Arbitrary divisions of angle aren't very interesting, since the radian is so fundamental. But there is an angle system using multiples of 100: gradians. 100 is a quarter-turn and the internal angles of a triangle add up to 200. A lot of electronic calculators still offer them (the DRG button = degrees, radians, gradians).
There are two cases where linear divisions of arclength-measured angles make sense: (1) the angles to make regular polygons of low numbers of sides, (2) repeated binary divisions, whose cartesian coordinates can be computed easily because we can just add two vectors and renormalize using an inverse square root which we know how to efficiently compute, and which are then nice for fixed point representations using binary integers in computers. For the first case, I think rational fractions of a turn work best; for the second case, I like binary angles (“brads”) https://en.wikipedia.org/wiki/Binary_scaling#Binary_angles ; losing the ability to precisely represent a thirds of a full turn is just one of those trade offs you sometimes need to make.
Otherwise, treating angle as a linear quantity with precise measurements (instead of some kind of approximations) is for most problems less useful than just using a cartesian coordinate representation for an angle (possibly keeping track of the coordinates squared if we want to stick to rational arithmetic.)
Radians are only really useful for solving calculus problems by hand, or writing academic math/science papers where it’s important to stick to established conventions. For practical computation radians are almost always an inferior choice, more computationally expensive and less precise (they’re built into lots of existing libraries, so can often be convenient despite the inefficiency).
Well just multiply your fraction with τ = 2π and you get radians. E.g. a whole revolution is τ radians (or 360°), half a revolution is ½τ radians or (180°), etc.
http://www.garywallace.net/wp-content/uploads/NF-SCI-0001-P0...
Each finger has three numbers, starting with the index and ending in the pinky, and you count pointing to the relevant division with your thumb.
I too wish base-12 were used more. Given how much a penny is worth, getting rid of decimal money and substituting a base-12 coin would solve a lot of pizza / dinner cost split problems.
The other day, right next to the problematic awg definition, I discovered the kcmil. Which is a kilo pi/4 Micro square inch.
four score for some things, six score for others, or for onions, a hundred was 15^2
The real reason why the US hasn't switched is that it is really painful to change something like this.
If you are brought up with metric units, the units are built around the number system.
A good way to understand the problem by for the HN audience is the problem of using IEEE floats where precise numbers matter in programming. The problem with IEEE floats is that we usually think about them in base 10 but they represent something in base 2, and no lossless conversion across the domain is possible. American units bridge that sort of gap much better than metric ones do because they usually focus on divisibility than on quick representation and conversion.
There are certain areas where metric makes sense. If I have prices in kg and I want to know how many g I can buy for a certain amount of money, because the money and weight systems coincide we are well optimized for that problem.
But, imagine calculating the angles of a triangle or a hexagon if your degree system was base 10.
In other words, American units are abstracted around actual application.
There is of course a happy medium. We could change to duodecimal numbers and come up with a duodecimal metric system. Then everyone would get what they want, right? ;-)