What is the algebraic difference between arithmetic operations, that prevents entities with different units from being summed or subtracted, but allows them to be multiplied or divided?

This looks more like a question for Physics, but lengths and areas, for example, are in the domain of pure mathematic.

Now, I cannot sum or subtract an area and a length, but I can multiply and divide an area with a length!

Reading Wikipedia, it looks like this is a property of the dimensions set. Does it just depend on the definition of the dimensions, or is it something intrinsic in the operations of add, subtract, multiply and divide?

Please explain with simple words, if possible.

**Answer**

Apples and oranges are actually a rather bad example. The reason why it doesn’t make sense to *add* quantities with different dimensions, but it *does* make sense to multiply (or divide) them is *scale invariance*.

Let U be the unit of some quantity u, and V be the unit of another quantity v. Now say we change the *scale* of U, i.e. we instead use a different unit U’ such that 1U=10U′. For V we do the same, only that there we choose V′ such that 1V=5V′. If we compute the sum s of u and v in units U,V we get s=u+v

If, instead, we compute the sum in units U′ and V′, however, we get s′=10⋅u+5⋅v

Note that s and s′ *don’t* just differ by a factor, i.e. we can’t convert s from unit U+V to s′ in unit U′+V′ without knowing the *original* values of u and v.

Compare this to the situation of a product. If we compute the product p of u and v in units U and V, we get p=u⋅v

If, instead, we compute it in units U′ and V′, we get p′=(10⋅u)⋅(5⋅v)=50⋅p.

So p′ is simply p, expressed in a different *unit* P’, with 1P=1UV=50P′=1U′V′.

So why do you *want* scale invariance? We want that, because the scale of physical units is usually completely arbitrary. There’s nothing fundamental about 1 meter, or 1 inch, or 1 Volt – we just picked *some* reference value. But since the reference value is *arbitrary*, the actual physics must not change if we replace it by a different one. Which it doesn’t, so long as we only *multiply* and *divide*, but not *add* or *subtract* values with different units, as the example above shows.

And this is also why apples and oranges are a bad example. We *don’t* expect scale invariance for these, because apples and oranges are *discrete* objects, so there’s a canonical definition of what “1 apple” means. So adding apples and oranges makes perfect sense, and we may e.g. assign the result the unit *fruits*.

**Attribution***Source : Link , Question Author : danza , Answer Author : BlueRaja – Danny Pflughoeft*