# Visual Presentation of Real Number System

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## Visual Presentation of Real Number System

 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       E-mail: [hidden email] -- 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

 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

 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

 In reply to this post by Jerry Becker A slow year at the math department I guess.Bob HansenOn 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       E-mail: [hidden email] -- 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]
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## Re: Visual Presentation of Real Number System

 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

 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

 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

 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

 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=18225Bob 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

 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

 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

 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

 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

 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

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## Re: Fwd: Re: Visual Presentation of Real Number System

 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

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## Re: Visual Presentation of Real Number System

 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

 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.:)