7 ms·

What are the 'real numbers', really?

107 points by dhammack 13y ago

snowwindwaves 13y ago
Along a similar vein you may also enjoy http://arxiv.org/pdf/1303.6576 http://arxiv.org/pdf/1303.6576 The foundations of analysis by Larry Clifton. I always enjoy checking out the references in his papers as they are often hundreds of years old or more.
- anatoly 13y ago
  What other papers did he author? This is a curious paper. It's a rigorous derivation of (positive) real numbers without the use of 0 or negative numbers anywhere. It isn't very useful, although the fact that this can easily be done is by itself interesting. I have sometimes thought about the possibility of us encountering an advanced alien civilization and trying to match our math to theirs. Someone told me recently that if aliens were able to get into space, we can take it for granted that they knew negative numbers (in addition to more advanced concepts). I disagreed. Negative numbers are very convenient, but all the math that's needed for modern physics can, I think, be built up without them in a way that's more bulky and awkward, but not an order of magnitude bulky. This paper is weak evidence of my position.
  - snowwindwaves 13y ago
    The only other ones I know of are on his website http://cliftonlabs.net/TechnicalArticles.html http://cliftonlabs.net/TechnicalArticles.html
Bahamut 13y ago
I do have an issue with this line "Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later)" Infinitesimals have been made rigorous with modern mathematics.
- igravious 13y ago
  I agree. It would be truer to say that infinitesimals are studiously ignored by modern mainstream mathematicians because they feel that Dedekind and co. have put the calculus on a firm footing way back when. Anybody with a small bit of curiosity or a dashing of non-conformity will be suspicious of this narrative. If anything, infinitesimals in their various guises carry a certain explanatory heft, and are quite beguiling little creatures if you take the time to get to know them. I'd be happy to elaborate or leave a few links here if anybody is interested.
  - protonfish 13y ago
    I loathed limit-based calculus in High School and College. Later I read Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html http://www.math.wisc.edu/~keisler/calc.html and it all came clear in a fraction of the pages. It's infuriating that most math curricula won't drop those old, bloated, overly formal calculus tomes to improve the clarity and effectiveness of the instruction method.
    - ColinWright 13y ago
      Added in edit to emphasise a point: If all you want to do is differentiate and integrate, then non-standard analysis is probably, for most people, a faster way to be able to do just that. Now read on ... Non-standard analysis has been put on a firm, formal footing. Theorems have been proven showing that (largely) it's equivalent to the regular form of analysis. Some things are easier to prove in standard analysis, some things are easier to prove in non-standard analysis, etc, etc. However, this is only really of use if all you want to do is calculus. If you want to go beyond calculus, almost everything (in this and related areas) is about sequences, limits, limiting processes, functions, and transformations. There, non-standard analysis tends not to help, and unless you've done calculus the standard way, you have to learn all this stuff in an unfamiliar and difficult-to-visualize, abstract area. One of the main reasons for continuing to learn calculus in the epsilon-delta limiting process manner is exactly because it's not only formally sound, it's also giving you tools for moving beyond the rather limited world of differential calculus. Speculating wildly from limited experience, it might also be the case that starting people with the non-standard approach in calculus is actually just as confusing. You may find that you really only got the insights you did because you had already struggled with the standard approach, and then were given something that made it all fall into place. Perhaps some people they think the non-standard approach is easier, but in fact it's only because they've actually got the foundations from the other. Just a thought.
igravious 13y ago
a real number is "a point on the number line" These posts are always stimulating. My understanding of a line is that it is delimited by two points, but does not contain any points. To elaborate, no point could be "on" a line because a point has no extension, whereas a line does. This is the crux of the matter. Therefore a line is not "made up of" points. (By analogy a plane could not be made up of lines.) This begs the question, what are lines made up of? Are they made up of anything? Is a point really where two (or more) lines would intersect if they could intersect. Is this what is meant by a Dedekind cut?
- dragonwriter 13y ago
  > My understanding of a line is that it is delimited by two points, but does not contain any points. A line is (or can be viewed as) an infinite set of points. > To elaborate, no point could be "on" a line because a point has no extension, whereas a line does. That seems to be a consequence of an unusual definition of "on".
- dhammack 13y ago
  Your argument is more philosophical than mathematical. Lines are traditionally defined as the set of all points which satisfy some critera. In this case, a line is precisely made up of points.
- gizmo686 13y ago
  The "point on the number line" definition has always been non-rigourous. It is meant to imply the intuition that real numbers are what we typically think of as "numbers", notably that they extend to infinity, are ordered, and are dense (for any two distinct real numbers, there exists a real number between them). Of course from a rigorous perspective, this does not even suggest a difference between the reals and the rationals. The line you are talking about in the rest of your post seems to be an 'unrelated' object that is used in geometry. I am not familiar with the formal definition of line that is used in geometry, but one way of defining a line is as the set of all points which satisfy "y=mx+b", for a given (m,b). A line segment would be the above definition with restrictions on the domain: x_0<x<x_f.
