# How would one make units work with a variable of unknown magnitude?

So, right now, I am working on setting various quantities up as functions of R. R will be varying later on, so right now I just have it set to the variable radius. It's breaking my dimensional analysis later on in the notebook though, because Mathematica assumes radius has units, when it should just be a scalar. Is there a better way to say the following so that I can have dimensional analysis work in the meanwhile?

R = Quantity[radius, "Meters"];


This is the output:

UnitSimplify[
"Seconds")^2 ("Teslas")^2)/("Kilograms")])]


It should just be in Joules! But, when I set R to a quantity, like so:

R = Quantity[1, "Meters"];


I get such a pretty result:

{7.30945*10^-12 J}


I think I need to strip the units off radius. I tried Quantity Magnitude, but Mathematica wasn't a fan.

• Probably if you edit to include a minimum working example of your code and data in formatted form we could see how its breaking a suggest a solution. Otherwise we can only guess. Oct 14, 2015 at 14:47
• MapAt[UnitSimplify, expr, -1] instead of UnitSimplify[expr]. Oct 14, 2015 at 15:21
• I am confused. In your sample expression R is never used, so its value cannot matter to the final result. What expression are you actually evaluating when you obtain the numerical result you show? Oct 14, 2015 at 21:22
• a = Solve[lorentz*m0*tempv/(q*B) == R, lorentz][[2]] T = ((lorentz /. a) - 1)*m0*c^2 // N Oct 16, 2015 at 14:44
• Perhaps you can make use of the available dimensionless unit: Quantity[radius, "DimensionlessUnit"]? Mar 18, 2016 at 16:57

General Remarks

UnitSimplify is expecting an expression with head Quantity, and the factor before the quantity prevents the evaluation. We have for instance:

UnitConvert[a Quantity[2, "Inches"]]
(* UnitConvert[a (Quantity[2, "Inches"])] *)

UnitConvert[Quantity[2, "Inches"]]
(* Quantity[127/2500, "Meters"] *)


Note that the expression a Quantity[2, "Inches"] does not evaluate itself to a Quantity object:

a Quantity[2, "Inches"]
(* a (Quantity[2, "Inches"]) *)


because a is not NumericQ. On the contrary, this returns a Quantity:

(1 + Sqrt[2]) Quantity[2, "Inches"]
(* Quantity[2 (1 + Sqrt[2]), "Inches"] *)


Some spelunking (<<GeneralUtilities; PrintDefinitions[Quantity]) shows the definition yielding this behavior. It starts with

Quantity /: Condition[HoldPattern[Times][...]


OP's question

Back to OP's question, a possible approach is to make the non-NumericQ factor of the expression enter the magnitude of the quantity. This can be done as follows:

makeQuantity[HoldPattern[Times[fact_, q_Quantity ?QuantityQ]]] :=
Quantity[fact QuantityMagnitude[q], QuantityUnit[q]]


Taking OP's expression

expr = Sqrt[1 - (1.408969*10^6 radius^2)/(1.380123*10^7 + 1.408969*10^6 radius^2)] *
Quantity[
(("Amperes")^2 ("Meters")^2 ("Seconds")^2 ("Teslas")^2)/("Kilograms")
]


we get the expected simplification after processing:

makeQuantity[expr]


UnitSimplify[%]


This returns the mentioned result for the special case radius = 1:

% /. radius -> 1
(* Quantity[7.30944*10^-12, "Joules"] *)


The pattern I used for the definition of makeQuantity` is a simple one, it may need to be enhanced depending on what the use is.