Visual Presentation of Real Number System

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Visual Presentation of Real Number System

Jerry Becker
Visual Presentation of Real Number System
Colleagues -

Prof. Hyunyong SHIN would like to share the attached article that deals with the number line as a visual representation of the real number system.

Abstract:

Abstract Number line is a visual representation of real number system. There was no arrow at either
end of the representation in the early figures of number line. But after the 19th century, number lines with one arrow on the right hand side have begun to appear. In USA, however, many textbooks of secondary school mathematics have number lines with arrows on the left hand side as well as on the right hand side. It is quite different from the global standard. We argue that it is mathematically reasonable for number lines to have an arrow only on the right side.

This is for your information, from

Professor Hyunyong Shin            
Department of Mathematics Education 
Korea National University of Education
Chungbuk 363-791, KOREA      


--
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL  62901-4610
Phone:  (618) 453-4241  [O]
            (618) 457-8903  [H]
Fax:      (618) 453-4244
E-mail:   [hidden email]

number-line-1.pdf (217K) Download Attachment
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Re: Visual Presentation of Real Number System

Dave L. Renfro
Jerry P. Becker posted/forwarded:

http://mathforum.org/kb/message.jspa?messageID=9330308

> Prof. Hyunyong SHIN would like to share the attached article that
> deals with the number line as a visual representation of the real
> number system.

[Abstract follows]

I suggest omitting the 1 page length axiomatic definition of an ordered field.
This is not used elsewhere and it seems unrelated to your topic.

Dave L. Renfro
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Re: Visual Presentation of Real Number System

Gary Tupper
 From a rather naive perspective, it seems that arrows visually
differentiate between segments, rays and lines.
I'm more interested in how one might draw a rational (or irrational)
number 'line'. (as distinct from the Real counterparts.)

Gary Tupper
Terrace BC



On 11/21/2013 10:37 AM, Dave L. Renfro wrote:

> Jerry P. Becker posted/forwarded:
>
> http://mathforum.org/kb/message.jspa?messageID=9330308
>
>> Prof. Hyunyong SHIN would like to share the attached article that
>> deals with the number line as a visual representation of the real
>> number system.
> [Abstract follows]
>
> I suggest omitting the 1 page length axiomatic definition of an ordered field.
> This is not used elsewhere and it seems unrelated to your topic.
>
> Dave L. Renfro
>
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Re: Visual Presentation of Real Number System

Robert Hansen
In reply to this post by Jerry Becker
A slow year at the math department I guess.

Bob Hansen

On Nov 21, 2013, at 10:28 AM, Jerry Becker <[hidden email]> wrote:

Visual Presentation of Real Number System
Colleagues -

Prof. Hyunyong SHIN would like to share the attached article that deals with the number line as a visual representation of the real number system.

Abstract:

Abstract Number line is a visual representation of real number system. There was no arrow at either
end of the representation in the early figures of number line. But after the 19th century, number lines with one arrow on the right hand side have begun to appear. In USA, however, many textbooks of secondary school mathematics have number lines with arrows on the left hand side as well as on the right hand side. It is quite different from the global standard. We argue that it is mathematically reasonable for number lines to have an arrow only on the right side.

This is for your information, from

Professor Hyunyong Shin            
Department of Mathematics Education 
Korea National University of Education
Chungbuk 363-791, KOREA      


--
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL  62901-4610
Phone:  (618) 453-4241  [O]
            (618) 457-8903  [H]
Fax:      (618) 453-4244
E-mail:   [hidden email]
<number-line-1.pdf>
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Re: Visual Presentation of Real Number System

Joe Niederberger
In reply to this post by Jerry Becker
>From a rather naive perspective, it seems that arrows >visually
>differentiate between segments, rays and lines.
>I'm more interested in how one might draw a rational (or >irrational)
>number 'line'. (as distinct from the Real counterparts.)
>
>Gary Tupper
>Terrace BC

It depends on context.

