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I have been doing highschool physics for about 3 years now and I am perfectly fine with it. However, Something tricked me off few days ago. It is how we treat units. I was kind of hesitant to post this thread because I have seen multiple threads about it before and some member really posted useful links about it. It is a very common one.

I will define a "Unit" from what I understand. Unit is a measurement of physical quantity. We define how much a Kg weighs and we use it in our mathematics. We say we have 5 Kg which means we have something that weighs 5 times bigger than the unit we chose.

We can add similar units together. It makes intuitive sense.

2 Kg + 3 Kg = 5 Kg

we can subtract, divide such as density.

Now if I think about different units.

I cant simply add:

2 apples + 3 oranges = ... (No physical meaning)

"Multiplication is repeated addition" or we can see it is scaling which consists of addition.

So 2 apples * 3 oranges = shouldnt give you 6 apples x oranges.

However division makes sense. For example speed when I say 5m/s

I am just saying that for every sec passes I move 5 m.

and what about area? I define an area of an object to be made out of a square meter for example which is just a square with length of 1 m

So when I find the area of a rectangle with length 5 and width 2 I am simply stating

that I have 5 square meters for 1 width so you have 10 square meters

We reach to the same result as if we multiply units

5 m x 2 m = 10 m^2

I have read books about this and they just simply state that the derived quantity is just the product of the units without giving explanation

A summary of all this is, Why do we treat units as if they were variables?

Hopefully, Someone can set me straight because it is really irritating that I am not able to figure this out and it is kind of embarrassing :/