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christophm

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More Points in a Line or in a Plane

Maybe I should be putting these questions in the puzzles section?  Just trying to maintain some interest until the REAL MATH guys discover the site.

OK (let's agree or at least pretend!)
  - a line contains an infinite number of points
  - a plane contains an infinite number of lines; that is a plane contains an infinite number of lines each line with an infinite number of points.

Seems like a plane should have an infinite number of points more than the infinite number of points in the line.  This is a comparison of two like kinds/classes of infinity so they (the number of points in a line and the number of points in a plane) are really-really equal.  If those two infinities are equal (equal means, "every point in a plane can be mapped to a single unique point in a line and the converse is true") than describe such a (isomorphic) mapping.

thanks - chris-m
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acerola

Once again, I just dont understand what you want.

Let me throw in some guesses:

First, you said "infinite number of points more" when it is really "infinite number of points times".

The number of points in a line is:
lim x->infinity (x)
The number of points on a plane is:
lim x->infinity (x^2)

You have y points in a line and x points in a plane. The number of points in a plane would be x*y. But that isnt important.

Second, you want to map every point in a plane to a point in a line and vice versa. The first is possible, but the second, I dont think so.

To map every point in a plane to a point in a line is a scalar function of two variables, such as, z=f(x,y). For example, z=x+y. Every point (x,y) in the plane is mapped to a point (z) in the line. And all points in line are mapped to. But the mapping isnt unique. The point z=5 in the line can be mapped by the points (0,5) or (2,3) or (10,-5), and infinte more.

To map every point in a line to a point in a plane is a vector funcion of one variable, such as, (x,y)=F(z). Every point (z) in the line is mapped to a point (x,y) in the plane, but there are many (infinite) points in the plane that are not mapped to.

For example: (x,y)=(z+1,2*z)
The point (5,8) in the plane is mapped by the point 4 in the line. But the point (5,9) cant be mapped. You may change the function to map (5,9), but then other numbers wont be mapped.

In conclusion, a plane has two degrees of freedom, a line has just one. Thus, you cant have a unique maping of every point in a plane to a point in a line and you cant map every point in a line to a point in a plane and cover the entire plane.
A line is a one-dimensional object; a plane is a two-dimensional object.

A line has an infinite number of points in one dimension, while a plane has an infinite number of points in each of two dimensions.

Hence, a line has an infinite number of points, but a plane has an infinite number of points *squared*.
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Hi guys,

I assume you comment seriously so I feel I owe you a serious explanation.

This question has to do with Georg Cantor's (also seen as Kantor) study of classes of infinity.  You may have heard the term Aleph-Null, Aleph-One, etc.  These were Cantor's terms for the different classes of infinity.  ((There is an Infinity automobile in my neighborhood with the tag "Aleph0", I asked the lady once if she was a mathematician and she told me that she wasn't but her husband was and was I asking her because of the license tag.  She said her husband got the tag and ha told her the other mathematicians would know it - she didn't know what it meant).

I just ran another instance of IE and used a search engine to look for "Aleph Null"  (perhaps you might try www.mathacademy.com/pr/minitext/infinity/).

Some infinities are (forgive my layman's term) 'denser' than others.  For example the number of points in a line is a denser infinity than the number of counting numbers.  One could establish an arbitrary point on a line and map the counting numbers as 1 inch to the right of the point, 2 inches to the right of the point, 3 inches to the right of the point, ....
 You see in this case the counting numbers {1, 2, 3, ...} are mathched in all their infinity to points on the number line but there are lots of points on the number line that have no matching natural number; for examples, the sqrt of 2 to the right of the zero point, PI to the right of the zero point, the tangent of 17.5 degrees to the right of the zero point, any irrational number you can think of - plus even all the counting numbers do not include any negatives!
  In this illustration the number of points on the number line represents a denser infinity than the number of counting numbers.  Yes, both are infinite but you can see (I hope!) from my example that the points on the real number line are denser/thicker/closer/whatever than the number of natural numbers.  The end result is not all infinities are created equal - sorry! - if you don't like that you will have to take it up with the great Mathematician in the sky.

So, onward and upward.  There are different kinds of infinities, some more 'dense' than others.  For example the set of imaginary numbers includes all the reals (as numbers with a zero coefficient on the imaginary term), so the reals can be matched to the complex numbers one-to-one and still there are infinities of imaginary numbers (those with a non-zero coefficient for the imaginary term) left over.

Now.... the game/problem/question/consideration is commonly to consider two infinities and ascertain which is 'denser'; that is, determine which is the higher Aleph number by Cantor's definitions.  

