Dimension analysis is a super power in applied math. I was first taught it in chemistry, but was able to use it to avoid tripping myself up on just about every real world math problem I did...

(And I too changed an implementation to add units of measure to a system. It took a few days, and caused some grief with downstream system developers. But we also avoided the Mars Climate Orbiter experience in production )

Many quantities that in the current official version of SI appear to have the same dimensions, in fact do not have the same dimensions.

The illusion of having the same dimensions has been caused by a vote of the General Conference on Weights and Measures, which has been exactly as stupid as deciding by vote that 2 + 2 = 5. It is really a shame that something like this has happened.

According to that vote, some quantities have been declared as adimensional, even if arbitrary units of measurements must be chosen for them, exactly like for any other physical quantity and their units enter in the definitions of many other units, so changing any such choice of a unit changes the values of many other SI units, which implies that any version of SI that does not explicitly include the fundamental units chosen for "adimensional" quantities is incomplete.

The system of units of measurement required for the physical quantities includes fundamental units a.k.a. base units, which are chosen arbitrarily, without any constraints, and also derived units, whose values are determined in such a way as to eliminate all constants from the formulae that relate the physical quantity for which the derived unit is determined to other physical quantities for which units have already been determined.

The fundamental units are of 3 kinds, which I shall call arithmetic units, geometric units and dynamic units.

The arithmetic units are for discrete quantities, i.e. quantities that are expressed in integer numbers, so they have natural units.

There are a huge number of such fundamental units for quantities that can be obtained by counting, e.g. the number of atoms of a certain kind, the number of ions of certain kind, the number of molecules of a certain kind, the number of objects of a certain kind, the number of apples, the number of oranges, the number of humans and so on.

While "1" is the natural unit for all such quantities, in physics and chemistry there are many such quantities for which the conventional SI unit is not "1", but the number of Avogadro, a.k.a. one mole of substance.

Besides numbers of various entities, if we neglect nuclear/"elementary" particle physics, there is one more discrete physical quantity, which is the electric charge a.k.a. the quantity of electricity. The natural unit for the electric charge is the elementary electric charge and the conventional SI unit, the coulomb, is defined as an exact multiple of the elementary electric charge.

Besides the units for discrete quantities, there are 6 fundamental units for continuous quantities, 3 geometric units & 3 dynamic units. The 3 geometric units are the units of logarithmic ratio, plane angle and solid angle (which are respectively associated with 1-dimensional, 2-dimensional and 3-dimensional spaces).

The 3 fundamental geometric units are chosen by a mathematical convention, but they are exactly as arbitrary as the fundamental dynamic units. This is proven by the fact that in practice many different choices are used for these 3 fundamental units. Like length can be measured in meters or in feet, logarithmic ratios can be measured in nepers or in decibels, plane angles can be measured in degrees or in right angles or in cycles or in radians, while solid angles can be measured in orthogonal trihedral angles, in whole spheres or in steradians. Changing the choice for any of these fundamental units requires value conversions for all physical quantities that have units derived from them, in the same way like when replacing Imperial units with metric units.

There are many physical quantities in SI that appear to have the same dimensional formulae, but this happens only because the units of logarithmic ratio or plane angle or solid angle are omitted from the formulae. This omission is especially frequent for the unit of plane angle, which enters in a much greater number of dimensional formulae than it is obvious. For instance, the magnetic flux also includes in its dimensional formula the unit of plane angle, though you will almost never see this taken into account.

The SI includes a greater number of conventional fundamental dynamic units than necessary, for historical reasons, which results in a greater number of "universal constants" than for an optimal system of units, which requires only 3 fundamental dynamic units.

While at the end of the 18th century, when the metric system has been designed, it was convenient to choose as fundamental dynamic units the units of length, time and mass, for which distinct physical devices were chosen to materialize the units, the current SI system is equivalent with choosing a single physical device to materialize all 3 fundamental dynamic units, which are the units of length, time and electric voltage (yes, now the unit of mass is derived from the unit of electric voltage, not vice-versa, despite confusing definitions that can be encountered in many texts).

