Mathematically Precise Definition of Unit

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Does anyone know exactly what kind of mathematical object a unit (like meters, coulombs, etc.) is? Or what kind of algebraic structure units are elements of?
 
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Units of measurement is not a mathematical quantity. We use them to calibrate our experiments. I guess you could try to create a structure for them.

"2 candlesticks" times "3 cabdrivers" = 6 [candlestick][cabdriver].

That has the same structure as

(2x) (3y) = 6xy,

and we can add and subtract if the units are the same: 3 cadlesticks + 2 candlesticks = 5 candlesticks is the same thing 3x+2x = 5x.

So polynomials. You could let "l" be length and then metre would be "lm" and inch would be "li" and define li = 0.0254 lm.

Edit: I hate the fact they spell it "meter" in the US.
 
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"Dimensional Analysis" is a branch of applied mathematics (or is it properly called physics?). It is based on the idea that the units in equations that describe physical laws must work out properly, but I don't know if it gives a formal definition of such units.
 

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