The co-ordinates of a point, that is, (x,y) are always pure numbers. The origin is (0,0). Every number on X-coordinate gives its distance from the origin or from zero. Thus in geometry a number on the axis directly gives its distance from the origin or from 0. When number directly gives distance from 0 then the problem is to find how far should we place 1 (unit) from 0. If X is a point on X-axis and the number marked on the point A is 'x', then length OX/x = 1. Although OX (distance from origin) and x (the number on the point X) are variables OX/x is always 1. If we decide to place 1 at a distance of, say, 1" from the origin, then the 'distance' OX is not 'continuos' because the number of 1s given by OX/x must be infinite if OX is compact and continuous. If number of points within 1 is infinite then we can never reach 0 or the origin from 1. (Zeno's paradox. 1=1/2+ 1/2^2 + 1/2^3 -------- has infinite terms.) I wish to know in geometry whether OX/x=1 or not. If OX/x = 1 also OX is continuous, then it means OX/x=1 then we can never know how far is 1 from 0 and every finite length is 1. If we want numbers on the axes of coordinates then we have to use a finite unit of length to mark the numbers and fractions would serve the same purpose as unit. In geometry numbers more than 1 do not have reciprocals!
In article <38af3945.0210281055.2bf30...@posting.google.com>,
vgopa...@rediffmail.com (V.Gopal) wrote: > The co-ordinates of a point, that is, (x,y) are always pure numbers. > .... Thus in geometry a number on the axis directly gives its distance > from the origin .... then the problem is to find how far should we place > 1 (unit) from 0.
It doesn't seem entirely clear where you're starting.
If you start with a Euclidean plane and then put a coordinate system on it, you can choose any two perpendicular lines as axes, and any point (except the origin) on the x axis as (1,0). Once you've chosen those things, you don't change them. Then it's possible to set up coordinates (x,y) for every point in the plane.
OTOH if you start with just the real numbers, you can define points to _be_ ordered pairs (x,y), and then introduce suitable Euclidean notions of length, angle, etc.
> .... If we decide to place 1 at a distance of, say, 1" from the origin, > then ....
This mention of inches suggests that you're mixing mathematics with physics. Certainly we draw graphs on paper using physical lengths, for two reasons. One is to apply our geometrical theory to the real world. The other (which seems closer to your concerns) is to aid our imaginations in thinking about the abstractly defined Cartesian plane. However, the pure theory can't involve any physical unit of length.
I hope these comments will at least help clear up some misunderstandings and perhaps clarify your question.
> In article <38af3945.0210281055.2bf30...@posting.google.com>, > vgopa...@rediffmail.com (V.Gopal) wrote:
> > The co-ordinates of a point, that is, (x,y) are always pure numbers. > > .... Thus in geometry a number on the axis directly gives its distance > > from the origin .... then the problem is to find how far should we place > > 1 (unit) from 0.
> It doesn't seem entirely clear where you're starting.
> If you start with a Euclidean plane and then put a coordinate system > on it, you can choose any two perpendicular lines as axes, and any point > (except the origin) on the x axis as (1,0). Once you've chosen those > things, you don't change them. Then it's possible to set up coordinates > (x,y) for every point in the plane.
> OTOH if you start with just the real numbers, you can define points > to _be_ ordered pairs (x,y), and then introduce suitable Euclidean notions > of length, angle, etc.
> > .... If we decide to place 1 at a distance of, say, 1" from the origin, > > then ....
> This mention of inches suggests that you're mixing mathematics with > physics. Certainly we draw graphs on paper using physical lengths, for > two reasons. One is to apply our geometrical theory to the real world. > The other (which seems closer to your concerns) is to aid our imaginations > in thinking about the abstractly defined Cartesian plane. However, the > pure theory can't involve any physical unit of length.
> I hope these comments will at least help clear up some > misunderstandings and perhaps clarify your question.
> Ken Pledger.
Don't hold your breath about this poster making anything clear. Check Google groups for previous attempts by other posters to communicate with him.
> Is this where the hyper-intelligent come to heap abuse on the ignorant? > Sounds like my kind of place :) > agb
In general the folks on this ng seem to be pretty tolerant. Especially of homework nerds. I like to see good homework stuff bandied about if it is clear that the poster has tried something and just doesn't want answers to copy and hand in. One of the best examples was where a MAT108 problem was answered at a MAT 500+ level. I hope the dork handed it in that way!!
On occasion dorkiness flounders about, and so I still have to use the Plonquer at times.