The single physical device that can materialize all 3 fundamental dynamic units is an atomic clock. The electromagnetic wave corresponding to its output signal provides the unit of length in its wavelength, the unit of time in its wave period, while the Josephson voltage corresponding to its frequency provides the unit of electric voltage. The conventional units of SI are obtained from the 3 units provided by an atomic clock by multiplication with conventional exact constants, in order to ensure continuity with the older units of measurement.

A few of the so-called base units of SI are no longer true base units, but they are derived from the true base units by inserting exact conventional constants in the formulae relating physical quantities, e.g. the units of temperature and luminous intensity.

I am using Unitful.jl for some complex electromagnetics code I am developing. Although it makes the definition of types and constructors more complicated, it has helped me find several subtle bugs in the equations. Forcing the type system to keep track of measurement units has pros and cons, but so far, the advantages are more significant than the cons.

Interesting, I am intrigued by the support for partial derivatives with typed dimensions. I'll have to check out OpenFOAM

There are Haskell packages for uncertainty (e.g. [0], based on automatic differentiation [1]). However, these packages don't support typed dimensions, because multiplication/division becomes more complex.

It is my goal to provide quantities with uncertainty AND typed dimensions one day.

[0]: https://hackage.haskell.org/package/uncertain

[1]: https://hackage.haskell.org/package/ad

Yes you can do that (easily) with Wolfram Language (aka Mathematica.)

Here is an example:

    mat1 = Table[
       RandomChoice[{
           Quantity[RandomReal[], "Meters"],
           Quantity[RandomReal[], "Seconds"],
           Quantity[RandomReal[], "Meters/Seconds"],
           Quantity[RandomReal[], "Meters/Seconds^2"]
         }] , 3, 3];
    mat1 // MatrixForm
    
    mat1 . Transpose[mat1]

This is something I always feel is missing from unit libraries.

That's a great question.

Haskell has arrays and matrices of homogeneous types, so it wouldn't work by default.

If you needed this kind of functionality, you would have to create your own Matrix type which would look very similar to a 9-tuple, and then define matrix multiplication. It would then be possible to encode the dimensional constraints in the type signature, but it's already quite involved for 2x2 matrices:

    data Mat2 a b c d 
        = MkMatrix a b c d
    
    matmul :: ???? -- good luck writing this type signature
    matmul (MkMatrix a11 a12 a21 a22) (MkMatrix b11 b12 b21 b22)
        = MkMatrix ((a11 * b11) + (a12 * b21))
                   ((a11 * b12) + (a12 * b22))
                   ((a21 * b11) + (a22 * b21))
                   ((a21 * b12) + (a22 * b22))

Note that Mathematica units impose a very large runtime penalty, making them unsuitable for a lot of applications

The `dimensional` library doesn't provide this, no. However, it's easy to 'tag' data in Haskell using phantom types. For example, for vectors in reference frames:

    newtype Framed f a = MkFramed (Vector a)
    
    add :: Framed f a -> Framed f a -> Framed f a
    add (MkFramed vec1) (MkFramed vec2) = ... add vec1 and vec2
    
    data ReferenceFrame1
    data ReferenceFrame2
    
    v1 :: Framed ReferenceFrame1 Double
    v1 = ...
    
    v2 :: Framed ReferenceFrame2 Double
    v2 = ...
    
    main :: IO ()
    main = print $ v1 `add` v2 -- will not type check
   

Something I briefly mention in the post is pint [0] for Python, but unfortunately, I don't think dimensions can be specified via type annotations.

At least you can check the input of functions at runtime [1].

[0]: https://github.com/hgrecco/pint

[1]: https://pint.readthedocs.io/en/stable/advanced/wrapping.html

Check out sympy.physics.units. I've been using it for unit-checking in my symbolic work for a few years.

https://docs.sympy.org/latest/modules/physics/units/index.ht...

Could this be done with an IDE plugin (for Pycharm or whatnot)? It is tedious to go though code and carefully annotate each variable and run so many compile test, read error, revise, cycles. It should be possible on a per function basis to statically check how each variable relates to the inputs and outputs, and just annotate those. Then the plugin could solve the units for each item, and only ask for an annotation where needed to resolve an ambiguous case.