Cheers,
Joe N
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Re: Visual Presentation of Real Number System

Robert Hansen

On Nov 24, 2013, at 10:50 PM, Joe Niederberger <[hidden email]> wrote:

>> From a rather naive perspective, it seems that arrows >visually
>> differentiate between segments, rays and lines.
>> I'm more interested in how one might draw a rational (or >irrational)
>> number 'line'. (as distinct from the Real counterparts.)
>>
>> Gary Tupper
>> Terrace BC
>
> It depends on context.
>
> Cheers,
> Joe N

In general, I think he is asking for a physical model[1] of rational (or irrational) numbers. Specifically, something like a line.

[1] Physical model - A physical object or drawing whereby there is an explicit mapping between elements of the theory and physical measures of the object or drawing, such as quantity, length, height, etc.

A troublesome (impossible) task given the nature of these sets. At a deeper level, probably similar to the fact that we cannot make certain constructions with compass and straightedge.

It is a very good question though, and an example of how much math exists that we cannot "see".

Bob Hansen
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Re: Visual Presentation of Real Number System

Joe Niederberger
In reply to this post by Jerry Becker
R Hansens says:
>It is a very good question though, and an example of how much math exists that we cannot "see".

I'd prefer to consider that perhaps its the reals that don't really exist (as well as being invisible.) However,
we know a bit of how to see such impossibilities -- we use logic and the mind's eye.

Cheers,
Joe N
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Re: Visual Presentation of Real Number System

Joe Niederberger
In reply to this post by Jerry Becker
To amplify a bit -- if we pick a real number (say in (0,1)) the odds we hit it with a randomly thrown dart is exactly zero. So it reasonable to assume that what goes for darts goes for light rays too. That randomly chosen real is "invisible". The real number line is almost, or completely, entirely invisible. Is the rational number line likewise?

What do we really see then?

Cheers,
Joe N
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Re: Visual Presentation of Real Number System

Robert Hansen
Well, an "actual" real number line would have a width (breadth) of zero and thus be entirely invisible.

And, if we take that the smallest actual "point" on a drawn line would be represented by the centers of the molecules or atoms of the ink/lead, then I would suspect that those locations are all at irrational distances from the end (transcendental in fact). So the number lines we know and love are actually irrational number lines with a finite number of points.:)

Obviously, number lines work, and we "see" the geometric version in our mind. Kids seem to see the geometric version pretty easily as well (it is the later formality that gives them problems). But if we drew them with dots or dashes that would probably play havoc with their senses.

I think the hardest aspect of the irrationals and rationals to portray in a tangible way is that between any two rationals there are an infinite number of rationals and an infinite number of "holes". Even though my son easily surmised (after learning fractions) that he can keep choosing a smaller and smaller number and get as close to zero as he wanted, I don't think he realized how empty that gap is. I get the distinct impression that he thinks he is filling the gap up.

If you want to see what this discussion looks like in baby talk, check out...

http://blog.mrmeyer.com/?p=18225

Bob Hansen


On Nov 25, 2013, at 1:05 PM, Joe Niederberger <[hidden email]> wrote:

> To amplify a bit -- if we pick a real number (say in (0,1)) the odds we hit it with a randomly thrown dart is exactly zero. So it reasonable to assume that what goes for darts goes for light rays too. That randomly chosen real is "invisible". The real number line is almost, or completely, entirely invisible. Is the rational number line likewise?
>
> What do we really see then?
>
> Cheers,
> Joe N
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Re: Visual Presentation of Real Number System

Joe Niederberger
In reply to this post by Jerry Becker
R Hansen says:
>And, if we take that the smallest actual "point" on a drawn line would be represented by the centers of the molecules or atoms of the ink/lead, then I would suspect that those locations are all at irrational distances from the end (transcendental in fact).

Really? You think physical distances are truly all irrational? And then what distance could ever be demarcated as 1?

R Hansen says:
>I think the hardest aspect of the irrationals and rationals to portray in a tangible way is that between any two rationals there are an infinite number of rationals and an infinite number of "holes".