- dnautics 13y ago
chowells 13y ago
What are "real numbers"? A horribly misnamed fiction. Nearly all of them cannot be represented with a finite amount of information. I strenuously object to naming an uncountable set "real" when only a countable subset (measure 0 of the full set) can be worked with in any way at all. We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.
- gizmo686 13y ago
  Do you have any idea of what set we should use to replace them with? The rational numbers can do a lot, but we have discovered that there are numbers worth talking about (and which can be described) that are not rational. Whatever replacement you propose must be usable where ever we would use real numbers, and must be at least as simple to use.
  - moyix 13y ago
    One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]). [1] http://en.wikipedia.org/wiki/Computable_number http://en.wikipedia.org/wiki/Computable_number [2] http://en.wikipedia.org/wiki/Computable_analysis http://en.wikipedia.org/wiki/Computable_analysis
    - gizmo686 13y ago
      Unfourtuantly, there exist numbers which are definable but not computable.
      - moyix 13y ago
        Sure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.
        deleted 13y ago
        [deleted]
graycat 13y ago
"Points on the line" is fine for the first, second, ..., tenth cut at a definition. Sure, completeness is the biggie for the reals compared with the rationals, algebraics, etc. Still, as in the OP, mentioning Dedekind cuts is okay since it is one way to establish completeness, but there is much more, e.g., as in John C. Oxtoby, Measure and Category. and even that doesn't fathom all that is special about the reals. E.g., for just a little more, there is the continuum hypothesis, that little thing! The OP wants to say that by mentioning Dedekind and completeness he is getting at what the reals really are; no, instead he is just cutting one layer deeper of something that has likely some infinitely many layers available. Yes, yes, yes, I know; I know; the reals are the only complete, Archimedean ordered field, okay, after we have defined completeness, Archimedean ordered, and field and explained why these are important. So, back to "points on the line" -- it's actually pretty good for a first cut.
DArcMattr 13y ago
I have a Master's in Applied Math. The comments about how "few students take [Real Analysis]" doesn't square with my experience and survey of an undergraduate mathematics education. Such a course is often called "Advanced Calculus", and is a required course for a Bachelors-level education in Math. I also understand in the European-style approach to teaching Math, students start off with a foundational approach to Calculus through Real Analysis, and not the hand-wavy & computation-driven Calculus course. The equivalence class approach attributed to Cantor is more generalizable in discussing sets. The theoretical foundation of Fourier Transforms lies in a similar completion of functions.
- pedrosorio 13y ago
  "I also understand in the European-style approach to teaching Math, students start off with a foundational approach to Calculus through Real Analysis, and not the hand-wavy & computation-driven Calculus course." Yes. Where I graduated, all engineering majors learn the axiomatic definition of the real numbers including the "supremum (least upper bound) axiom" at the beginning of the first calculus class.
bglazer 13y ago
Wow, vector multiplication suddenly makes sense. I had never seen it described with polar coordinates. It's wonderful to have this little insight now. It's unfortunate that my math knowledge is so filled with holes.
totemizer 13y ago
" It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same. Unfortunately there is no such theory." http://njwildberger.wordpress.com/2012/12/02/difficulties-with-real-numbers/ http://njwildberger.wordpress.com/2012/12/02/difficulties-wi...
- GFK_of_xmaspast 13y ago
  Guys like that in general have never seemed all that convincing to me.
  - totemizer 13y ago
    I admit, some of his ideas are a bit.. well, I don't like when people talk about God seriously, and he sometimes mentions it, very rarely. But aside that, everything I can understand from what he says is true. It's a philosophical debate and if you are on the "real numbers" bandwagon (where most people are), you would lose integrity and your reputation might suffer even if you would speak to Wildberger about real numbers. It's a shame really how people don't see why it's bad to use abstractions which are so general that they can be fit for any kind circumstances. Even if you real this article about the real numbers, there is wishful thinking (where he says that the real numbers would look like a line even at infinity but the rationals wouldn't. well, I don't see why the rationals would stop especially given what he says later...), cherry picking / the whole axiom selection stuff for proving it... and yeah, the axiom idea is generally bad anyway. etc.
    - GFK_of_xmaspast 13y ago
      So I have a phd in math and I do tend to think less of other mathematicians who argue against infinite sets or uncountable sets and such, the argument's been over for a hundred years, you lost, deal with it. It's mathematical geocentrism.
snake_plissken 13y ago
"Since (a,0)+(c,0)=(a+c,0) and (a,0)×(c,0)=(ac,0), the points along the horizontal axis have an arithmetic just like "ordinary" numbers" Holy hell that is clear, concise and compelling. If only my professors would have explained it like this more often in my freshman calc class which was so much more abstract and proof based than anything I had encountered before. The only thing I remember form that time is hellishly long study groups late into the night with my classmates.
gweinberg 13y ago
