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In my calculus class, I've learned a lot about finding areas/volumes of various shapes by summing up infinitely small slices of 'something' and adding them all up. This is very interesting to me, but I'm confused about how different dimensions relate with one another. Let me give an example of what I mean:

A 2x2 square has an area of 4. A line has length 4. These quantities are the same, unitless quantities, yet they are fundamentally very different. This elementary notion has become quite strange to me. One could potentially split the 2x2 square into infinitely small pieces (dx by 2) and string them together and get a rectangle which is infinitely long and infinitely thin. Even when made as 'line-like' as possible, the shape seems very different than the line of length 4 examined before.

I suppose my question is how and why do quantities relate between different dimensions and is there any meaning behind these connections? Furthermore, how does one geometrically convert a quantity from one dimension to another? Additionally, do there exist methods of summing infinite series geometrically using this idea?

I'm not sure if this question belongs here, but I'd really like to understand how it all works.

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  • $\begingroup$ The short answer is that these areas and lengths aren't really unitless -- we've just neglected to specify the units. It still doesn't make any more sense to compare them than it does to compare 4 square inches with 4 inches. $\endgroup$ – Rahul Feb 20 '14 at 20:53
  • $\begingroup$ @Rahul What units are implied? Or are they all unitless within their own distinct realms? I'm sorry, I'm just trying to figure it out. $\endgroup$ – Jackson Feb 20 '14 at 21:05
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They have units within distinct realms. The fact that your square and line have the same numeric value for area and length is an artifact of the length unit you have selected. Say the square has a side of 2 inches, so an area of 4 square inches, and the line has a length of 4 inches. If I change my unit to feet, the square has an area of $\frac 1{36}$ square foot and the line has a length of $\frac 13$ foot. It makes no sense to say the square and line are the same size.

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Length and area have different dimensions. Length is a fundamental dimension and $Area=[Length]^2$. A line has zero area. I hope I have made things clearer for you.

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