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To give you the context of this question, before I had a question about why an area of a square equals to its side squared, and how to see it geometrically, because multiplying a length by a length to give you an area is something hard (impossible?) to see.

I've seen some interesting posts here and here. And I had the conclusion that for:

$$Area\_of\_square = side \times side$$

It is in fact:

$$Area\_of\_square = \frac{side}{unit\_length} \times \frac{side}{unit\_length} \times area\_of\_unit\_square$$ Here we just deal with scales of sides and we don't need the fact that $unit\_area = unit\_length \times unit\_length$, or that the area unit $m^2$ is actually the multiplication $m \times m$. And we can see this geometrically, we divide a square by unit squares and count how many we have. So far, for me if we take meters as unit, then $m^2$ is just a unit of area and doesn't have to do with multiplying $m$ by itself.

After this, I had a question that makes me rethink the previous explanations. It is the fact that dividing a unit area by a unit length gives a unit length. $m^2 / m = m$.

Actually we can divide an area by a scale to get another area, or we can divide an area by an area and get the scale required to move from the first to the second. But dividing an area by a length doesn't have any sens for me ! (like comparing an area with a length)

Does this simply require another axiom (unit area / unit length = unit length), or does $m^2$ really have something to do with $m \times m$ ?

Quade
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1 Answers1

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The notation we use for dimensional measures is consistent with the arithmetic rules

$$1 m\times 1m=1 m^2$$

and

$$\frac{1 m^2}{1 m}=1 m.$$

In terms of pure dimensions,

$$[m]\times[m]=[m]^2=[m^2]$$ and $$\frac{[m^2]}{[m]}=\frac{[m]^2}{[m]}=[m]$$

so you can very well say that a square meter is the square of a meter without introducing inconsistencies. Whether this corresponds to some "physical" reality is speculative and need not be answered in operational conditions.

Deeper topics are dimensional analysis https://en.wikipedia.org/wiki/Dimensional_analysis#:~:text=In%20engineering%20and%20science%2C%20dimensional,kilometres%2C%20or%20pounds%20vs and the pi theorem https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem.