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I was thinking about the physical units (g, m, s, K, J, A, V, etc). I know we got a thing in physics called dimensional analysis, so you analyse some variable that you know it depends on other variables and the two sides of the "equation" (I mean a equation with the proportional sign instead of equal sign) have to have the same "dimension" (the same units).

We also use to talk: "The dimension of velocity is the dimension of space over the dimension of time.".

Why do we call "dimension" the units of physics? Has this term "dimension" the same meaning of when we talk that time is the fourth dimension in space-time, of when we talk that a equation like "x + y + z = 0" describes a plane?

What's the correct way to imagine the dimensions of the physical units in a graph? What about a complicated concept that has lots of base units like electric current (measured in A), or even a simple one (Joule = kg * m^2 / s^2)? How should I imagine this complicated combination of dimensions? Does it has really intuitive and physical meaning, or is a pure result of the mathematics?

Other questions...

I know how to understand a unit like Watt: Watt is equals to Joules per second, so if we have a resistor that dissipates 2W of power we can talk that this resistor converts 2 Joules of electrical energy into thermal energy (heat) in one second.

But what about units like N*s (unit of impulse), or Joules (N*m)? How do I understand that? I have a understanding of these units but I don't know if it's right so check it please: I understand that if we have 5J or 5N*m I have two values encapsulated in this number: we of course don't know what are the values separately but we know that they multiplied is 5, and I can also conclude that we have two physical concepts encapsulated on unit Joule (force in N and space in m), that's right? Am I thinking right?

Any contributions are welcome... Thanks