The problem with "points on a number line" as a definition for real numbers is that it's not clear how you can tell if you have all of them. You can populate a number line as densely as you care to using just rational numbers, but that's not all of them, you're missing out on numbers like the square root of two. You can toss in the non-intergral powers of rational numbers, but you still won't have all of them, you're missing out on col numbers like pi (or tau, if you prefer). Even after you toss in every solution to every differential equation you can name, and every number you can generate using well defined finite or infinite serieses, there's probably some horrible diagonaliztion proof that says you still don't have all of them.
- ColinWright 13y ago
  If you assume that there is no number bigger than zero but smaller than every positive number (basically the Archimedean property) then you can prove that "you've got them all." You use Dedekind cuts. Suppose there's a location on the line that's somehow missing - call it x. Let A be all the numbers less than x, let B be all the numbers greater than x, and that gives you your Dedekind cut. That Dedekind cut is, in a very real sense, x, and that means x is a real. QED. That needs tidying up and formalising, but it does work.
  - lmm 13y ago
    If you're using the Dedekind cut definition why use the line at all? Just say a real is any set of rationals bounded above, with arithmetic defined the obvious way; defining equality is slightly fiddly but it's fiddly with a number line too. What does the line visualization gain you?
    - ColinWright 13y ago
      Because it was asked how we knew we "got them all", referring to points on the line. The reals are a way of modelling the line, the line is a way of visualising the reals. Each is complementary to the other. And besides, the rationals are totally ordered, and their completion is totally ordered, so it makes sense to think of them as arranged in a line. The problem is that the reals are very, very strange in some ways, and people do get seduced into thinking they understand them, whereas usually it's just a case that they've got used to them.
SourPatch 13y ago
The contents of the linked page were the first lecture I had in my undergraduate calculus course. At the end of the lecture, we all looked around at each other wondering what we had just signed up for.
mherrmann 13y ago
This is a great article but unfortunately has one thing horribly wrong: Democracy far preceded the Age of Enlightenment. A form of democracy was already in place in ancient Greece at around 500 BC. Newton and the Age of Enlightenment were much later, at 1600+ AD. See Wikipedia: http://en.wikipedia.org/wiki/Democracy#History http://en.wikipedia.org/wiki/Democracy#History, http://en.wikipedia.org/wiki/Age_of_enlightenment http://en.wikipedia.org/wiki/Age_of_enlightenment, http://en.wikipedia.org/wiki/Isaac_Newton http://en.wikipedia.org/wiki/Isaac_Newton. Other than that, a great article!
- ddebernardy 13y ago
  Hehe. I balked at that too... It also states that you cannot order the field of complex numbers. Whereas I seem to recollect that there are ways to do so. For instance, z1 < z2 if x1 < x2 or x1 = x2 and y1 < y2.
  - claudius 13y ago
    By your definition of <, 0 < i and i^2 < 0, however, the OP requires that 0 < p AND 0 < q => 0 < p * q, which is not fulfilled by your < for p = i = q.
bjornsing 13y ago
I have an issue with this (albeit parenthesised) line: "It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces." In-between any two rational numbers there's an infinite number of other rational numbers. So, in any reasonable sense and at any level of "magnification", if you can "see" two dots representing two rational numbers then they are connected by a line of other little dots (just like the reals). Perhaps you could argue though that at "infinite magnification" there are no rational numbers to be seen, it's just empty space, whereas the reals of course still make a nice line.
- gizmo686 13y ago
  I don't think that works. The rational numbers are a dense subset of the real numbers. Informally this means every real number is either a rational number, or is arbitrarily close to a rational number. This means that at any magnification, if there was a hole that is filled by a real number, then their would also be a rational number that is arbitrarily close to that real number.
  - bjornsing 13y ago
    Well, I'm not a big fan of this "infinite magnification" idea in the first place, but "arbitrarily close" is typically one of those things that infinity can beat. (Compare e.g. with the fourier transform of a function. It consists of a sum series which comes "arbitrarily close" to the function, but "at the limit" when the number of terms approaches infinity the function and its fourier transform is one and the same.)
- thaumasiotes 13y ago
  Well, consider the ruler function[1], which is continuous on the irrationals and discontinuous on the rationals. The real numbers really are denser than the rationals; that's why something like the ruler function is possible (notably, a conceptual reverse, continuous on the rationals and discontinuous on the irrationals, cannot exist -- the rationals are too far apart). I'm pretty sure this is precisely the phenomenon the quote you extract is referring to: if you were standing, infinitely magnified, at a point on the ruler function, then the function would be continuous ("look like a line") if your point was irrational, but if your point was rational, there would be a measurable gulf separating you from the rest of the function. [1] http://en.wikipedia.org/wiki/Thomae%27s_function http://en.wikipedia.org/wiki/Thomae%27s_function
Stal3r 13y ago
Can someone explain the setup of the 0=1 exercise? It's poorly worded. Is it saying find: (Y,1,+,1,×) or is it saying find what "1" has to be to make it a valid field?
- ddebernardy 13y ago
  Given the field (Y, 0, +, 1, *), you need to show that either 0 != 1, or 0 = 1 is the only element in the set. I don't remember the precise proof, but if memory serves it derives from the existance of opposites and inverses, and 0 and 1 being unique in the set, due the commutative properties of abelean groups.

