.. namespace:: units Magnitudes ========== The ratio of two Units of the same Dimension---say, `inches` and `centi`---is some constant number, which is known at compile time. It's a positive real number---a _Magnitude_. We also use Magnitudes for _Dimensionless_ Units. `percent` has a Magnitude of $1/100$, and `dozen` would have a Magnitude of $12$. Interestingly, it turns out that the usual numeric types are not up to this task. We need a Magnitude representation that can do everything Units can do. This means, among other things: 1. We need _exact_ symbolic computation of the core operations of Quantity Calculus (i.e., products and rational powers). 2. We must support _irrational_ Magnitudes, because they frequently occur in practice (e.g., consider the ratio between `degrees` and `radians`). 3. We should _avoid overflow_ wherever possible (note that `std::intmax_t` can't even handle certain simple SI prefixes, such as `yotta`, representing $10^{24}$). Integers' inadequacies are clear enough, but even floating point falls short. Imagine if we implemented all angular units in terms of `radians`: then both `degrees` and `revolutions` pick up a factor of $\pi$. The arithmetic with _its floating point representation_ is unlikely to cancel _exactly_. Another common alternative choice is `std::ratio`, but this fails the first requirement: rational numbers are ([rather infamously](https://hsm.stackexchange.com/a/7)!) _not_ closed under rational powers. The only viable solution we have yet encountered is the _vector space representation_. The implementation is fascinating---but, for purposes of this present page, it's also a distraction. _Here,_ we're more focused on how to _use_ these Magnitudes. One type per Magnitude, one value per type ------------------------------------------ Each typical numeric type (`double`, `int64_t`, ...) can represent a wide variety of values: the more, the better. However, Magnitudes are **not** like that. Instead, they comprise a _variety_ of types, and each type can hold only _one_ value. .. tip:: A given Magnitude represents the _same_ number, whether you use it as a _type_, or as a _value_ (i.e., an _instance_ of that type). Use whichever is more convenient. (In practice, this is usually the _value_: especially for end users rather than library developers.) If these types can only represent one value each, why would we bother to instantiate them? Because _values are easier to use_. - `mag()` gives the Magnitude value corresponding to any integer `N`. - You can combine values in the usual way using `*`, `/`, `==`, and `!=`, as well as `pow(m)` and `root(m)` for any Magnitude value `m`. Traits: integers and rational Magnitudes ---------------------------------------- If you have a Magnitude instance `m`, we provide traits to help you reason about integers and rational numbers, or manipulate integer or rational parts. - `is_integral(m)`: indicates whether `m` represents an _integral_ Magnitude. - `is_rational(m)`: indicates whether `m` represents a _rational_ Magnitude. The above traits indicate what kind of Magnitude we already have. The next traits _manipulate_ a Magnitude, letting us break it apart into _constituent_ Magnitudes which may be more meaningful. (For example, imagine going from `inches` to `feet`. Naively, we might multiply by the floating point representation of `1.0 / 12.0`. However, if we broke this apart into separate numerator and denominator, it would let us simply _divide by 12_, yielding **exact** results for inputs that happen to be multiples of 12.) - `numerator(m)` (value): a Magnitude representing the "numerator", i.e., the largest integer which divides `m`, without turning any of its base powers' exponents negative (or making any previously-negative exponents _more_ negative). - `denominator(m)` (value): the "numerator" of the _inverse_ of `m`. These traits interact as one would hope. For example, `is_rational(m)` is exactly equivalent to `m == numerator(m) / denominator(m)`. Why these particular definitions? Because they are equivalent to the numerator and denominator when we have a rational number, and they are compatible with how humans write numbers when we don't. Example: - $m1 = \frac{27 \pi^2}{25}$. Then `numerator(m1) == mag<27>()`, and `denominator(m1) == mag<25>()`. - $m2 = \sqrt{m1}$. Then `numerator(m2) == mag<3>()`, and `denominator(m2) == mag<5>()`. Note that this is consistent with how humans would typically write `m2`, as $\frac{3\sqrt{3} \pi}{5}$. Getting values out ------------------ Magnitude types represent numbers in non-numeric types. They've got some amazing strengths (exact rational powers!), and some significant weaknesses (no support for basic addition!). So what if you just want to turn a Magnitude `m` into a traditional numeric type `T`? You call `get_value(m)`. This does what it looks like it does, and it does it at compile time. Any intermediate computations take place in the "widest type in category"---`long double` for floating point, and `std::intmax_t` or `std::uintmax_t` for signed or unsigned integers---before ultimately being cast back to the target type. For `T = float`, say, this means we get all the precision we'd have with something like `long double`, but without any speed penalty at runtime! `get_value(m)` also has the protections you would hope: for example, if `T` is an integral type, we require `is_integral(m)`. How to use Magnitudes --------------------- - First, start with your basic inputs: this will typically be `mag()` for any integer `N`, or the built-in Magnitude constant `pi`. (Again, these are all _values_, not types.) - Next, combine and manipulate these using the various "Magnitude math" operations, all of which are **exact**: `*`, `/`, `pow`, `root`, `numerator()`, `denominator()`. - If you need to translate a Magnitude `m` to a "real" numeric type `T`, call `get_value(m)`.