What I see with my mind's eye is not holes but rather that the minds concept of mathematical "line" and "point" and the relationship between them is just not similar to pebbles laid in a row. And yet a lot of common talk in mathematics is always leaning on that conception. As if the line is made of of a bunch of points (or points and holes).

In my mind at least line and point are distinct species with a curious but easily visualized game that can be played - I can always zoom in to the space between any two points on a line, and find a line segment between them. That line segment is similar in certain respects to any other segment, and so I can play the game over and over. The only thing "tangible" is this notion of a never ending process.

Its also quite "tangible" (in the imaginary sense of "tangible! What??) to view incommensurable lengths in a similar way: imagine (or image ;-) multiple copies of two segments of two incommensurable lengths being laid end to end -- and the frustration that they never, ever line up on the cracks.

Cheers,
Joe N
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Re: Visual Presentation of Real Number System

Robert Hansen

On Nov 25, 2013, at 2:51 PM, Joe Niederberger <[hidden email]> wrote:

R Hansen says:
And, if we take that the smallest actual "point" on a drawn line would be represented by the centers of the molecules or atoms of the ink/lead, then I would suspect that those locations are all at irrational distances from the end (transcendental in fact).

Really? You think physical distances are truly all irrational? And then what distance could ever be demarcated as 1? 

Not only irrational, but transcendental (not algebraic). I am talking about the actual positions of the molecules and atoms, although they are moving, but you know what I mean. The argument is not much different than what you said about the improbability (impossibility) of a ray of light ever hitting a particular number. I am saying that said ray will not even hit any algebraic number. We know the ray of light will hit a number, I am saying that number will always be transcendental. But don't ask me for a formal proof.:)

Bob Hansen

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Re: Visual Presentation of Real Number System

Robert Hansen

On Nov 25, 2013, at 3:27 PM, Robert Hansen <[hidden email]> wrote:

Not only irrational, but transcendental (not algebraic). I am talking about the actual positions of the molecules and atoms, although they are moving, but you know what I mean.


Let me generalize the Niederberger-Hansen conjecture...

If you take a snapshot of the universe at an instant in time, then, in that snapshot, every dimension of every object or collection of objects will be a transcendental number.

Another way to say this is that you can never pick *the* instant in time where any dimension is algebraic.

Bob Hansen
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Fwd: Re: Visual Presentation of Real Number System

Gary Tupper
In reply to this post by Gary Tupper
We tend to think primarily about visual algebraic models in terms of
marks on a sheet of paper.
Which makes difficult any attempt to draw the model of a Real line as
distinct from a Rational or Irrational line.

However, if we consider a computer display, then we can introduce the
possibility of animation. On a computer screen, the model of the
Real number line would appear much as a drawn model on paper. In order
to display a model of the Rational number line, the line might blink
(red, say),
and an Irrational line might blink blue. If that capability were
implemented, then a line that blinked alternately red & blue would be an
additional
model of the Real line.

Now - what about a line minus a point? On paper usually shown by a small
circle on the line. However a computer screen model might consist of a
blinking green pixel.
So - what would we make of a line that that blinked red & blue, with a
single pixel blinking green?

Gary Tupper

========================================================================

 From a rather naive perspective, it seems that arrows visually
differentiate between segments, rays and lines.
I'm more interested in how one might draw a rational (or irrational)
number 'line'. (as distinct from the Real counterparts.)

Gary Tupper
Terrace BC
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Re: Visual Presentation of Real Number System

Robert Hansen

On Nov 25, 2013, at 4:47 PM, Gary Tupper <[hidden email]> wrote:

So - what would we make of a line that that blinked red & blue, with a single pixel blinking green?

After my epileptic fit or during?:)

I don't understand what you are suggesting by color. That sounds more like a symbol than a model. For example, the open circle denoting a missing point is a notational device or symbol. A model of a missing point would a very tiny gap in the line, but that would be a poor model and wouldn't work at the end of the line, thus the use of symbols instead.

Bob Hansen 

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Re: Visual Presentation of Real Number System

Gary Tupper
On 11/25/2013 1:59 PM, Robert Hansen wrote:

On Nov 25, 2013, at 4:47 PM, Gary Tupper <[hidden email]> wrote:

So - what would we make of a line that that blinked red & blue, with a single pixel blinking green?