107 points by dhammack 13y ago

Along a similar vein you may also enjoy http://arxiv.org/pdf/1303.6576 http://arxiv.org/pdf/1303.6576 The foundations of analysis by Larry Clifton. I always enjoy checking out the references in his papers as they are often hundreds of years old or more.

anatoly 13y ago
What other papers did he author? This is a curious paper. It's a rigorous derivation of (positive) real numbers without the use of 0 or negative numbers anywhere. It isn't very useful, although the fact that this can easily be done is by itself interesting. I have sometimes thought about the possibility of us encountering an advanced alien civilization and trying to match our math to theirs. Someone told me recently that if aliens were able to get into space, we can take it for granted that they knew negative numbers (in addition to more advanced concepts). I disagreed. Negative numbers are very convenient, but all the math that's needed for modern physics can, I think, be built up without them in a way that's more bulky and awkward, but not an order of magnitude bulky. This paper is weak evidence of my position.
- snowwindwaves 13y ago
  The only other ones I know of are on his website http://cliftonlabs.net/TechnicalArticles.html http://cliftonlabs.net/TechnicalArticles.html

Bahamut 13y ago

I do have an issue with this line "Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later)" Infinitesimals have been made rigorous with modern mathematics.

igravious 13y ago
I agree. It would be truer to say that infinitesimals are studiously ignored by modern mainstream mathematicians because they feel that Dedekind and co. have put the calculus on a firm footing way back when. Anybody with a small bit of curiosity or a dashing of non-conformity will be suspicious of this narrative. If anything, infinitesimals in their various guises carry a certain explanatory heft, and are quite beguiling little creatures if you take the time to get to know them. I'd be happy to elaborate or leave a few links here if anybody is interested.
- protonfish 13y ago
  I loathed limit-based calculus in High School and College. Later I read Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html http://www.math.wisc.edu/~keisler/calc.html and it all came clear in a fraction of the pages. It's infuriating that most math curricula won't drop those old, bloated, overly formal calculus tomes to improve the clarity and effectiveness of the instruction method.
  - ColinWright 13y ago
    Added in edit to emphasise a point: If all you want to do is differentiate and integrate, then non-standard analysis is probably, for most people, a faster way to be able to do just that. Now read on ... Non-standard analysis has been put on a firm, formal footing. Theorems have been proven showing that (largely) it's equivalent to the regular form of analysis. Some things are easier to prove in standard analysis, some things are easier to prove in non-standard analysis, etc, etc. However, this is only really of use if all you want to do is calculus. If you want to go beyond calculus, almost everything (in this and related areas) is about sequences, limits, limiting processes, functions, and transformations. There, non-standard analysis tends not to help, and unless you've done calculus the standard way, you have to learn all this stuff in an unfamiliar and difficult-to-visualize, abstract area. One of the main reasons for continuing to learn calculus in the epsilon-delta limiting process manner is exactly because it's not only formally sound, it's also giving you tools for moving beyond the rather limited world of differential calculus. Speculating wildly from limited experience, it might also be the case that starting people with the non-standard approach in calculus is actually just as confusing. You may find that you really only got the insights you did because you had already struggled with the standard approach, and then were given something that made it all fall into place. Perhaps some people they think the non-standard approach is easier, but in fact it's only because they've actually got the foundations from the other. Just a thought.

igravious 13y ago

