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Dimension analysis is a nice tool to create functions using physics dimensions that are desirable for our lives. I know, as an axiom, of sorts to me, that addition and subtraction must conserve units, and thus are actually relatively uncommon in physics equations.

Now, for why that is, I would tell someone it's the same reason why you only can add terms with the same qualities as that term in math as well (such as $x^2$ only being able to be added with scalar multiples of $x^2$, the same applying to $y^5$ or $ e^{-2x}\cosh(3x)$, for example). Except in the case of inputs, we have... (and here is where my explanation gets shady), vector units or magnitudes with magnitude 1 of things such as $\text{meters/second}$, $\text{Newtons}$, $\text{Tesla}$, $\text{Joules/Coulomb}$ etc.

However, I can't seem to necessarily explain why I couldn't say, add $x^2$ to $x$ or a velocity to a force other than my body simply not letting me out of habit, and my only explanation is: "You just can't", or "It'd look bizarre."

I need a better explanation for this. Could someone enlighten me? My two questions:

  • Why does adding and subtraction break down at things with different dimensions?

  • Why does multiplication/division allow for it?

Qmechanic
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sangstar
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1 Answers1

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I'm not sure I really understand what you're getting at but here goes: Say you add 2 metres to 3 kg - what have you got? I'd say you have just got 2 metres and 3 kg, which is not telling you anything. You can't say you have 5 of something, although if the metres and kilograms were apples and oranges, respectively, you could say you had 5 pieces of fruit.

Dr Chuck
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