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Units are standards of measurement used for different types of quantities.

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Is it possible to prove that units can be manipulated algebraically?

I suppose that the best way to argue for this is to consider the units as indicative of the vector space from which the quantities in question originate. … $[m] \times [v]$, which is indicative of that quantity belonging to a different vector space than either of the two original spaces (specified by units $[m]$ and $[v]$). …
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