a real number is "a point on the number line" These posts are always stimulating. My understanding of a line is that it is delimited by two points, but does not contain any points. To elaborate, no point could be "on" a line because a point has no extension, whereas a line does. This is the crux of the matter. Therefore a line is not "made up of" points. (By analogy a plane could not be made up of lines.) This begs the question, what are lines made up of? Are they made up of anything? Is a point really where two (or more) lines would intersect if they could intersect. Is this what is meant by a Dedekind cut?

dragonwriter 13y ago
> My understanding of a line is that it is delimited by two points, but does not contain any points. A line is (or can be viewed as) an infinite set of points. > To elaborate, no point could be "on" a line because a point has no extension, whereas a line does. That seems to be a consequence of an unusual definition of "on".
dhammack 13y ago
Your argument is more philosophical than mathematical. Lines are traditionally defined as the set of all points which satisfy some critera. In this case, a line is precisely made up of points.
gizmo686 13y ago
The "point on the number line" definition has always been non-rigourous. It is meant to imply the intuition that real numbers are what we typically think of as "numbers", notably that they extend to infinity, are ordered, and are dense (for any two distinct real numbers, there exists a real number between them). Of course from a rigorous perspective, this does not even suggest a difference between the reals and the rationals. The line you are talking about in the rest of your post seems to be an 'unrelated' object that is used in geometry. I am not familiar with the formal definition of line that is used in geometry, but one way of defining a line is as the set of all points which satisfy "y=mx+b", for a given (m,b). A line segment would be the above definition with restrictions on the domain: x_0<x<x_f.
dnautics 13y ago

chowells 13y ago

What are "real numbers"? A horribly misnamed fiction. Nearly all of them cannot be represented with a finite amount of information. I strenuously object to naming an uncountable set "real" when only a countable subset (measure 0 of the full set) can be worked with in any way at all. We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.

gizmo686 13y ago
Do you have any idea of what set we should use to replace them with? The rational numbers can do a lot, but we have discovered that there are numbers worth talking about (and which can be described) that are not rational. Whatever replacement you propose must be usable where ever we would use real numbers, and must be at least as simple to use.
- moyix 13y ago
  One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]). [1] http://en.wikipedia.org/wiki/Computable_number http://en.wikipedia.org/wiki/Computable_number [2] http://en.wikipedia.org/wiki/Computable_analysis http://en.wikipedia.org/wiki/Computable_analysis
  - gizmo686 13y ago
    Unfourtuantly, there exist numbers which are definable but not computable.
    - moyix 13y ago
      Sure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.
      - deleted 13y ago
        [deleted]

graycat 13y ago

"Points on the line" is fine for the first, second, ..., tenth cut at a definition. Sure, completeness is the biggie for the reals compared with the rationals, algebraics, etc. Still, as in the OP, mentioning Dedekind cuts is okay since it is one way to establish completeness, but there is much more, e.g., as in John C. Oxtoby, Measure and Category. and even that doesn't fathom all that is special about the reals. E.g., for just a little more, there is the continuum hypothesis, that little thing! The OP wants to say that by mentioning Dedekind and completeness he is getting at what the reals really are; no, instead he is just cutting one layer deeper of something that has likely some infinitely many layers available. Yes, yes, yes, I know; I know; the reals are the only complete, Archimedean ordered field, okay, after we have defined completeness, Archimedean ordered, and field and explained why these are important. So, back to "points on the line" -- it's actually pretty good for a first cut.