My question is to examine the infinity of points in a line and the infinity of points in a plane.  Talking about a plane being an infinity of lines is a smokescreen, yes, that is true but it is disguising the real question.  Lines and planes are the same in that they are sets of points.  The REAL question is "Is the number of points in a plane greater than the number of points in a line?"  Offhand one would say "Ridiculous!, of course there are more points in a plane than in a line".

Welllllll......    Can you establish a one-to-one correspondence between the points in a plane and the points in a line.  I'm not trying to 'trick' you with the one-to-one stuff.  I mean can you come up with a formula/procedure/method/approach where you can match every point in a plane uniquely to every point in a line?, and will this procedure uniquely match every point in a line uniquely to a point in a plane.

These are serious mathematical questions, not logic or trick games.  Unfortunately you are pushing the limits of my know-how of Cantor's infinities - too many more questions and I will have run out of answers.

So... think about it, are those two infinities of different order or not?

have fun ! - chris-m
As I said in my previous comment, the answer is no.

You cannot match every point in a line to EVERY point in a plane.

You cannot match every point in a plane UNIQUELY to every point in a line.

A line has 1 degree of freedom (or 1 dimension) and a plane has 2 degrees of freedom (2 dimensions).

When you go from a plane to a line, you lose 1 dimension, thus 1 point in a line must correspond to more than 1 point in the plane.

When you go from a line to a plane, you increase 1 dimension. So a point of the line would have to match more than 1 points in a plane.

Let's take my example again. Supose the function z=f(x,y) which maps every point in a plane (x,y) to a point z in the line. Let't now make x=1 and only let y vary. We would have z=f(1,y). y can assume infinite values. to be unique, each value of y must correspond to one and only one value of z. So, we "spent" all possible values of z just varying y. If we now take x=2, there are no more avaliable values of z to be used, because they were all used when we made x=1.

Ok, now about the density of infinity. This is an old question that dates back to Aristotle. The original problem was "if space can be divided in infinite parts, how can you go from point A to point B in a finite time?" The answer is that time can also be divided in infinite parts. And there comes the diference between someting beein infinite in LENGTH and infinite in DIVISIONS. The integer numbers are infinite in length, but not in divisions. And if we take rational numbers between, lets say, 1 and 2, we have infinite numbers between them. And even more if we consider real numbers. The real number group is denser than the rational numbers group which is denser than the integers group.

I think you could say that a plane is a denser infinity, but as a phisicist, I prefer to say that they have different degrees of freedom or dimensions.
I think the answer is already given by Cantor. A line could be said to represent the set of Real numbers. A plane represents R2. Both these sets have the same size (according to Cantor), so the number of points in each is identical.

Look for Cantor's method of mapping R to R2 and you will see that the mapping does work both ways, counter-intuitive as it might seem.

From the Wolfram Research page:

"Curiously enough, n-dimensional space has the same number of points (c) as one-dimensional space, or any finite interval of one-dimensional space (a line segment), as was first recognized by Georg Cantor."
I had to cheat and look it up on the web.

any point x,y on the plane can be directly mapped to a single point on the line and the mapping is reversible;
write down X and Y in decimal,
 x = 2.5 6 7 8
 y =3.3 5 7 1
interleave their digits, 32.35567781
The mapping can then be reversed by taking every even positioned digits of 32.35567781 as representing the original x and the odd digits as representing the original y.

Similar for N dimentional space although you'd take every 3rd digit to un-map x,y,z exc.

x =  2 .5  5  6  7
y = 3 .3  6  7  8
z =1. 5  6  1  4

maps to 132.535665176487 and the mapping is again reversible.

Note you might have to copy this into notepad since EE's proportional spacing will mess up the columns in the example.
 



Thank you 'gbentley' and 'andyalder' - you are both right.  I give the points to 'andyalder' because he detailed the mapping.  BTW - Some years since I just sat around and thought about this kind of stuff.  One would have to extend the mapping example to account for all four quadrants of the XY plane because the each point in any quadrant has evil triplets (points with the same numeric values for X and Y) in each other plane.  This leads us to conclude that a single quadrant of a graph is the same order of infinity as the entire graph - Mathematics of infinity is really fun!! - You see here we have a part that is equal to the whole!!

Hi acerola  - You said, "I think you could say that a plane is a denser infinity, but as a phisicist (sic), I prefer to say that they have different degrees of freedom or dimensions."  With this statement you are getting close, you need a REAL MATHEMATICIAN (I'm just a hobbyist) now to guide you.  It would be easy for me to say that what you call "degrees of freedom" are what Cantor called "orders of infinity" but I have a gut feeling that may leave us with some wrong impressions.  I don't believe that is acceptable to associate physical (or even more than three) spacial dimensions with Cantor's classes.  Perhaps even associating various sets of numbers (integers, naturals, reals, irrationals, ...) with different classes is at the deepest plane (pun, get it 'plane') not 100 percent appropriate.

