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Why are rational numbers, alone, unsuitable for making a coordinate system? Can somebody provide me with some reasons or some book/website at which I can read?

These are the only ones I can come up with

  1. Unable to plot points which have an irrational coordinate
  2. Unable to find solutions to some set of equations which have irrational solutions
mathnoob123
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    They are suitable as coordinate system in the rational vector space $\Bbb Q^2$ – Hagen von Eitzen Sep 16 '17 at 18:48
  • In practice it does not matter because we cannot mark, for example, EXACTLY the position of $\sqrt{2}$. If we determine values graphically, they are therefore always rational. Graphically, we cannot distinguish rational and irrational numbers. – Peter Sep 16 '17 at 18:50
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    @HagenvonEitzen Can you please elaborate a little, for example, how would 2^(1/3) be rational in your suggested space? – mathnoob123 Sep 16 '17 at 18:52
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    In general, you can define a coordinate system with two copies of any set, however, the rational numbers alone are not complete, so some methods of calculus are not well defined. See here: https://math.stackexchange.com/questions/1880741/why-cant-calculus-be-done-on-the-rational-numbers –  Sep 16 '17 at 18:55
  • If you mean coordinate systems in theory : If we do not allow irrational entries, we miss (to only give an example) the roots of $x^2-2$. Even worse, almost no points of the function $\large f(x)=e^x$ have the property that both coordinates are rational. We would have only the point $(0/1)$ – Peter Sep 16 '17 at 18:56
  • @Alex Thank you very much for your insight. Since I am writing a paper, can you please cite more sources or keywords which I should search for? It would be very helpful – mathnoob123 Sep 16 '17 at 18:59
  • They are suited enough for basic linear algebra. They are unsuited for analysis because of the missing completeness and the reasons you gave yourself. E.g. $x^4+y^4=1$ contains only four points. In $\Bbb R^2$ this curve is closed like a circle. – M. Winter Sep 16 '17 at 19:03
  • @FaiqRaees Any introductory real analysis text book should cover this in detail. I like Analysis with an Introduction to Proof, 5th Edition by Steven Lay –  Sep 16 '17 at 23:03

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As many comments said, there are ways of analyzing $\mathbb{Q}^n $ as a vector space. However, they do not form a "coordinate system" for Euclidean space as we normally think about it. You can make a coordinate system which includes only rational points, but just like you said, it will be missing irrational points. It will also have odd problems, such as not forming a metric space. For instance, if you have a right triangle with legs of length 1, its hypotenuse will in fact not have a length, since it should have length $\sqrt 2$ (an irrational number) by the Pythagorean Theorem. Thus, not every pair of points has a distance between them which can be described by a rational number. Similarly, a circle of radius one would not have a rational circumference. In fact, you can't create a circle with a rational circumference in a rational only system, since those have a radii which are irrational. Also, the fact that irrational points are missing actually means MOST points are missing. This is a bit more advanced, but since the reals are an uncountable set while the rationals are countable (you can easily look up what this means and see proofs of this fact, pick your favorite). That means that nearly every point in space has irrational coordinates.

In the end, rational coordinates are simply not useful in most cases for the reasons I gave above and many more. You can create a rational coordinate system, but many things which form the foundation of modern math and interesting results, like all of calculus, would be impossible.

MathTrain
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  • Thank you very much for your insight. Since I am writing a paper, can you please cite more sources or keywords which I should search for? It would be very helpful – mathnoob123 Sep 16 '17 at 19:00
  • Sure, I will update with some helpful sources :) – MathTrain Sep 16 '17 at 19:02
  • Will you need more rigorous proofs in these sources, or just introductory materials to the ideas? – MathTrain Sep 16 '17 at 19:14
  • A little of rigorous material but more of introductory. (Basically anything that provides some thoughts on the provided question) – mathnoob123 Sep 16 '17 at 19:16
  • Hello. Apologies for disturbing you again but can you please tell me if you're providing some sources? It would really lay off sone burden. – mathnoob123 Sep 16 '17 at 19:40
  • Sorry for the wait, hopefully this is still helpful! My apologies. Here is a quick intro video to the idea of Cantor's diagonal argument: https://youtu.be/elvOZm0d4H0 and here is a link to a bit more in depth description:http://www.mathpages.com/home/kmath371/kmath371.htm – MathTrain Sep 18 '17 at 04:21