|WikiProject Mathematics||(Rated C-class, High-importance)|
Geez, Axel, we were just talking about the reconfiguring of Topology and Topological_space, and here I can't figure out how to use them correctly! Thanks for catching me. — Toby Bartels, Tuesday, June 11, 2002
Merging "Number line"
I suggest that "number line is merged with this article. They cover almost the same idea. Actually, I think the article should be "Real number line", which is currently a redirect to "Real line", because it seems to be more popular in actual usage than just "Real line". Jason Quinn (talk) 15:50, 22 January 2010 (UTC)
- I agree with the merge but I think this should be merged into 'Number line'. The references I'm looking at that phrase as the primary one with 'Real line' given as a synonym if at all.--RDBury (talk) 17:50, 12 May 2010 (UTC)
As a teacher using the number line page, I hope you do not merge this into the real line page. The real line page is written at a level appropriate for college students. The number line page is usable in my high school classroom. Please keep them separate. -- Jeffrey Elkner, Arlington Public Schools. Jelkner (talk) 14:26, 18 October 2010 (UTC)
- I strongly disagree with the merge -- these two articles are about different things. Number line is about the idea that the real numbers can be arranged in a line, something that is familiar to any high-school student. This article is about the real line as a geometric object in higher mathematics, including its role as a topological space and metric space. The name is appropriate -- no mathematician would ever say "the number line" when asked to give an example of an uncountable connected metric space, though they might say "the real line", which is some indication that the two concepts are distinct. Similarly, if asked to explain why 3 + −5 = −2, a mathematician might say "well, think about it on the number line", and would be less likely to say "think about it on the real line".Jim (talk) 00:27, 24 October 2010 (UTC)
Another maths teacher here, voting not to merge. I see the number line, as distinct (discrete) points for each of the integers. Almost like beads on a thread. They are separate concepts that only seem the same in hindsight. 21 December 2010 —Preceding unsigned comment added by 126.96.36.199 (talk) 18:48, 21 December 2010 (UTC)
- How about combining both article under number line and organizing it in increasing level of sophistication? Is it really so that you picture the number "line as beads on thread" only containing the beads? I suppose at least rational numbers belong to your curriculum. Lapasotka (talk) 11:54, 15 January 2011 (UTC)
Merging real line with line (geometry) should also be considered. In fact the real line is the unique line of the Euclidean space of dimension 1 and all lines in any Euclidean space are isomorphic to it. As real line is mainly devoted to the mathematical properties shared by all these lines, it would make sense to merge this article into line (geometry). If it is done number line could remain as an article of explicit basic level, with explicit references to the other articles for "further properties". D.Lazard (talk) 12:15, 15 January 2011 (UTC)
I agree not to merge it too, since "real line" is obviously more abstract, and "number line" is labelled with specific numbers. Also, could someone add material to demonstrate the geometric construction for adding fractions to a number line? I looked at real line, number line, rational numbers, and fraction (mathematics) but none of these pages show how fractions can appear on a number line. I think it could help people understand quantities if they could see this. Wikivek (talk) 19:56, 26 January 2011 (UTC)
Yet another 'maths' teacher, though this time from The States. I support the merge. While they are certainly written at different levels, we could place the early-level, understandable content at the beginning of the article, and move forward in complexity as do most mathematics articles. I see no reason for students to view the number line as "as distinct (discrete) points for each of the integers. Almost like beads on a thread." and think this is a wrong concept for students to have. I strive to remind my students when they look at a number line, that it is not only the numbers that are labeled that are one the line. Otherwise it's not a line at all, but a string of points............. MERGE. Cliff (talk) 04:32, 1 June 2011 (UTC)
- And another thing,(rant coming, skip if you want :), i think it is inappropriate to try to protect students from higher level knowledge and material. If a student doesn't understand something that is written, they'll probably forget it, but to try to keep the articles separate so that your students cannot see material that could help them, or that they might use in the future is not historically helpful. Textbooks of mathematics today provide faulty conceptions (beads on a line) and do not give students any glimmer of what else there might be to learn. Students who might teach themselves and move very quickly into higher mathematics, and who might be the next Euler or Descartes, are not given the chance to do so when we protect them from learning by compartmentalizing knowledge in this way. Cliff (talk) 04:45, 1 June 2011 (UTC)
I have changed my mind about the merge. I am persuaded by the argument that "number line" and "real line" have different audiences. Will remove merge discussion template after a grace period if there's no opposition. Jason Quinn (talk) 05:20, 22 July 2011 (UTC)
Redirect to REAL NUMBER LINE
FOLKS, PLEASE TAKE THE TIME REQUIRED TO FULLY HEAR ME OUT:
I didn’t have enough time to carefully thread my way through the labyrinth of misunderstandings that littered the discussion page I visited yesterday. Something happened between then and now. And I apologize in advance to those well-meaning contributors to that other page, for my being blunt. I also apologize up front for its length, and for not posting this as a link to a file (I didn’t know how). I desire to tell you an interesting story about NUMBERS, and hope the emperor of this page will find my story not just compelling, but memorable. With that said, I will begin:
The history of mathematics, and I have read a fair amount of it, teaches us how unbelievably hard it was to get humanity to understand what today we flippantly call the real numbers. Numbers are REAL, aren't they? Respectfully, I point out that this article says absolutely NADA about the history I am about to provide. I do wonder why.