In article <38af3945.0210281055.2bf30...@posting.google.com>,
V.Gopal <vgopa...@rediffmail.com> wrote: >The co-ordinates of a point, that is, (x,y) are always pure numbers. >The origin is (0,0). Every number on X-coordinate gives its distance >from the origin or from zero. Thus in geometry a number on the axis >directly gives its distance from the origin or from 0. When number >directly gives distance from 0 then the problem is to find how far >should we place 1 (unit) from 0. If X is a point on X-axis and the
1/10 inch, according to the little squares on my graph paper.
Put it wherever you like, and wherever you put it will define a unit distance.
>number marked on the point A is 'x', then length OX/x = 1. Although OX >(distance from origin) and x (the number on the point X) are variables >OX/x is always 1. If we decide to place 1 at a distance of, say, 1" >from the origin, then the 'distance' OX is not 'continuos' because the >number of 1s given by OX/x must be infinite if OX is compact and >continuous. If number of points within 1 is infinite then we can never >reach 0 or the origin from 1. (Zeno's paradox. 1=1/2+ 1/2^2 + 1/2^3 >-------- has infinite terms.) I wish to know in geometry whether
For some reason, Zeno chose to approach his destination taking smaller and smaller steps in each time interval. It works better if you just keep walking, 1+1+1+1... -- "A nice adaptation of conditions will make almost any hypothesis agree with the phenomena. This will please the imagination but does not advance our knowledge." -- J. Black, 1803.
andi babian wrote: > Is this where the hyper-intelligent come to heap abuse on the ignorant? > Sounds like my kind of place :)
No. It's where those of any intelligence come to heap abuse on those who think that ignorance of physics makes them physics experts, and anybody who doesn't think so is closed minded and a slavish unthinking lapdog of the orthodoxy.
> > The co-ordinates of a point, that is, (x,y) are always pure numbers. > > .... Thus in geometry a number on the axis directly gives its distance > > from the origin .... then the problem is to find how far should we place > > 1 (unit) from 0.
> It doesn't seem entirely clear where you're starting.
> If you start with a Euclidean plane and then put a coordinate system > on it, you can choose any two perpendicular lines as axes, and any point > (except the origin) on the x axis as (1,0). Once you've chosen those > things, you don't change them. Then it's possible to set up coordinates > (x,y) for every point in the plane.
> OTOH if you start with just the real numbers, you can define points > to _be_ ordered pairs (x,y), and then introduce suitable Euclidean notions > of length, angle, etc.
> > .... If we decide to place 1 at a distance of, say, 1" from the origin, > > then ....
> This mention of inches suggests that you're mixing mathematics with > physics. Certainly we draw graphs on paper using physical lengths, for > two reasons. One is to apply our geometrical theory to the real world. > The other (which seems closer to your concerns) is to aid our imaginations > in thinking about the abstractly defined Cartesian plane. However, the > pure theory can't involve any physical unit of length.
> I hope these comments will at least help clear up some > misunderstandings and perhaps clarify your question.
> Ken Pledger.
Along an axis of coordinate we assume that 1"=100 feet or 1"= 1 second. It meas that along every axis of coordinate we assume that either length/length is constant or length/time is constant. Therefore we assume that along every axis of coordinate TanA is constant. please let me know if I am wrong. May the noble minded scholars instead of cherishing ill-will kindly correct any error here committed through dullness of intellect in the way of wrong statements and interpretations.
In article <38af3945.0210291411.16fe4...@posting.google.com>,
vgopa...@rediffmail.com (V.Gopal) wrote: > .... > Along an axis of coordinate we assume that 1"=100 feet or 1"= 1 > second. It meas that along every axis of coordinate we assume that > either length/length is constant or length/time is constant. Therefore > we assume that along every axis of coordinate TanA is constant....
Now here you're definitely mixing mathematics with physics. The expression tan(A) always represents a pure number, never anything like "one second per inch". I find it helpful always to keep absolutely clear when I'm thinking of mathematical objects which are entirely in the mind, and when I'm applying them to physical objects such as graphs on paper using units of distance, time, etc.
> > .... > > Along an axis of coordinate we assume that 1"=100 feet or 1"= 1 > > second. It meas that along every axis of coordinate we assume that > > either length/length is constant or length/time is constant. Therefore > > we assume that along every axis of coordinate TanA is constant....
> Now here you're definitely mixing mathematics with physics. The > expression tan(A) always represents a pure number, never anything like > "one second per inch". I find it helpful always to keep absolutely clear > when I'm thinking of mathematical objects which are entirely in the mind, > and when I'm applying them to physical objects such as graphs on paper > using units of distance, time, etc.
