Why isnt this value imaginary? Or is it? Lets suppose we have a square, 50m², and another square, 25m². Subtracting the second from the first we get:
$$50\textrm{m}^2 - 25\textrm{m}^2 = 25\textrm{m}^2$$
something that makes absolute sense, so why cant we say that 5m could make up a negative square? I.e. $-25\textrm{m}^2$, as in $50\textrm{m}^2$ and ($-25\textrm{m}^2$)
Obviously two negative "real" numbers cant be multiplied to be a negative number, but in practice, we can have negative squares, is $-25\textrm{m}^2$ only imaginary? Since both its sides would be negative compared to a normal square.
Maybe its a silly question, but why cant negative squares exist, when they are just as "real" as negative integers, i.e. $50+(-25)$, or even $-1$, a value we can only assume to be real (as it would make no sense to take a negative amount from say a stack of well anything).
edit: Is this really a bad question?
 A: Ultimately the sign on the area isn't indicating an area. The sign on $-25m^2$ is indicative of a procedure where you are removing area. 
On the other hand there is a notion of "net signed area." This is comparing area above or below a separating line by attaching a sign to one or the other. 
So negative are isn't imaginary, so to speak, it is merely an assignment of measure. 
A: I'll try to give several points of view as to why constructing a square with "negative sides" (that is, a square with negative coordinates) and assigning it a negative area isn't a good idea.
And then restore your faith in your system, because someone has already thought up something similar.

From a number theory point of view, we're essentially answering the following question:
Why can't we have $(-x)(-x)=-x^2$?
The answer can range from very simple to very complicated. To begin with, we should generalize the square to a rectangle. So our question becomes:
Why can't we have $(-x)(-y)=-xy$?
I think you can begin to see the problems here. First of all, you no longer have a "number $1$", which you could multiply with any number and give that number. Because $(-1)\times1=-1 \neq1$. So multiplication between the natural numbers can't be easily extended to the negative numbers.
You also have problems when solving equations. The equation $(-2)x=-4$ have two solutions, $x=2$ and $x=-2$. This might not seem important, but this means that the function $f(x)=(-2)x$ don't have an inverse anymore, but $g(x)=2x$ has.
And I won't even mention how problematic the Pythagorean Theorem becomes.

From a geometric point of view, you create a problem with reference systems. 
The negative area is dependent on the origin. But you could put the origin in the corner of a room, while I put it in the center of the Earth. Thus, you might have a negative area, while I have a positive one. Imagine what happens when trying to sum the areas.
Also, the sign of the area becomes dependent on the direction you measure its length.

That said, not all hope is lost in your system: There is actually a way to have your square with "negative area" using vectors. 
But they rely on entirely different concepts and rigorous proofs (for example, by agreeing upon a direction of measure and position of origin) than just making up a new multiplication rule for all numbers.
And yes, it still respects the old rule $(-x)(-x)=x^2$.
