Dimensionless quantities

Question

I must admit that I don't think this problem is really new. However, I haven't found a post dealing exactly with this aspect of the issue.

I am doing calculation using Quantities, but I have problems dealing with dimensionless quantities. Suppose you define:

h = Quantity[b, "m"]

Now I would like to multiply h by a dimensionless parameter a, so to obtain the equivalent of the expression:

Quantity[a b, "m"]

Simply multiplying by a doesn't work (as you can check by trying to evaluate QuantitymMagnitude[a h]); this is understandable, since a is just an undefined symbol and as such has "unknown" units. I though, however, that multiplying by Quantity[a,"DimensionlessUnit"] would do the trick. Nope (same check).

The only solution I have found is to use

Quantity[a QuantityMagnitude[h], QuantityUnit[h]]

Rather involved... any idea or consideration in regard to this? Is there a profound reason I am not seeing for not supporting dimensionless symbols?

Answer 1

You could try something like the following:

(A) Create a replacement Rule:

rule=a_*b_Quantity:>  MapAt[#*a&,b,1]

and then apply this rule:

a*Quantity[b, "Meters"] /. rule
(* Quantity[a b, "Meters"] *)

or (B) live more dangerously and Unprotect Quantity. After this you can use TagSetDelayed:

Unprotect[Quantity];
Quantity /: Times[a : Except[_Quantity], b_Quantity] := MapAt[#*a &, b, 1]

and now you don't need the rule anymore.

(1/c) Quantity[b, "Volts"]
(* Quantity[b/c, "Volts"] *)

What are the risks of Unprotecting Quantity? I am not sure, but if you start getting strange results/errors after doing this you likely know the culprit.