> Ken Pledger.
Pure numbers are of two types - one that gives quantity and the other that give level. The number that gives level is like ordinal numbers and ordinal numbers are exactly like TanA. Although angle A is also like ordinal numbers A returns to its starting point after 360 degrees. Since ordinal numbers are endless we have to represent ordinal numbers by TanA. The Ordinal numbers are also like frequencies, therefore any discussion on pure numbers seems to be a discussion on physics. If axes of coordinates only show 'consistent quantities' then in geometry we would only get straight lines. In geometry ordinal numbers enter because geometry also gives position of every number. Therefore in geometry 0 occupies ONE point and 10^-100 also occupies ONE point and 1, 10 etc also occupy ONE point (this is how we get hyperbola). It is strange that sometimes we assume that all points are of the same size (in hyperbola) and some times we assume as the number decreases the size of the point also decreases. In A vs TanA curve although we are showing all numbers from 0 to 1, space occupied by each point depends on the value of TanA. It seems that, when every ordinal number, irrespective of its 'value' occupy same amount of space then then line shows a^x (where a<1), where a is the space between two consecutive values of a^x. When the space occupied by each ordinal number is equal to its value then it is equivalent to X^X where X approaches from 1 to 0. We must remember that ordinal number is TanA and unless TanA or slope 'm' is a comtinuous variable we cannot get a curve. In geometry we are compelled to show that a number X and its reciprocal 1/X both begin to increase from 0 (zero) and X and 1/X occupy the same point whatever be the value of X. Does it mean that in geometry numbers do not have their reciprocals?
> > In article <38af3945.0210291411.16fe4...@posting.google.com>, > > vgopa...@rediffmail.com (V.Gopal) wrote:
> > > .... > > > Along an axis of coordinate we assume that 1"=100 feet or 1"= 1 > > > second. It meas that along every axis of coordinate we assume that > > > either length/length is constant or length/time is constant. Therefore > > > we assume that along every axis of coordinate TanA is constant....
> > Now here you're definitely mixing mathematics with physics. The > > expression tan(A) always represents a pure number, never anything like > > "one second per inch". I find it helpful always to keep absolutely clear > > when I'm thinking of mathematical objects which are entirely in the mind, > > and when I'm applying them to physical objects such as graphs on paper > > using units of distance, time, etc.
> > Ken Pledger. > Pure numbers
As opposed to "impure" numbers? An example of each would be helpful.
> are of two types - one that gives quantity and the other that give level.
Huh? Some examples of "quantity" and "level" would be very helpful.
> The number that gives level is like ordinal numbers
By "exactly like" do you mean isomorphic? Are you saying that "quantity" v. "level" numbers is the same distinction as cardinals v. ordinals? If so, why not just use the standard the terminology? If not, what ARE you saying?
> and ordinal numbers are exactly like TanA.
Rubbish. You really do need to give an example here.
> Although angle A is also > like ordinal numbers A returns to its starting point after 360 > degrees.
And tan(A) doesn't?!?
> Since ordinal numbers are endless
Are you talking about transfinite ordinals here? If not, what ARE you talking about?
> we have to represent ordinal numbers by TanA.
Utter rubbish. What is your proposed mapping between the (countable) ordinals and the (uncountable) values of tan(A)?
> The Ordinal numbers are also like frequencies,
With sufficient imagination a raven is also like a writing desk. In WHAT SENSE are they "like"?
> therefore any discussion on pure numbers seems to be a > discussion on physics.
That depends on the seemer. I would give you long odds that for almost all the readers of these posts, "pure" numbers (whatever exactly you might mean by that) and ordinal numbers seem nothing at all like physics.
> If axes of coordinates only show 'consistent quantities'
Is this a technical term, or just more rubbish?
> then in geometry we would only get straight lines.
How so?
> In geometry ordinal numbers enter because geometry also gives position > of every number.
If you are referring to Euclidean co-ordinate geometry then positions are represented by reals, not by ordinals. What (if anything) ARE you referring to?
> Therefore in geometry 0 occupies ONE point and 10^-100 > also occupies ONE point and 1, 10 etc also occupy ONE point
At last, something I agree with (assuming we are restricted to 1-dimensional co-ordinate geometry, which is a rather uninteresting beast).
> (this is how we get hyperbola).
Say WHAT?!? Please provide the intermediate steps between the real number line (your "points") and hyperbolae.