One last long-winded explanation cause I feel like I owe you.   Consider the closed numeric interval of all real numbers from 1 to 3, I believe the proper notation is
[1, 3]  (the '[ ]') indicates the end points are included in the interval).  There are only three integers in that range however there are an infinite number of rational numbers in the same interval!  Certainly it would seem to the reasonable man that though both integers and rational numbers are sets containing infinite members that the real numbers are "bigger? - no they are both infinite", "come by faster? - no we are looking at a line it's not going anywhere". My bud Cantor says to say they are different orders of infinity.  BTW - Cantor was Russian (I'm sure he's long dead now)  I hope the answers - and links easy to find on the web - will satisfy your questions some.

Thanks all for playing.  This is going to cost me a fortune in points, but what the heck, I'll think up another question.
There are two types of mathematics. One that is concrete (the one physicists use) and another one that is more abstract (probably what you would call "real" math).

This mapping is not a function. It can't be expressed with algebra or calculus. It's more like playing with the digits as if they were a string. But I agree it maps all the points uniquely.

In nature, things follow certain functions (most of them unknown). And the biggest problem of physics is that when you have too many degrees of freedom (or dimensions, or numbers of variables) you just cant solve the problems anymore. For example, you can solve the problem of a hydrogen atom, which has 1 proton and 1 electron, but it is impossible to solve, without approximations, the helium atom, which has 2 protons and 2 electrons and maybe neutrons.

So, if you could reduce the dimensions of a problem to 1 dimension, as this mapping does, all problems in physics could be solved exactly. The problem is you can't do it in a usuable way (with a function for example). You cannot put this mapping in a mathematical language.

So now I, as a scientist, must change my statement as new facts came to my knowledge. You can map every point uniquely, but you cannot find a mathematical FUNCTION that does it. It would be so good if we could...
Not sure I want the points, Not sure I accept the proof.

I looked it up on the web rather than work it out and anyway, I can see that it works for rational numbers where there is a limit to their length or a recurring which you can allow for with the positional interspersing trick but I can't see that it works for irrational numbers since you can never represent them with a string of digits.

Are we allowed zero and PAQ like the other TAs if we can't solve it? Can you seperate maths from philosophy?
You can represent an irrational number with an infinite string of digits. And some rational numbers also have infinite decimal places (such as 1/3=0.33333333...).

Enven though, this procedure would take numbers with infinite digit lenght to infinite digit lenght numbers. It is not practical or useful, but as an abstract concept it works.

This reminds me of another question: is 0.9999... = 1??? I've heard "real" mathematicians say they are not!
I have no problem with "nought point three recurring" since it's a lexical proof and I have used only about 30 letters to describe that value. But the proof requires me to accept that that two numbers that cannot be described with a finite number of words can be intermeshed simply by merging their description.

Convince me that it is valid for the irrational numbers.
Take Pi as an example. If you do the interleaving algorithm, you simply end up with two different irrational numbers. The same applies to any other irrational number.

The important point is not "can I actaully carry out the task in the physical universe". Maths says nothing about how long it might take to do something, an algorithm that takes an infinite time to run is still valid as it will produce it's result at time equals infinity. Similarly there are math algorithms that require an object to be cut into an infinite number of pieces. Not something we can do in the physical world!

I think one of the problems people have with this stuff is trying to understand it intuitively. There is no reason to expect maths to be intuitive. It is a formal system with fixed rules.

The same applies to physics, the world does not necesarily work in an intuitive fashion.

But this stuff is fun to think about!
Regards
Gordon
Welllll... you guys need somebody more knowledgable than I to take you further.  I think gbentley's last comments are very good.

1st - The cavail about mathematics not being very useful (in its more esoteric forms!) for 'real life' problems.  May I mention Karl Gauss who celebrating his discovery of 'groups' in number theory proposed a toast to "Pure mathematics' It was some years (100 and then some) before we found out that atomic particles under certain conditions follow the rules of mathematical groups.

