing against your statement that if .333… = 1/3, then .999… != 1.)
Let me just repeat one thing: intuition does not apply to infinities.
except for the fact that the 0’s behind the 3 or the 1 have no value, and therefor are omitted. the repeating 9’s and 3’s do have a value, and must be there…
1/3 is a definite value – how can any number, such as 0.333… or 0.999… that never ends have a definite value, it never ends, how do you assign a value to it?
It appears that you have the conception that the expressions “1/3” and “0.3rep” describe different numbers. Although I can understand the confusion resulting from 0.9rep, I’m not exactly sure how to explain this. Do you really think that that 1/3 can’t be expressed as a decimal? How about 1/4 and 0.25? Do you believe those are the same thing?
How about a base 3 representation where one-third = 0.1 (no repeating) and one-fourth = 0.020202rep? Do you believe those are equal?