About small ancient cultures (and some that are still around), we know that some less stressed, or less developed cultures actually counted one, two, many; that's it! And it is said of olden days that as shepherds took sheep from their village into the fields each morning, someone would drop pebbles into a bowl, one for each, so when the sheep returned home at night, they could make sure the correct number of sheep returned, by removing pebbles, one at a time, one for each. An empty bowl must have given the villagers a kind of satisfaction.
More compulsive humans began developing counting numbers, actual symbols that could be written down and used to represent numbers, quite unlike pebbles or sticks or marks cut into sticks, so we humans could more easily count sheep, or anything else, like, say, the number of hectares of land one owned. These counting numbers were eventually given the name positive whole numbers, also known as the positive integers, After what seems like eons of time passed, man gave in and admitted the seemingly ridiculous (number?) zero (like there might have been zero caves available to live in, way back then), but the simple addition of zero to the then existing set of counting numbers lead man to the set of natural numbers (the set of counting numbers plus zero). Each slow step in this steady advance mankind made in understanding numbers seemed the last time man would ever have to make an annoying accommodation to his (consensus) number system.
But then one day some idiot came along who wanted to be able to subtract a larger positive integer from a smaller positive integer, and to make any sense of such an operation, discovered the negative integers, which like doubled the number of elements we then had in the set of integers that man had to work with. But man didn't stop there, he showed genuine cleverness by inventing fractions, numbers representable as p/q, where p and q are elements of the set of integers, and again man thought he was done. Lest I offend anyone, women doubtless figure into this as well!
And later someone wanted to determine the square root of 2, or determine pi, and OMG it turned out there were more of those stupid numbers than there were of the more acceptable fractions. That problem got fixed in man's normally backassward way by deciding to name the stupid numbers irrational (god forbid they had gotten called imaginary; what a fix people would have been in today if that had happened). Correspondingly, the not stupid fractions were given the name rational numbers (as if to say, fractions were somehow qualitatively better, or at least more rational than the irrational numbers). However long all of this took to evolve, voila', man finally arrived at what was thought to be the complete set of numbers that we now call the set of real numbers.
This story (though simplistic, imperfect, and far from complete) doesn't end here, however. Over and beyond the real numbers, man went on to discover complex numbers, in answering the problem of determining the square root of negative real numbers. After that even more extraordinary mathematical constructs were discovered.
But it appears that something like a certain line of human inquiry pretty much ended with the completion of the discovery of the set of real numbers. It must have felt to the participants similar to what humans need so much of now, a thing called closure. The real numbers are so fundamentally important to our concept of number that anyone who today dares to carelessly toss around nomenclature that took centuries to develop, at this moment in our history, when there still exist people who know better, should be hung from a yard-arm! (figuratively) :(
I am attempting to argue, strongly I hope, that the topic of this Wiki page should be the REAL NUMBER LINE. I do not claim to have my history absolutely correct; I'm not even trying to make the actual chronology work; but I know the rough outline of my story, the essence of which I am trying to communicate, is essentially correct. If it were my goal to teach this as real history, I would have peppered this with citations.
No, I feel that what I am about is far more important than relating a perfect history of the development of the REAL NUMBER LINE. I am telling you that there is a tremendously exciting learning experience awaiting eager students of mathematics who decide to learn about the real number system, and its representation as a real number line, and any compelling story of it must be entertwined with the telling of the correct history of how mankind got to where we now believe we know that the Real Numbers are at least fully categorized.
But this story is incredibly interesting as, for example, we know we can never write down all of the real numbers because there are infinities that bar such enumeration. The set of natural numbers is infinite, as is the set of all squared numbers, or the even numbers, or the odd numbers, or the rational numbers, or the irrational numbers. In fact, if one looks up Georg Cantor's work, one will become truly amazed, because he showed us that there is a hierarchy of these infinities, the so-called transfinite numbers. Georg’s work explored the notion of infinity, as applied to the set of real numbers, and its subsets, and his work spawned set theory, as well. I cannot imagine a student that cannot be jostled to life in order to understand what infinity is. Perhaps that interest might lead that student into areas of mathematics that are yet to be explored.
And that is only part of the excitement of the story, because at this point you can tell the student how amazing it must have been when man discovered that in between any two given real numbers there exists an infinity of real numbers they could not write down but, nonetheless, were able to prove existed. Now, how is that for telling them an exciting story? Here we have the REAL NUMBER LINE, envisioned incorrectly by the untutored as a "string of beads" extending to minus and plus infinity in going from the left and right sides of the point marked ZERO.