After my epileptic fit or during?:)

I don't understand what you are suggesting by color. That sounds more like a symbol than a model. For example, the open circle denoting a missing point is a notational device or symbol. A model of a missing point would a very tiny gap in the line, but that would be a poor model and wouldn't work at the end of the line, thus the use of symbols instead.

Bob Hansen 

Hi, Bob:

Actually, the gap would be smaller than 'tiny' since its width would be zero. We should remember that any physical representation of a mathematical object is a model. The objective of our model is essentially to impart information - the open circle on a curve is interpreted to mean that a single point is missing.

Take, for example, the graph of y=x^2 /(x-1) :
Usually drawn as a parabola with a hollow dot at (1,1).

On a computer screen, a blinking pixel at (1,1) can impart the same information. If we "zoom in" to that pixel on the computer screen, the display will remain - a blinking pixel. (unless we exceed the computer's display resolution, which I imagine depends on the computer's numerical capabilities.)

BTW, although we are usually interested in the graph of the solution of y=x^2/(x-1) over a region of the plane, and therefore draw the hollow dot at (1,1), we should realize that the division by x-1 affects the whole plane - ie the whole line x=1 is 'removed'. Maybe the easiest way to visualize this is to ask about 'truth':
So, for y=x^2/(x-1), what about (2,4)? It satisfies the equation - ie 'true', What about (2,3)? - it generates 'false'. What about (1,1)? And what about (2,1)?

Gary Tupper
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Re: Fwd: Re: Visual Presentation of Real Number System

Joe Niederberger
In reply to this post by Jerry Becker
Gary Tupper says:
>Which makes difficult any attempt to draw the model of a Real line as distinct from a Rational or Irrational line.

Might as well rely on the old symbols R and Q to differentiate. I vaguely remember the Rs were generally not ordinary Rs, but had a funny double vertical line on the left. Maybe the Qs had that funny doubling too ... no doubt very significant ;-) Anyway, its not hard to add an R or a Q to the picture.

Cheers,
Joe N

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Re: Visual Presentation of Real Number System

Robert Hansen
In reply to this post by Gary Tupper

On Nov 25, 2013, at 6:07 PM, Gary Tupper <[hidden email]> wrote:

Actually, the gap would be smaller than 'tiny' since its width would be zero. We should remember that any physical representation of a mathematical object is a model. The objective of our model is essentially to impart information - the open circle on a curve is interpreted to mean that a single point is missing.

This might be your definition, but that would include algebraic expressions and d/dx and everything written. I prefer some granularity. By my definition, to be a physical model, or just model for short, it must appeal to our innate physical senses. If we put two segments next to each other and one is longer than the other, that is sensed, without any pre knowledge of anything needed.

Symbols and notation on the other hand are not innate at all. If I showed you a line with a hollow dot, and you had no previous instruction, you wouldn't figure out what that meant in a million years. Some people confuse "symbols that make sense" with innate sense. Hollow dots existed long before someone serendipitously came upon the idea to represent closed points with solid dots and open points with hollow dots. In effect, they invented a use of symbols that already existed, and that part makes sense. But until you are told what they are for, they make no sense at all.


Take, for example, the graph of y=x^2 /(x-1) :
Usually drawn as a parabola with a hollow dot at (1,1).

I don't think that expression is a parabola, it is a hyperbola, but let's go with your visual, a parabola with a hole at (1,1). The position of the hole (its location) satisfies my definition of "model" because there are several (physical) properties that the student can feel innately. If (1,1) is a minimum then it will be felt as being at the bottom. But the hollow dot satisfies only my definition of symbol. Without being told what a hollow dot stands for, the student will never get it. The whole graph itself is a diagram and contains elements that model and elements (symbols) that communicate but only with pre agreed upon meaning. Similarly, drawing hashes across congruent segment in a geometric figure (diagram) are symbols. They communicate, but only after being told what they mean. That two sides "look" equal in length doesn't require any pre-instruction at all.


