Why is length often a unitless quantity in mathematics?

Question

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.

Answer 1

Mathematics itself is self-sufficient without units. The “objects” of mathematics are complete abstractions that are rigorously regulated by sets of agreed-upon axioms from which theorems can be proven. As far as mathematics goes, strictly speaking, that’s all.

When mathematics (or, rather, some mathematical theory) is used to model a physical theory, one establishes a one-to-one correspondence between some of the abstract objects involved and so-called “physically observable quantities”, which can be measured with physical instruments that return a specific “value”: an element of $\mathbb R$. Each observable has its own “associated unit”, and units should follow “the same laws” as observables, in the sense that if an object corresponding to an observable is expressed within the mathematical theory as the “multiplication” (however defined) of two other objects corresponding to observables, then the units of the former should be the product of the units of the latter two, and so on. Of course, units come in systems that are completely organized according to these relationships between observables; if we gloss over natural units, no system is a priori privileged over another, meaning that slightly modifying the reference value for some unit in a given system leads to a different system that is just as valid as the first one.

So it makes no sense to ask what the units of $x\in \mathbb R$ are, independently of a physical model and a system of units. It does make sense to ask what its units are (within some system) if $x$ is interpreted as the outcome of the measurement of a certain physical observable, say the distance between two physical objects. If the physical model is formulated in such a way that the two objects are abstracted as, say, points in the plane $p,q\in\mathbb R^2$, then $x$ acquires a precise mathematical meaning as a third object in the same mathematical theory, namely $d(p,q)$, the standard Euclidean distance between the two points.

One could “make the stretch” and say that $d(p,q)$ itself is measured in (unit of length), as is frequently done in this situation, but we should keep in mind that the abstract object $d(p,q)$ exists a priori, independently of the correspondence we drew with some experimental outcome. (This is the reason why people working on mathematical models often don’t bother with units unless they are describing a specific application.) In another mathematical model, describing a different physical situation, that distance could have the physical meaning of a value of energy, time...

Answer 2

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).

Answer 3

Ideally, you should be using units and most practical-based questions do in fact have units assigned to the variables.

But yes, when doing geometry or calculus the units are ignored because it is not required and any unit will suffice. If its really bothering you, you may just add $\text{units}$ to the end of the length, $\text{units}^2$ to the end of an area quantity and so on. I do this all the time and no teacher will mind.