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September 17th, 2017, 09:57 PM  #1 
Member Joined: Nov 2015 From: USA Posts: 96 Thanks: 6  Unitless, but not dimensionless. What do I call it and has been studied?
Okay, so I've been looking at things, and figuring out how to measure geometry independent of units. For example, the length of a side of a triangle measured as a percentage or ratio of the perimeter rather than each side measured in some sort of abstract unit of measure. Another example is measure the angles of a triangle as ratios/percentages of the total angle of the three angles in a triangle. I was trying to see if I could find articles or info related to this, but I haven't found any. I figured it would likely be called unitless, but any searches for that comes up as dimensionless numbers (even had one result go "What is unitless numbers? Answer: Dimensionless numbers are..." Kinda funny isn't it.) So does anyone know what this is called, or if it is even studied at all? I figured it would be studied cause the potential implications in trig and geometry, but I haven't figured out what else it might called. 
September 17th, 2017, 10:19 PM  #2 
Senior Member Joined: Aug 2012 Posts: 1,661 Thanks: 427 
I must not be understanding because what you want is standard math. The unit interval has length 1. There's no unit. A rectangle with sides 3 and 5 has area 15. No units. Likewise angles. Everything's in terms of the portion of the unit circle swept out by a given angle. If you go 1/8 of the way around we call it pi/4. We take everything as a fraction of 2pi rather than pi. Cue the tau versus pi "controversy" but it's only a linear scaling factor and makes no difference. But in math nothing has units and angles are in fact fractions of 2pi. Triangles come in all shapes and sizes so I'm not sure what you mean by defining the length in terms of the perimeter. If it's an equilateral triangle the side is always 1/3 of the perimeter. Last edited by Maschke; September 17th, 2017 at 10:22 PM. 
September 18th, 2017, 02:42 AM  #3 
Member Joined: Nov 2015 From: USA Posts: 96 Thanks: 6 
On the contrary, a rectangle with sides of 3 and 5 are units, 3 units long on one side and 5 units long on the other and 15 square units of area. those are units. How do you get that a side is length three? You get that by having some external reference length, and comparing the side of the rectangle to that external reference and finding that the side of the rectangle is 3 times longer than that external reference, even when the external reference is implied (such as by simply telling you the final result of the side being length 3 without explicitly mentioning the reference length itself). However, if you describe the rectangle purely in terms of ratios of different lengths of the rectangle itself, that is a different matter. For example, a rectangle with long sides that are .3125 of the total perimeter, that is not using an external reference for unit length. It is a ratio of one aspect of the rectangle compared to a different aspect of the same rectangle. For example, any triangle can be describe by its three sides being a percentage of the total perimeter and the three angles being a total percentage of the total angles. Last edited by skipjack; September 18th, 2017 at 02:54 PM. 
September 18th, 2017, 02:27 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,398 Thanks: 546 
Your point as far as the sides are concerned is well taken. Angles start out as unit less, so what you are suggesting is pointless, especially since for all triangles the sum of the angles is the same.

September 18th, 2017, 03:38 PM  #5  
Senior Member Joined: Feb 2010 Posts: 633 Thanks: 104  Quote:
For a triangle with sides 3, 4, 5, you are proposing that the sides are $\displaystyle 0.25,$ $\displaystyle 0.333\ldots,$ $\displaystyle 0.41666\ldots$ These are called dilations. In the first case you multiplied all lengths by 1/10. In the second case you multiplied all lengths by 1/12. A high school student would say that you are making similar figures with a constant of proportionality equal to 1/10 or 1/12.  
September 18th, 2017, 03:53 PM  #6 
Senior Member Joined: Jun 2015 From: England Posts: 704 Thanks: 202 
I suppose there are things that are specified by number alone, without units. For instance Avogadro's number is just that  a particular number. I don't know whether you would accept (for geometry) that the number of sides (or vertices) in a hexagon is just six, without units? 
September 18th, 2017, 10:09 PM  #7 
Member Joined: Nov 2015 From: USA Posts: 96 Thanks: 6 
I didn't multiply anything, I simply describe such things by proportion to universal aspects of geometry. Any polygon has a perimeter, thus perimeter is a universal constant of polygons. A polygon can have it's sides all described by the proportion of that side compared to the total perimeter of the polygon. This describes shape but without scaling to external factors. Angles could be described as a proportion of one full turn, since polygons would have different totals for total angle, or as a proprtion of the largest possible angle (assuming you always measure on the smaller side, such as the interior angles of convex polygons, but for concave it might be weird measure some angles on the interior while others are measure on the exterior), but it works simply as measuring each angle as a proportion of the total angle of all vertices of the polygon. The point is, is that a number is just a ratio between a measure and a reference. The methods I describe use only references to the geometry itself. If you were to describe two polygons as I describe, and they were "similar" (as in like congruent except different sizes), then theg would be described in my method as being identical. Then in comparing two such similar polygons that differ only in size, you can describe one, then simply give a scaling value for how much larger/smaller the other is in comparison. It also has the advantage of allowing you to remove scale from any functions working on polygons, perhaps to simplify them, then if you really need scale, you can add it to the final result. I asked about this because I actually quite curious how removing scale from triangles might affect trigonometry formulas, and perhaps to see if any of the other triangle ratios that are not normally used would become useful when scale is removed. 
September 18th, 2017, 10:16 PM  #8 
Senior Member Joined: Aug 2012 Posts: 1,661 Thanks: 427 
Aren't you just talking about similar polygons? I'm assuming all your polygons are regular. So yes you can identify the polygon by the ratio of its perimeter to its side. All similar polygons will have that same ratio. Those are similar polygons. They have the same shape but differ from one another by a scaling factor. What next? What does that mean? It wouldn't affect trigonometry. If a right triangle (but now that's not regular so I'm not sure how this fits your idea) is 11sqrt(2) then the cosine of one of the acute angles is 1/sqrt(2). And if the triangle is 222sqrt(2) the ratio is still 1/sqrt(2). The trig functions themselves are defined in terms of ratios so scaling the triangle doesn't change the function. 
September 19th, 2017, 02:50 AM  #9 
Member Joined: Nov 2015 From: USA Posts: 96 Thanks: 6 
If you are using ratios, you wouldn't have cos(degrees/radians). Also, trig isn't always used on right triangles. A right triangle can be defined as a triangle with two sides perpendicular to each other, and perpendiculer can defined as when two lines cross, or extended till they cross, then the interception point results in four equal angles. There are many more ratios in a triangle than the 6 often used in trig. Also, without scale you might find it useful to use other polygons and not just triangles. Also, yes this gives unscaled geometric descriptions, which may or may not be useful in a number of ways. 
September 19th, 2017, 04:31 AM  #10  
Senior Member Joined: Feb 2010 Posts: 633 Thanks: 104  Quote:
I will however accept $\displaystyle 0.31+0.0025$.  

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