DArcMattr 13y ago

I have a Master's in Applied Math. The comments about how "few students take [Real Analysis]" doesn't square with my experience and survey of an undergraduate mathematics education. Such a course is often called "Advanced Calculus", and is a required course for a Bachelors-level education in Math. I also understand in the European-style approach to teaching Math, students start off with a foundational approach to Calculus through Real Analysis, and not the hand-wavy & computation-driven Calculus course. The equivalence class approach attributed to Cantor is more generalizable in discussing sets. The theoretical foundation of Fourier Transforms lies in a similar completion of functions.

pedrosorio 13y ago
"I also understand in the European-style approach to teaching Math, students start off with a foundational approach to Calculus through Real Analysis, and not the hand-wavy & computation-driven Calculus course." Yes. Where I graduated, all engineering majors learn the axiomatic definition of the real numbers including the "supremum (least upper bound) axiom" at the beginning of the first calculus class.

bglazer 13y ago

Wow, vector multiplication suddenly makes sense. I had never seen it described with polar coordinates. It's wonderful to have this little insight now. It's unfortunate that my math knowledge is so filled with holes.

totemizer 13y ago

" It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same. Unfortunately there is no such theory." http://njwildberger.wordpress.com/2012/12/02/difficulties-with-real-numbers/ http://njwildberger.wordpress.com/2012/12/02/difficulties-wi...

GFK_of_xmaspast 13y ago
Guys like that in general have never seemed all that convincing to me.
- totemizer 13y ago
  I admit, some of his ideas are a bit.. well, I don't like when people talk about God seriously, and he sometimes mentions it, very rarely. But aside that, everything I can understand from what he says is true. It's a philosophical debate and if you are on the "real numbers" bandwagon (where most people are), you would lose integrity and your reputation might suffer even if you would speak to Wildberger about real numbers. It's a shame really how people don't see why it's bad to use abstractions which are so general that they can be fit for any kind circumstances. Even if you real this article about the real numbers, there is wishful thinking (where he says that the real numbers would look like a line even at infinity but the rationals wouldn't. well, I don't see why the rationals would stop especially given what he says later...), cherry picking / the whole axiom selection stuff for proving it... and yeah, the axiom idea is generally bad anyway. etc.
  - GFK_of_xmaspast 13y ago
    So I have a phd in math and I do tend to think less of other mathematicians who argue against infinite sets or uncountable sets and such, the argument's been over for a hundred years, you lost, deal with it. It's mathematical geocentrism.

snake_plissken 13y ago

"Since (a,0)+(c,0)=(a+c,0) and (a,0)×(c,0)=(ac,0), the points along the horizontal axis have an arithmetic just like "ordinary" numbers" Holy hell that is clear, concise and compelling. If only my professors would have explained it like this more often in my freshman calc class which was so much more abstract and proof based than anything I had encountered before. The only thing I remember form that time is hellishly long study groups late into the night with my classmates.

gweinberg 13y ago

The problem with "points on a number line" as a definition for real numbers is that it's not clear how you can tell if you have all of them. You can populate a number line as densely as you care to using just rational numbers, but that's not all of them, you're missing out on numbers like the square root of two. You can toss in the non-intergral powers of rational numbers, but you still won't have all of them, you're missing out on col numbers like pi (or tau, if you prefer). Even after you toss in every solution to every differential equation you can name, and every number you can generate using well defined finite or infinite serieses, there's probably some horrible diagonaliztion proof that says you still don't have all of them.

