# Why can’t you add apples and oranges, but you can multiply and divide them?

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.

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
If, instead, we compute the sum in units $U'$ and $V'$, however, we get
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
If, instead, we compute it in units $U'$ and $V'$, we get
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.