Why is length often a unitless quantity in mathematics? This is something that's bothered me for a long time: Why is length often not given any units (e.g. inches or meters) in mathematics? The unit circle is said to have a "radius of 1," with no unit given. A vector is said to have a magnitude of, say, 27, with no unit given. A triangle is said to have a hypotenuse length of, say, 10, again with no unit given. 
Is the reason no unit is given simply because one could substitute any unit, so long as the proportions between quantities remain the same? I searched around for a little on here but couldn't find any questions directly addressing my concern here. I'd appreciate any clarification. 

Distance in a Cartesian coordinate plane is a function which inputs two points $p_1 = (a_1,b_1)$ and $p_2 = (a_2,b_2)$ and outputs their distance $d(p_1,p_2) = \sqrt{(a_1-a_2)^2 + (b_1-b_2)^2}$, which is a real number. So you can think of the unit of distance abstractly as the numeral $1$, which is the multiplicative identity of the real number system. This abstraction is actually useful, because it enforces a focus on mathematical concepts that are independent of the real world units that may occur in applications (e.g. inches or meters).