ColinWright 13y ago
If you assume that there is no number bigger than zero but smaller than every positive number (basically the Archimedean property) then you can prove that "you've got them all." You use Dedekind cuts. Suppose there's a location on the line that's somehow missing - call it x. Let A be all the numbers less than x, let B be all the numbers greater than x, and that gives you your Dedekind cut. That Dedekind cut is, in a very real sense, x, and that means x is a real. QED. That needs tidying up and formalising, but it does work.
- lmm 13y ago
  If you're using the Dedekind cut definition why use the line at all? Just say a real is any set of rationals bounded above, with arithmetic defined the obvious way; defining equality is slightly fiddly but it's fiddly with a number line too. What does the line visualization gain you?
  - ColinWright 13y ago
    Because it was asked how we knew we "got them all", referring to points on the line. The reals are a way of modelling the line, the line is a way of visualising the reals. Each is complementary to the other. And besides, the rationals are totally ordered, and their completion is totally ordered, so it makes sense to think of them as arranged in a line. The problem is that the reals are very, very strange in some ways, and people do get seduced into thinking they understand them, whereas usually it's just a case that they've got used to them.

SourPatch 13y ago

The contents of the linked page were the first lecture I had in my undergraduate calculus course. At the end of the lecture, we all looked around at each other wondering what we had just signed up for.

mherrmann 13y ago

This is a great article but unfortunately has one thing horribly wrong: Democracy far preceded the Age of Enlightenment. A form of democracy was already in place in ancient Greece at around 500 BC. Newton and the Age of Enlightenment were much later, at 1600+ AD. See Wikipedia: http://en.wikipedia.org/wiki/Democracy#History http://en.wikipedia.org/wiki/Democracy#History, http://en.wikipedia.org/wiki/Age_of_enlightenment http://en.wikipedia.org/wiki/Age_of_enlightenment, http://en.wikipedia.org/wiki/Isaac_Newton http://en.wikipedia.org/wiki/Isaac_Newton. Other than that, a great article!

ddebernardy 13y ago
Hehe. I balked at that too... It also states that you cannot order the field of complex numbers. Whereas I seem to recollect that there are ways to do so. For instance, z1 < z2 if x1 < x2 or x1 = x2 and y1 < y2.
- claudius 13y ago
  By your definition of <, 0 < i and i^2 < 0, however, the OP requires that 0 < p AND 0 < q => 0 < p * q, which is not fulfilled by your < for p = i = q.

bjornsing 13y ago

I have an issue with this (albeit parenthesised) line: "It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces." In-between any two rational numbers there's an infinite number of other rational numbers. So, in any reasonable sense and at any level of "magnification", if you can "see" two dots representing two rational numbers then they are connected by a line of other little dots (just like the reals). Perhaps you could argue though that at "infinite magnification" there are no rational numbers to be seen, it's just empty space, whereas the reals of course still make a nice line.

gizmo686 13y ago
I don't think that works. The rational numbers are a dense subset of the real numbers. Informally this means every real number is either a rational number, or is arbitrarily close to a rational number. This means that at any magnification, if there was a hole that is filled by a real number, then their would also be a rational number that is arbitrarily close to that real number.
- bjornsing 13y ago
  Well, I'm not a big fan of this "infinite magnification" idea in the first place, but "arbitrarily close" is typically one of those things that infinity can beat. (Compare e.g. with the fourier transform of a function. It consists of a sum series which comes "arbitrarily close" to the function, but "at the limit" when the number of terms approaches infinity the function and its fourier transform is one and the same.)
thaumasiotes 13y ago
Well, consider the ruler function[1], which is continuous on the irrationals and discontinuous on the rationals. The real numbers really are denser than the rationals; that's why something like the ruler function is possible (notably, a conceptual reverse, continuous on the rationals and discontinuous on the irrationals, cannot exist -- the rationals are too far apart). I'm pretty sure this is precisely the phenomenon the quote you extract is referring to: if you were standing, infinitely magnified, at a point on the ruler function, then the function would be continuous ("look like a line") if your point was irrational, but if your point was rational, there would be a measurable gulf separating you from the rest of the function. [1] http://en.wikipedia.org/wiki/Thomae%27s_function http://en.wikipedia.org/wiki/Thomae%27s_function

Stal3r 13y ago

Can someone explain the setup of the 0=1 exercise? It's poorly worded. Is it saying find: (Y,1,+,1,×) or is it saying find what "1" has to be to make it a valid field?

ddebernardy 13y ago
Given the field (Y, 0, +, 1, *), you need to show that either 0 != 1, or 0 = 1 is the only element in the set. I don't remember the precise proof, but if memory serves it derives from the existance of opposites and inverses, and 0 and 1 being unique in the set, due the commutative properties of abelean groups.