2nd - yes indeed!, I would agree, mathematics is more targeted toward building elaborate (logically valid) consistent structures upon unprovable assumptions.  Euclidean geometry for example has two elements that are basically undefined, 'points' and 'lines' - I think we would all agree that Euclidean geometry (and its cousins Trigonometry and Analytic Geometry and ....) have been found useful over time.  Do parallel lines never meet?? Do you know that apparently Euclid was uncomfortable with that idea and held off using it in his 'Elements' until it was impossible to proceed without that assumption?  Is space curved like Euclidean geometry? Maybe it's curved like Lobachesvky said.  (dang those Russians and their names!, I think my spelling is at least close) If space is curved like Lobachesvky has postulated then parallel lines cross at some point.  Suppose we pitch out Lobachevsky because it doesn't help us with how a 'lever' or a 'motor' works and space really is curved that way - won't we look like dopes when the Alpha Centaurans finally notice us!

3rd - you want some weird stuff from physics?? How about Hawkings description/sepculation (in - I believe it was - "A Brief History of Time, it's been a 'long history' of time since I read that book!) that the universe as we know it may well be a single black hole and we are inside the event horizon.  Hawking describes a black hole as a certain ratio of mass to volume and our universe (as best we know it) meets his mass/volumen criteria.  This explains some physical phenomena, such as why light seems to curve as it travels toward the outer reaches of known space.  This effect accounted for in his speculation by the fact that light cannot escape a black hole. (Long time since I read the book - but I recall that the most interesting point was what lies beyond our event horizon??, Is there a froth/bubbles of black holes each an entire universe with its own laws of physics??

4th - this kind of stuff is fun to play with.  It was mentioned above that irrational numbers never repeat.  Hmmm... what's 'never' mean, is that like, uh..., 'infinity'?  Are there more digits in an irrational number then there are numbers in the set of integers (can you determine that

and finally, i'm trying to close this question - I feel obliged to answer every comment and my wife wants me to GET TO WORK AND STOP PLAYING!

thanks - chris-m
Avatar of Mike McCracken
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Infinity is infinity, period.

When you try to establish any kind of equivalence you are talking about something less
than infinity, such as engineers do when they reduce the scale of reality to fit what
they can actually construct.

So, for example, if you reduced the scale there would supposedly be a point when
some real number of points would be equivalent to an infinite number of points,
which is absurd.
Read "Goedel Escher and Bach" by Hofstadter.  He discusses this notion of entrapment and absurdity, and how to progress past it.  Time and time again, real advances are made by first accepting a paradox as real, and the "stepping out" of the system to find a new language of concepts where the paradox does not occupy center stage.  The fuller understanding of the original paradox only comes about through eluciation of its brothers and sisters, and to extend the analogy - the family tree, and at some time in the future when finally genetics replaces genalogy...
Newton solved the problem of calculus by "stepping out" of the constraint that a curved line can never be a straight line.  He simply ACCEPTED that a curved line WAS a straight line at any particular point, or at least that it could be parallel to a straight line (the tangent) at any particular point, and then went on from there. You could say that on a semantic level, he ALLOWED the word parallel to apply to a previously disallowed class, and in the process learned a lot more about the potential in a concept called "parallel".  He stepped out and found himself in a whole new territory.  
That is exactly the kind of thing that Cantor did too.  
>>I have no problem with "nought point three recurring" since it's a lexical proof and I have used only about 30 letters to describe that value. But the proof requires me to accept that that two numbers that cannot be described with a finite number of words can be intermeshed simply by merging their description.

Convince me that it is valid for the irrational numbers.
<<

You use the word "valid" when you should use the word "consistent".  You seem to look at math like some objective external reality, rather than a discovery of self-consistent sets of propositions.

If you seriously look at infinities, you can identify qualitatively different classes of them.  And you can see that infinities have a particular calculus of there own, and even their own brand of arithmetic.  There are pleasing symmetries etc.  You can complain that this cannot "be", but then you must be forgetting that numbers are not really beables in the first place.  (that is a nice word, be-ables, used by Bell of Bell's theroem fame).  The maths of infinity do be in one important sense, because it works!
Same reasoning as square root of minus one, and that has proven to be absolutely essential to make any progress understanding wave phenomena.  You can argue till you are blue in the face that the square root of minus one is a fiction, but if you stop and think about it the same criticisms ultimately apply to all numbers.  Remember it took centuries to convince a skeptical world that zero was a number, and just as long to get the negative numbers into circulation.

