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In mathematics it is said that a "dot" has no dimension. On a different context it is said that a line is made by joining different "dots". Then how do line have only one dimension when its made up of something which has no dimension??

Caleb Stanford
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user304327
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  • Since when was a line made up of just its two endpoints? The dimension $1$ stuff is what's in between. – Milo Brandt Jan 12 '16 at 04:34
  • @milo brandt....i know but how is it even getting that one dimension since its made up of "dots" which has no dimension...... – user304327 Jan 12 '16 at 04:41
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    Answering this question is mostly about resolving language: what is the definition of dimension you are working with? – Simon S Jan 12 '16 at 04:45
  • I think itz upon us if we accdpt a dot to be $dx$ circle then we can say it to 1D – Archis Welankar Jan 12 '16 at 04:47
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    How is it possible for "cat" to refer to a small, furry mammal, when neither "c" nor "a" nor "t" has any of those qualities by itself? It's an "emergent property". Put things together the right way, the collection can have properties that the individual components don't have. – Gerry Myerson Jan 12 '16 at 05:01
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    How can the set of natural numbers be infinite when every element is finite? – Thomas Andrews Jan 12 '16 at 05:03

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The first thing you need for this question to make sense is a definition of dimension. One convenient and commonly used notion is the Hausdorff dimension. The definiton may be too technical and beyond the level you seem to be asking at, but here it is anyway: the Hausdorff dimension of a metric space (or say, a subset of $\mathbb{R}^n$ for some $n$) is the smallest possible $d > 0$ such that your set can be covered with balls of radius $r_1, r_2, \ldots$, where the sum $\sum_{i} r_i^d$ is finite. (Actually, rather than "smallest possible", it's the infimum, in the case that there is no smallest $d$.)

Under this definition, it turns out that a point has Hausdorff dimension $0$ and a line has Hausdorff dimension $1$. How is this possible? Well, in your question, you seem to be assuming some principle like

The union of any collection of sets of dimension $d$ must have dimension $d$.

Assuming "dimension" = "Hausdorff dimension", this is true for a finite union. It is even true for a countably infinite union. But a line is composed of uncountably many points, and in the case of an uncountable collection the principle above is simply false.

Caleb Stanford
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