Of course it’s a valid endpoint, it would be drawn as a hollow circle instead of a filled in circle like we’d have at zero … you really don’t know the difference between closed and open sets, do you?
The length of the line segment [0, 1) is _____ ?
If you can’t answer this, then you should probably stop using this notation.
nm
As noted above, if it were true, then 0.000~ would be greater than zero. Makes zero sense.
You’d think these things would be persuasive!
Neither of those questions make sense when dealing with infinities. You’re starting with a conclusion and cherry picking what you think is evidence for it. When faced with obvious and logical mathematical proofs of your errors you simply ignore them. I’m going to repeat on such proof as my parting shot.
Every one of the numbers 0.3, 0.33, 0.333, … differ from 1/3 by a defined number.
1/30, 1/300 1/3000, …
As we approach infinity, what value do we approach?
[0, 1) is simply [0, 1] without point 1.
If you remove point 1 then all the other points are left, e.g. 0, 0.9, 0.99, 0.999, …
There is no largest number left in the set. Since we cannot define a largest number, we cannot define an endpoint. So, the correct answer to your question is:
The length of the line segment [0, 1) is undefined.
That’s true. It’s obviously also true to say “Every one of the numbers 0.3, 0.33, 0.333, … differs from 0.333…”
Nevertheless you can find every segment [0, 0.3]∪[0.3, 0.33]∪[0.33, 0.333]∪… in one of the numbers
0.3
0.33
0.333
…
already, if you represent them by the segments
[0, 0.3]
[0, 0.3]∪[0.3, 0.33]
[0, 0.3]∪[0.3, 0.33]∪[0.33, 0.333]
…
So, what is your answer to the question:
Do you agree that for a real number to be greater than a set of smaller numbers we need to show that the real number contains a segment which cannot be found in one of the smaller numbers?
Why do you say this? The length of [0, 2] is 2, and the length of [1, 2] is 1.
So given that [0, 2] = [0, 1) ∪ [1, 2] it looks to me that [0, 1) has length 1.
Of course, I can see why you want it to be undefined. But if anything you should be arguing that the length of [0, 1] is undefined, because it has that extra mysterious little bit at the end (the 1) – like counting the distance between fenceposts, that interval includes two fenceposts and you should only be counting one.
This is another point upon which you are in conflict with mainstream mathematics. See Lebesgue Measure.
Thank you, The Great Unwashed, excellent response to this ridiculous claim. I was going to use the fact that a point has a length of zero, by strict definition; therefore the lengths of [0, 1) and [0, 1] differ by just the length of a single point, or zero. If two numbers differ by zero, and again by strict definition, they are equal.
This comment wasn’t directed towards me, but I’ve some comments about it, my apologizes to naita if I’m stepping on your toes here a bit …
‘It might not be “logical” …’ — I agree here, you’re not following any manner of logical deduction, intuition or observation … this leaves only a faith-based belief system … or in other words, religion … not that this is a bad thing; after all, everyone know the hard fact that I make a size 12 dress look hot … but it is never used in mathematics … we must always deduce higher hard facts from lower hard facts in a logical way. [sup]note 1[/sup]
‘… it might not be “reasonable” …’ — Again I agree, there is absolutely no reason for the claims you are making, they are unreasonable and you are being unreasonable about your claims … thus the complete lack of an answer to the many times you’ve been asked “why” … there is no answer as your claims have no reason of any kind.
‘… it might not be what “makes sense” …’ — There is much about the universe that doesn’t make sense to me … as far as I know, every prediction made by Quantum Mechanics that can be experimentally tested as been shown to be true … now that’s some crazy-assed shit for there not to be any counter-examples … (every electron I shoot through a slit will hit my target … in a perfectly straight line goddamit !!!)
‘… it might not be “useful” …’ — Here we have the truth of your claims … they are useless … and by that I mean they have no use of any kind what-so-ever … you can’t even find the length of a simple line segment with your claims … that I would say is the most useless condition I can think of … however … using the established definition of infinity is of amazing utility … with this we can correctly predict the effects of gravity and her associated instantaneous acceleration … look around yourself … see all the things effected by gravity? … using your dogmatic definition of infinity will always give you wrong results … every time …
In conclusion, there is no arguing your faith that (0.999…) ≠ 1 … but it is good you freely admit this faith is illogical, unreasonable, senseless and completely useless … for if and when you look out, you will see in the universe that in every place, at all times and at every temperature … (0.999…) does equal 1 …
Note 1 = If a mathematical principle is logically deduced, we call it a theorem … we do have a few mathematical principles that are not deduced, and we call these conjectures … here are six widely known mathematical conjectures … note the seventh here has actually been proved and is now considered a theorem …
Nice …
More important than the niceness is that mathematicians do spend a lot of time thinking very precisely about these things.
