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Canonicalization is tough, and requires you to define some common ordering on your units. Systems will use an array of unit powers, so that if the array were defined as <Joules, Seconds, Newtons, Meters>, then acceleration would be <0,-2,0,1> and watts would be <1,-1,0,0>. Addition and subtraction require that your arrays are equal, and multiplication and division are pairwise additive/subtractive. |
Just represent every unit in terms of them then it's good.
One problem with the array you came up is that the units are not orthogonal, since Joules = Newtons * Meters.