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This question is related to the Geometry course.

On a quiz requiring an answer of always/sometimes/never, I was asked this question:

"A coordinate can __ be paired with a point on a number line"

My answer was "always", however the teacher's answer on the quiz was "sometimes". I thought my answer was true because of the ruler postulate.

However, I wanted to hear your opinion. Is my answer correct, or is the teacher's? If I am wrong, could you give me an example of when a coordinate cannot be paired with a point on a number line?

P.S. I googled the question, believing that my answer was correct, and found the same question that agreed (thus disagreeing with the quiz results) with me on this site.

http://quizlet.com/12951617/plane-questions-flash-cards/

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What is a coordinate? –  Mariano Suárez-Alvarez Sep 12 '12 at 19:26
    
How are you defining coordinates? –  Kris Williams Sep 12 '12 at 19:27
    
I'm not sure, but I think it means a point? –  ZERO Sep 12 '12 at 19:28
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It is impossible to know which answer is correct unless we know what the terms in the question mean. –  Mariano Suárez-Alvarez Sep 12 '12 at 19:31
    
Which definition of "coordinate" would cause the answer to be "sometimes"? Which definition will cause the answer to be "always"? –  ZERO Sep 12 '12 at 19:34
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1 Answer

up vote 1 down vote accepted

I will assume that by geometry you mean two-dimensional Euclidean geometry. A "point" would have two "coordinates", each of them being real numbers. So, if the question really does refer to a "coordinate", that is, one half of a "coordinate pair" defining the location of a point, the answer is "always"; every coordinate is a real number, and so it can be plotted on a one-dimensional number line.

SUBSTANTIAL EDIT:

Now, two coordinates, making up a coordinate pair, are both real numbers. As such, any one real number can be paired with any other in this manner, and only the points having coordinates $(x, f(x))$ for a deterministic, continuous $f(x)$ can be plotted along a line (this is what you learned in algebra). The function $f(x)$, in set theory, defines a "bijection" - a method of transformation between elements of two sets, for which every element in the "destination" set can be produced using one and only one element of the "source" set.

It was originally thought that no $f(x)$ existed such that the set of its solutions for all real values of $x$ contained all points in $\mathbb R^2$; equivalently, that no bijection existed mapping all $\mathbb R$ to $\mathbb R^2$ existed. Thus, it was thought that this meant that the set of all two-dimensional points, known in set theory as $\mathbb R^2$, was larger than the set of all real numbers, $\mathbb R$, and specifically that $|\mathbb R^2| = |\mathbb R|^2$.

However, it turns out this is not true. Georg Cantor proved in 1877, using the "set theory" branch of discrete mathematics he helped create, that there are as many points on the unit line segment (basically, as many real numbers between 0 and 1) as there are points in any "finite-dimensional space", including $\mathbb R^2$, by creating just such a bijection. He is famously quoted in a letter to fellow mathematician Richard Dedekind, "I see it, but I don't believe it!". He had previously proven that there are as many real numbers within any finite range as there are real numbers by a similar method. Taking those two together, we can say that $|\mathbb R| = |\mathbb R^2|$, or in plain English, that there are an equal number of numbers on the real number line as there are points in two-dimensional space, and therefore every point in two-dimensional space must be mappable to a single point on the one-dimensional real number line.

As such, the answer, from discrete math, to the question "A coordinate pair can __ be paired with a point on a number line" is also "always".

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The assertion that $\mathbb{R}^2$ and $\mathbb{R}$ have different cardinalities is not correct. The fact that the cardinalities are the same was first proved by Cantor. –  André Nicolas Sep 12 '12 at 20:36
    
(52,0) should create 520. A slightly better example would be (5,20) and (52,0) both create 520. –  Arkamis Sep 12 '12 at 20:45
    
@AndréNicolas - You are right, edited. Though I'm still a little sketchy on how a Hilbert curve, iterated countably infinite times and thus having countably infinite points, can have uncountably infinite points if $\aleph_0 \neq \beth_1$. –  KeithS Sep 12 '12 at 21:31
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@KeithS: The bijective map we can get should not be confused with geometric mappings, where we impose continuity conditions of some kind. The classical space-filling curves are not bijections. –  André Nicolas Sep 12 '12 at 21:41
    
@AndréNicolas - Can you point us to any actual online source of Cantor's proof? Wikipedia merely asserts he did prove it, but there's no reference to his letter to Dedekind or the subsequent 1878 Crelle paper. –  KeithS Sep 12 '12 at 21:57
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