In truth, all real numbers could NOT be encountered by a counting mite as it slid itself slowly along that “string of beads” some people imagine a REAL NUMBER LINE, because in taking its first minute step between the number zero and zero point one, the mite would be lost, seemingly forever, going deeper, and deeper, and deeper, down in-between those two specified numbers. Don't believe it? Do a little reading. Perhaps in an amount of time that is on the order of the age of the universe, that mite might actually make a little progress across the REAL NUMBER LINE; at least consider the possibility! Maybe you could calculate its progress.
Again, REAL NUMBER LINE. Please, say it slowly enough that it finds its proper place in your schema, for it is a simple geometric way of encapsulating, concisely expressing, without enumeration, each and every member of our hard won set of Real Numbers.
The mathematical terms line and set, as applied to the real numbers, come from two different philosophical approaches to knowing and naming things. The term line as a representation of the real numbers, such as in the real number line, descended from geometry (Euclid), whereas the use of the term set as a representation of the real numbers, descended from algebra, and specifically set theory, introduced by Cantor. Both terms turn out to be very helpful in visualizing and using the real numbers for purposes practical or pedantic. And both mathematical pursuits—geometry and algebra—are still there, underpinning the skills of pure mathematicians world-wide.
The REAL NUMBER LINE, all by itself, without embellishment, correlation, or other high sounding technical baggage or blather, is so fundamentally important, yet so simple, for young people to learn about that there should be a war if anyone dare upset this mathematical apple-cart, even if a claim is made that “damage, if any, would be minimal”. To diminish, in any way, the concept of the REAL NUMBER LINE, by calling it, say simply NUMBER LINE, while really meaning the REAL NUMBER LINE is to beg the question "What about the Complex Numbers, why can't there be a COMPLEX NUMBER LINE, all complex numbers expressed as a line? Or how about higher order systems of numbers, or abstract mathematical entities like matrices, n-manifolds, even general mathematical structures as complicated as one can imagine under the categorical name number"?
History has shown us that the fractalization of mathematics never stops. Yet here we find one mathematical funda that is something we hold solidly in hand, and it's called the REAL NUMBER SYSTEM, and it is about a set of numbers that can be simply represented as the REAL NUMBER LINE. What else is the Cartesian coordinate system but two such REAL NUMBER LINES drawn to orthogonally cross one another at what is called the origin, used to visualize the mapping of a real function from one real number line called x to another “real number line” called y.
Upon this magnificent mathematical achievement, the rest of mathematics is supported like a great building is supported by its rock-solid foundation. We are talking about the solid cement FOUNDATION OF MATHEMATICS here, nothing less. The REAL NUMBER LINE is as significant a development as, say Pythagoras's Theorem, and is unconsciously passed around even more frequently. But, it ain’t Calculus, so don’t make it any more complex than it is.
And notice that I did not have to stoop to calling it the REAL LINE. The notion of a REAL LINE is an absurdity. No real line is REAL, for starters. Paradoxically, no actual REAL NUMBER LINE exists in our 'real world', for only a crude facsimile of one can be used as one, for a mathematical "line" is so thin (vanishingly thin thing explained as the points of contact between two planes) a construct that no human being has ever been able to actually draw one! A line has no thickness because a line is a mathematical construct conceived as a tightly strung string of points formed by the junction of two planes, each individual point being so tiny (vanishingly tiny) that no human being has ever been able to actually draw one!
Mathematics is all about the precision of carefully defined idealizations that don’t actually exist in our “real world”, other than by approximations, yet this idea of a line that we take to actually BE a REAL NUMBER LINE, even though it is impossible to draw one, is immensely useful to us, despite what might seem to be a horrible drawback. Daily we draw this figment from our mathematical imagination, a figment which can only represent or stand for, a REAL NUMBER LINE, yet we all use them quite effectively; and thus from something imaginary, even Platonic (See Roger Penrose, for example), we get to something extremely practical. Imagine the curious, interestedly asking “How could that be?” Isn’t that a part of the excitement of “learning mathematics?”
I must have been just lucky when I thought to type "REAL NUMBER LINE" on Wikipedia yesterday, for it looks, if I am to believe what I see, someone changed all of this mathematical brilliance by simply changing what they consider to be just a name for something, nothing that should really matter, so long as it makes sense to them, or MERGE it with some other concept. HEY, this isn't a corporation looking to become a monopoly, this is MATHEMATICS; the language of physics and other sciences! The difficulty that many young people have with math, excluding the damage done them by (some) dispassionate or poorly schooled math teachers, is > 50% about the precision of math definitions and terminology, about mathematical concepts, knowledge of which is the price they have to pay for entrance into the priesthood, and about which includes knowing about how those concepts got defined, historically, in the ways they did; so please, please, please, don't screw this up. Put it back the way it was!!! Langing (talk) 17:30, 25 May 2011 (UTC)
- TL;DR. The history information you are talking about can be found at the article real number If you want that content and the description of the line in the same place I recommend you attempt to merge real line into real number. It could have it's own section. That said, I wouldn't be involved in that battle. You could also make sure that this article links to real number in a way that you feel is appropriate, be carefule not to start a Farm Cliff (talk) 04:38, 1 June 2011 (UTC)