Basically, it's in the title, but I was just reading my ODE book and the author (Pollard or Tenenbaum) writes (referencing area the $A$ of a square as a function of side length $l$):
… $\, A = l^2$ …
… The relationship between the two variables expressed mathematically by equation (a), is, however, not rigidly correct. It says each value of the length $l$, $A$, the area, is the square of $l$. But what if we let $l = -3$
? The square of $-3$ is $9$; yet no area exists if the side of a square has length less than zero. Hence we must place a restriction on $l$ …
Until reading that I've always thought, yep that's correct. No such thing negative lengths, maybe squares can described by negative coordinates but alas they still have positive side length. But then I thought, wasn't the same thing said about negative numbers? Eventually we realized that negative numbers were useful. And that thinking is not just confined to an abstract representation. We use negative numbers to represent real situations, like debt, liquid levels, temperature (for certain scales), etc. Why would length, or even area, stand apart from negative numbers and real world situations.
Someone might say, "well have you ever seen a stick with negative length?" to which I would retort, "no, but I haven't seen negative 50 dollars either."
I believe then, it would be fine to define length as being negative, depending on context. Unless there's something I'm missing here.
Best Answer
Actually, there are such things called pseudo-scalars, which change their sign according the orientation of the system related to them.
An area can be negative if the line integral is taken in the opposite sense.
A line integral also can be negative if taken on the opposite direction.
A question of convention, finally. Read below again.
Having said that.....
No. A magnitude cannot be negative because it is said to be positive or equal to zero between every points (elements). That is a Metric Space, on its very first rule. This inspires the Norm (metric on norm spaces). Most of those objects require an absolute value $|.|$ to make them positive. That is suspicious don't you think?. Read above back.