`std::quantity` as a numeric type

[mpusz]

• identities (dimension_one, dimensionless, one)
• magnitude (ratio on steroids)
• quantity_spec vs units/dimension equations
• quantity_spec conversions and common_quantity_spec
• quantity constructors and conversions (value preserving approach)
• exceptions and special cases for dimensionless quantities
• quantity arithmetics + quirks of modulo arithmetic
• custom representation types (improve safety, provide additional info like measurement, linear algebra, minimal requirements)
• representation type constraints
• vector quantities and their operations
• affine space and operations
• Math CPOs

[JohelEGP]

I understand the structure.

Please, clarify the intent. Is it also a wording paper?

For the non-wording portions, I think we'll be able to reuse a lot of mp-units' documentation, which is in pretty good shape. How do you think they'll differ? Here's what I'm thinking:

• Clearly distinguish domain abstractions (e.g., units) from invented abstractions (e.g., quantity_spec). The former must exist in the design. The latter can be refactored (but see the next points).
• Clearly explain the relation between domain requirements (e.g., quantities of the same kind are comparable) and invented abstractions.
• Link back to sections of the other papers where relevant.

[mpusz]

Is it also a wording paper?

No. It is meant to solve all of the math-related issues and concerns before we provide the first paper for LEWG with interfaces and, eventually, wording in later revisions.

I think we'll be able to reuse a lot of mp-units' documentation,

That's the point :-) I was writing the documentation with that in mind. If not copy-and-paste, documentation still highlights most things we need to describe there. The audience is different, so pure copy-paste may not be enough,

Here's what I'm thinking:

Agree.

[mpusz]

We also need to describe that as long as number * unit to get the value of a quantity is defined in the SI Brochure, quantity */ unit and number / unit is added by us for the engineering convenience.

[JohelEGP]

That's just math. Let $quantity = number unit$, $quantity / unit = number unit / unit = number$. We had removed that possibility in code because [see https://github.com/mpusz/mp-units/pull/262#issuecomment-802493851].

[mpusz]

1. Potentially more expensive when Rep is expensive to copy/move (quantity with a numeric value copied/moved many times before the final unit is obtained)
2. Multiplication is commutative so should m * 42 be a thing? This seems wrong to me.
3. Should 42 / s construct a quantity? If so, how to define a quantity<1 / s> or define hertz with:

inline constexpr struct hertz : named_unit<"Hz", 1 / second, kind_of<isq::frequency>> {} hertz;

It seems that only multiplication after a number has a sense, or we would have to heavily complicate the notation of inverses for units, dimensions, and quantity_specs (i.e. using std::integral_constant).