> It is strange that sometimes we assume that all > points are of the same size (in hyperbola)
What do you mean by the "size" of a point, and what does that have to do with hyperbolae?
> and some times we assume as > the number decreases the size of the point also decreases.
What is "the number" here, and for goodness sake what sort of non-standard geometry are you using where "points" not only have a non-zero "size", but that size VARIES?
> In A vs > TanA curve although we are showing all numbers from 0 to 1, space > occupied by each point depends on the value of TanA.
It sure looks like you are trying to use ordinary Euclidean co-ordinate geometry here. But you are failing miserably. In ordinary Euclidean co-ordinate geometry, "each point" occupies exactly NO space. Got that? NO space. None. Nil. Nichts. Nada.
[further flatulence snipped]
You post a fair bit here, and to me (and others) it all seems like rubbish. You do seem to be trying to make some point, but (1) it is thorougly unclear what that point is, and (2) your half-assed use of mathematics certainly fails to make any point whatsoever.
Please, WITHOUT attempting to use mathematical analogies, tell us the point you are trying to make.
-- --------------------------- | BBB b \ barbara minus knox at iname stop com | B B aa rrr b | | BBB a a r bbb | | B B a a r b b | | BBB aa a r bbb | -----------------------------
> It is strange that sometimes we assume that all > points are of the same size (in hyperbola) and > some times we assume as the number decreases the > size of the point also decreases.
<snip>
"A point is that which has no part." -- Euclid, Elemetns: Book I Definition 1
I don't know what you are drinking but you should patent that stuff.
As for the Zeno's paradox stuff, it's Physics and not mathematics. To paraphrase Bertrand Russel: "Zeno made an implicit assumption that A MOVING BODY IS IN A STATE OF MOTION. This is now known to be wrong."
As for the "axes of co-ordinates": a co-ordinate system and its physical representation are two different things.
It is customery and useful to use 'real lines' for the axes (of say a Cartesian co-ordinate system.) But you can assign numbers to points on the line in any way that satisfies your purpose or fancy.
It is not necessary that the physical distance between 0 and 1 should be the same as that between 1 and 2.
You can, for example, define the PHYSICAL representation of a real line such that the distance between points denoted by integers halves as one moves away from point 0.
Perhaps this may not be very useful in real world applications but as a representations this is just as valid as any other scheme.
So, in effect, a "real line" need not be physically unbounded in length.
s...@sig.below (Barb Knox) wrote in message <news:see-3110021036090001@192.168.1.2>... > In article <38af3945.0210300651.5b9ab...@posting.google.com>, > vgopa...@rediffmail.com (V.Gopal) wrote:
> > > > .... > > > > Along an axis of coordinate we assume that 1"=100 feet or 1"= 1 > > > > second. It meas that along every axis of coordinate we assume that > > > > either length/length is constant or length/time is constant. Therefore > > > > we assume that along every axis of coordinate TanA is constant....
> > > Now here you're definitely mixing mathematics with physics. The > > > expression tan(A) always represents a pure number, never anything like > > > "one second per inch". I find it helpful always to keep absolutely clear > > > when I'm thinking of mathematical objects which are entirely in the mind, > > > and when I'm applying them to physical objects such as graphs on paper > > > using units of distance, time, etc.
> > > Ken Pledger.
> > Pure numbers
> As opposed to "impure" numbers? An example of each would be helpful.
> > are of two types - one that gives quantity and the other that give level.
> Huh? Some examples of "quantity" and "level" would be very helpful.
> > The number that gives level is like ordinal numbers
> By "exactly like" do you mean isomorphic? Are you saying that "quantity" > v. "level" numbers is the same distinction as cardinals v. ordinals? If > so, why not just use the standard the terminology? If not, what ARE you > saying?
> > and ordinal numbers are exactly like TanA.
> Rubbish. You really do need to give an example here.
> > Although angle A is also > > like ordinal numbers A returns to its starting point after 360 > > degrees.
> And tan(A) doesn't?!?
> > Since ordinal numbers are endless
> Are you talking about transfinite ordinals here? If not, what ARE you > talking about?
> > we have to represent ordinal numbers by TanA.
> Utter rubbish. What is your proposed mapping between the (countable) > ordinals and the (uncountable) values of tan(A)?
> > The Ordinal numbers are also like frequencies,
> With sufficient imagination a raven is also like a writing desk. In WHAT > SENSE are they "like"?
> > therefore any discussion on pure numbers seems to be a > > discussion on physics.