And the trace of prejudice is still there, because we call them irrational and imaginary still.
Infinite is an abstract cocept if you can´t digest it nobody can for you.
Just another note to add, topology is the place to really get into this stuff, and category theory is an even better way to understand things.  Get this, it is possible to dissect an object like an orange, mathematically, and then reassemble it in such a way as to get more than you started with.  Thats topology for you.  But dont ask me for the proof, although I could google up a reference maybe.
the number of points in the line is not the same in the plane. to illustrate, get a cartesian plane or a simple graphing paper.  locate your origin. consider the x-axis as your line and the whole graphing paper as your plane. get any point on the x-axis. draw a vertical line from it. it is now clear to see that every point on the x-axis is mapped on more than one point. not two, not three, but infinite number of points. the relation is one to many correspondence. does this answer your question? i think yes :)
An infinite set can be put into one to one correspondence with a proper subset of itself, so the existance of a many correspondence does not show that the number of points is different.
And anyway you can also do the reverse procedure.  Take the first vertical line and map each point on that line to a different horizontal point.  Since you will be a long time finishing the vertical line, might as well do another one in parallel while u waut.  Just be careful to always choose an x that is not in use.  Dont worry, there is always an infinite number to chhose from.

The above posts are excellent, but perhaps another discussion summary will help.  You must always apply simple logic.  

What "countable" means is that an element in a countable set is always reachable by some kind of counting process.  So we have to understand natural numbers 1,2,3,4,5,6,7.... before we start out on more complex things like the reals.

Now think clearly and logically:-  When you multiply the values of a finite set by two, then you do not change the number of values in the set!  MULTIPLICATION DOES NOT ALTER THE CARDINALITY OF A FINITE SET.  OK, now we extend this simple reasoning to an infinite set of ntural numbers.
When you multiply the infinite set of natural numbers by two, the size of the set does NOT change.  This is a perfectly reasonable way to proceed.

But wait a minute, when you multiply all the natural numbers (1,2,3,4,5,...)  by two, you get the set of even numbers.  Therefore there are just as many even numbers as there are natural numbers !!!!!!

Likewise there are just as many odd numbers as there are even numbers as there are natural numbers.  Follow?  

These sets all have the same cardinality, and a set that has the cardinality of the natural numbers is called "countable".  I will say it again, these particular sets have infinite cardinalities, but that are also called "countable" sets because they correspond to the natural numbers.

Now we are starting to see differences between finite and infinite sets, so we can expect that the idea of subsets follows different intuitions, the earlier posters explained this perfectly.

The kind of infinity produced here is actually the smallest kind of infinity, if you read back about Cantor earlier.

Interesting point: DIGITAL COMPUTING IS COUNTABLE!  All possible programs, all possible data programs belong to this set.

Now for the reals.  Read my lips: "THE REAL NUMBERS ARE NOT COUNTABLE".  The logic is simple once again, and you go back to reasoning from finite sets.

Take a finite set of decimals:

0.123
0.563
0.267

It is always possible to make a new number that is NOT listed, but should be.  In this case we can see that the diagonal one 0.167 is not there.  That is always going to be a problem!
There is always a way to make these kind of problem values no matter how big we make the set!  Infinity DOES NOT HELP!
Or to put it another way, if you assume that the reals are countable you will always get a contradiction, so the assumption cannot hold.  DROP IT!

Interesting conclusion:  THERE ARE UNCOMPUTABLE PROBLEMS!   To prove that you just have to show that the problem corresponds to a real number, (just about any decimal will do) because reals are incomputable by digital methods if you followed the logic.

So now you see we begin to build up the family tree for infinities, using simple logical methods.

So then we have two special new numbers, that are obviously not naturals themselves, two different kinds of infinities.  The next question is, are there more? The answer is yes, so we ask, are they countable?
And there you go.
ok here are my two cents you are corect in saying that an infinite number of points lie on a line and a plane but yu canot match a point x,y on a plain wiht a point on a line becuse (hpotheticy a line is one d) becuse a point on a line is at coordinite x so there would be more (mathamaticle) points on a plane but becuse infinity means infinity there are an equal number of points both on a line and a plane you can do the math and count them if you want but that wouldtake a while
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> The concept of "more" or "less" is for countable sets.
> The points on a line are not countable!
> You cannot reach any point by a counting process!

Hum... You can have more milk, or less milk. Can you count milk?
You can have more light or less light. Can you count light?
You can have more magnetic field intensity or less magnetic field intensity. Can you count it?
A girl can be more beautiful or less beautifull. Can you count beauty?

Discrete variables are countable. Continuous variables are not countable, but you can still compare them. Even infinite valued variables can be compared. Sometimes (it is weird, I know) you can say that one infinity is bigger than the other, or even equal. Depends on the situation.

Counting is usually related to the natural numbers group. But you can also compare rational an irrational numbers, although they are not "countable".

I would say that a plane has more points than a line, but that is still a topic discussed by mathematicians, so who am I to argue?

Just my habit of thought deeply ingrained in everyday thought, taken for granted but not logically justified. I am human after all, not vulcan.