BTW, although we are usually interested in the graph of the solution of y=x^2/(x-1) over a region of the plane, and therefore draw the hollow dot at (1,1), we should realize that the division by x-1 affects the whole plane - ie the whole line x=1 is 'removed'. Maybe the easiest way to visualize this is to ask about 'truth':
So, for y=x^2/(x-1), what about (2,4)? It satisfies the equation - ie 'true', What about (2,3)? - it generates 'false'. What about (1,1)? And what about (2,1)?

Now you are talking about mental visualization, not physical. Other than that, I agree that this is one of the mental exercises you should go through. It plays off the notion of sets and we know how well that helps sort this stuff out. But I would not call this or any single method the "easiest" way to visualize. Some methods are good and some very bad, and this may be one of the best, but no single method, model or analogy covers enough of the many facets of these constructions well enough alone.

The mapping is very easy to state formally, as is the case in most of mathematics. y = x^2 for all x except x = 1. But to comprehend and fathom its consequences takes time and thinking about it in different ways.

Bob Hansen
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Re: Visual Presentation of Real Number System

Robert Hansen
In reply to this post by Gary Tupper

> On Nov 25, 2013, at 6:07 PM, Gary Tupper <[hidden email]> wrote:
>
> If we "zoom in" to that pixel on the computer screen, the display will remain - a blinking pixel

I meant to add that I do agree that "zooming" appeals to our physical senses. I.e. It is modeling. And animations are also a medium with which to do modeling.

Bob Hansen

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Re: Visual Presentation of Real Number System

Robert Hansen
In reply to this post by Robert Hansen
Here is something like a proof. It seems reasonable (as Joe stated) that a ray of light would never hit a particular number, like 12 for example, on the number line. My extension is that a ray of light would never hit a whole number for the same reason, the odds are zero. Likewise, the ray of light would never hit a number divisible by 6, or by 30, or a number that is the square root of any of these numbers. In fact, the ray of light would never hit any "particular" (constructed?) number because there are infinitely more non particular numbers for it to hit.

Bob

> On Nov 25, 2013, at 3:27 PM, Robert Hansen <[hidden email]> wrote:
>
> We know the ray of light will hit a number, I am saying that number will always be transcendental. But don't ask me for a formal proof.:)
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Re: Visual Presentation of Real Number System

Joe Niederberger
In reply to this post by Jerry Becker
R Hansen says:
>Not only irrational, but transcendental (not algebraic). I am talking about the actual positions of the molecules and atoms, although they are moving, but you know what I mean. The argument is not much different than what you said about the improbability (impossibility) of a ray of light ever hitting a particular number. I am saying that said ray will not even hit any algebraic number. We know the ray of light will hit a number, I am saying that number will always be transcendental.

I think if you model a light ray as a 1 dimensional line,
that its just an approximation good for some purposes and not for others. Likewise, number lines and points and infinitely sharp darts makes a nice mental image, but the absurd result that says the dart does hit a particular point, but that it has zero % chance of hitting any point in particular, tells you something is not quite right with that picture. Probability is an area of mathematics where the difference between models, physics, and math is often murky, though I don't think it has to be.

As far as the distance between molecules, if you believe Heisenberg then surely you know the uncertainty is finite > 0 and greater than the tail end of any transcendental number. But the math of physics still uses the real number system, so one could argue the exact center of that uncertain area was exactly some transcendental number. Now I'm not even sure what that means, but to me it looks like faith at work (if one believes such things,) not anything that can be verified or proved.

R Hansen says:
>In fact, the ray of light would never hit any "particular" (constructed?) number because there are infinitely more non particular numbers for it to hit.

I find that insight interesting. It seems to touch on the delightful "paradoxes" G. Chaitin like to point out in "How Real are Real Numbers" and his book "Meta Math".
I would suggest that by the very fact that the dart hits some number (one of the anonymous, invisible "reals") then by that very fact, that number instantly becomes very particular, and therefore, unhittable.


Cheers,
Joe N

1234