> That depends on the seemer. I would give you long odds that for almost > all the readers of these posts, "pure" numbers (whatever exactly you might > mean by that) and ordinal numbers seem nothing at all like physics.
> > If axes of coordinates only show 'consistent quantities'
> Is this a technical term, or just more rubbish?
> > then in geometry we would only get straight lines.
> How so?
> > In geometry ordinal numbers enter because geometry also gives position > > of every number.
> If you are referring to Euclidean co-ordinate geometry then positions are > represented by reals, not by ordinals. What (if anything) ARE you > referring to?
> > Therefore in geometry 0 occupies ONE point and 10^-100 > > also occupies ONE point and 1, 10 etc also occupy ONE point
> At last, something I agree with (assuming we are restricted to > 1-dimensional co-ordinate geometry, which is a rather uninteresting > beast).
> > (this is how we get hyperbola).
> Say WHAT?!? Please provide the intermediate steps between the real number > line (your "points") and hyperbolae.
> > It is strange that sometimes we assume that all > > points are of the same size (in hyperbola)
> What do you mean by the "size" of a point, and what does that have to do > with hyperbolae?
> > and some times we assume as > > the number decreases the size of the point also decreases.
> What is "the number" here, and for goodness sake what sort of non-standard > geometry are you using where "points" not only have a non-zero "size", but > that size VARIES?
> > In A vs > > TanA curve although we are showing all numbers from 0 to 1, space > > occupied by each point depends on the value of TanA.
> It sure looks like you are trying to use ordinary Euclidean co-ordinate > geometry here. But you are failing miserably. In ordinary Euclidean > co-ordinate geometry, "each point" occupies exactly NO space. Got that? > NO space. None. Nil. Nichts. Nada.
> [further flatulence snipped]
> You post a fair bit here, and to me (and others) it all seems like > rubbish. You do seem to be trying to make some point, but (1) it is > thorougly unclear what that point is, and (2) your half-assed use of > mathematics certainly fails to make any point whatsoever.
> Please, WITHOUT attempting to use mathematical analogies, tell us the > point you are trying to make.
I am referring to Euclidean geometry with simply X and Y coordinates. My answers to your questions: 'Pure numbers' are numbers without units of measurement. In [2](inches) 2 is a pure number. we can write 2"= 1/6 foot. 2 is not equal to 1/6 in pure maths but here it is. In pur mathematics number gives 'quantity' without unis. I used the word 'quantity' to remind that if you remove 1 from 100 you cannot know which one you have picked up. First number 1 or any number from in between. Level is given by ordinal number. 1st. 2nd. 3rd --- 100th are ordinal numbers. If you remove any one number from a series of ordinal numbers the gap is not filled by the preceding or the succeeding number; the gao remains, you cannot use ordinal numbers in any calculations. You can use any terminology to indicate these two types of numbers. The word 'quantity' immediately tells that a number has no particular position in a quantity. Ordinal number give 'potential difference' therefore all 'point functions' are ordinal numbers. You cannot add pressures, temperatures, frequencies, velocities etc. Since we cannot add ordinal numbers, only a 'physical process' can change the ordinal number. Angle A increases from 0 to 360 degrees and returns to the same POSITION. Tan A increases from 0 to any number as big as you can imagine and never returns to 0 again. There are as many ordinal numbers as there are cardinal numbers. I use the term 'quantity' because we cannot know how many 'cardinal numbers' or units are there within 1. If the divisions on the axes of coordintes are equally spaced then it means we are using a homogeneous and cosistent unit of measurement. if both the axes of coordinates are marked by homogeneous or consistent units. that is if the 'scales' on the axes are fixed then we can only get straight line. Certain things cannot be explained you have feel it. Thanks God, you accept that 0, 10^-100, 1, 10 all occupy one point each. Then, when can we reach 0 from 1. In fact we never reach 0 or 1nfinity in a hyperbola. Unless 0 occupies no space and each 'number' in its own position occupies only as much space as its distance distance from 0 we can never reach 0 from 1. On Y=mX+c there is only one number equal to 'm' and at every point on the line it occupies the same space. Some body has insisted that a point does not occupy any space. If X and Y coordinates do not occupy any space then how can we imagine a curve to 'exist'? The purpose of this post is to prove that geometry is dimensionless if the lines are supposed to be continuous and the moment numbers enter in geometry all lines becomes discontinuous. Continuous lines require continuous change in length. What it means is beyond the scope of present discussion. May the noble minded scholars correct the errors here committed through dullness of intellect.
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