You could redefine measure in such a way that the length of [0, 1) is undefined, but you would have to give up at least one one of:
[ul]
[li]The length of [0, 1] is 1[/li][li]The length of a single point is 0[/li][li]The length of the union of two disjoint sets is the sum of the lengths of the individual sets[/li][/ul]
Arguably we pick these three properties for reasons of usefulness, which is one of the things netzweltler dreads, but it is very hard to imagine a definition of “length” that doesn’t have these properties.
@The Great Unwashed
@watchwolf49
@leahcim
Can we do arithmetic with points?
[0, 1] - 1 point = [0, 1)?
[0, 1] - 0 = [0, 1)?
[0, 1] - 0 = [0, 1]?
[0, 1] - 2 points = ?
[0, 1] + 10 points = ?
Please teach me the rules of arithmetic with points to be able to judge about it.
Until then I stay with the definition that distance can only be defined between two well-defined points on the number line.
The only point that is zero distance away from point 1 is point 1 itself. If you remove this point from [0, 1] the length cannot be the same as before, because for none of the points left in [0, 1) is valid that it is zero distance away from point 1.
If by this you mean that only closed intervals have a length, then there is a whole branch of mathematics called Measure Theory dedicated to disagreeing with you. If you wish to chose another definition, then you would have to pick which of the axioms of measure theory you want to give up.
(Incidentally measure theory does admit the concept of non-measurable sets, but they are a lot more complicated than half-open intervals.)
This is a point you will find broad agreement on.
That is incorrect. Even though every individual point in [0, 1) has a distance from 1 that is larger than zero, the infimum of the distance is still zero.
We have an “apples” and “oranges” thing going on here …we’re using arithmetic with length … so …
We are asking about length … which is a spatial dimension … points have no spatial dimensions … line segments do …
One rule of arithmetic is that we can only add together like values … 7 apples plus 4 apples equals 11 apples … we can never add together values that are unalike … 7 feet plus 15 acres equals nothing, adding feet to acres is meaningless …
So, we define a line segment with [0, 1) … the endpoints of this line segment are very clearly stated in the definition, they are zero and one … your quibble seems to be that the endpoint itself is not included in the line segment … however the endpoint itself has no spatial dimension … whether the endpoint is a member of the line segment or otherwise, it is not considered when determining length … it is length we are asking about, the inclusion of endpoints are not relevant to the question …
If we were measure the length of a board … do we care if we include the edges … never … the tape measure reads exactly the same whether we include the edges or not …
No worries. I wasn’t planning on replying anyway. I considered writing something about the length of an interval, but I figured netzwelter would reply with non-math, and so he has.
So I’m just popping in with a question that occurred to me looking at visualisations of the formula for the area of a disk.
I believe netzwelter has accepted the existence of the number π, but how does he feel about the equal sign in A[sub]disc[/sub]=πr[sup]2[/sup]
Any proof of that formula does after all depend on adding infinitely many sub-units of the disc together. Any finite number of sub-units gives us an area that is not equal to πr[sup]2[/sup]
Oh, and you can choose to define π as four times the inverse ratio between the area of a square and its inscribed disk, but then the question becomes “Is the Circumference C = 2πr ?”
Is this a “none of the elements is as close as zero distance to point 1, but all together can do that” thing?
Absolutely not. [0, 1) doesn’t have an endpoint. There is no largest number in [0, 1) that could serve as an endpoint.
I think it’s time we all stopped trying now. 
What an odd thing to say … if this is a “line segment”, then by strict definition it has two endpoints … if an object only has one endpoint, it is called a “ray” … it’s like you’re saying the notation [0, 1) is invalid and doesn’t describe a line segment exactly …
Are you seriously claiming the endpoint doesn’t exist?
If you define the distance between a set and a point, as is conventionally done, as the infimum of the distances between the points in the set and the given point, then yes.
I.e. for any point x ∈ [0, 1), distance(x, 1) > 0
BUT
distance([0,1), 1) := inf {distance(x, 1) | x ∈ [0, 1) } = 0
Incidentally, a fundamental property of the real numbers, the greatest lower bound property, guarantees the existence of this infimum in the real numbers.
[0, 1) doesn’t contain (one of) its endpoints. It still has both endpoints, but one of them is not an element of the set.