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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?

Andrei
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Charlie
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  • A multiplicative value that together with a real number defines a negative square..? Or am I misunderstanding the imaginary value-definition – Charlie May 04 '17 at 02:17

2 Answers2

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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.

Chickenmancer
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  • The square we are making, is a mirror of the original, but half. Its x-axis is negative (1/2) same as its y side. It would likewise, have an area that is negative, and (1/2)^2 of the original squares sides. Doesnt this mean I can construct a negative square: (-x/2)^2=-y ? or, (-x/2*-y/2)=-z, where "-" denotes a value being less than 0. Or does it say somewhere that squares cant be negative? Isn't even -1 imaginary? To a certain degree – Charlie May 04 '17 at 02:53
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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 $(-x)(-x)=x^2$.

AspiringMathematician
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  • Both sides of the square I'm subtracting, is negative compared to the original, as it mirrors the former. It will be, relatively, -(1/2) in the x-axis, and -(1/2) in the y-axis. Compared to the original it will be a negative square. Doesnt this suggest you could make a negative square with -x^2? Maybe I'm just silly – Charlie May 04 '17 at 02:46
  • Let me see if I understood it right. You're building a square with side $-\frac{1}{2}$ and area $1$? Or is it a square with side 1 whose side "begins" at $-\frac{1}{2}$ and ends at $\frac{1}{2}$? – AspiringMathematician May 04 '17 at 03:05
  • Again, very informally speaking. – AspiringMathematician May 04 '17 at 03:06
  • also, If I had to defend a stance like the one I'm asking here, I would argue, as reply to you square of "posive integers" part, that 2^2 doesnt actually have to be 2+2, it might as well be -2+-2, and, that by omitting that, you introduce an error to your calculations, or algebraic bag-of-tricks – Charlie May 04 '17 at 03:06
  • Well, your second part is true in the usual system, but then $(-2)+(-2)=2\times (-2)$. – AspiringMathematician May 04 '17 at 03:09
  • the square is -1/2 x times -1/2 y, with the area -25m^2. Im arguing this is easily imaginable. A square with a negative sum. – Charlie May 04 '17 at 03:09
  • if you were to take -2 twice, you would get the same, sure. Am I unwillingingly arguing -2*-2 could be -4 here? I was not intending to do so, I just happened to imagine a negative square, if that makes sense – Charlie May 04 '17 at 03:12
  • I'll edit my answer then to accomodate it (and delete the rest, since you seem to understand the basic concepts I wrote there) – AspiringMathematician May 04 '17 at 03:12
  • Check the new answer. Also, if this addresses what you wanted to ask, I'd like to ask you to be clearer in your question next time. ;) – AspiringMathematician May 04 '17 at 03:48
  • I'll give you a mark for effort, but I dont see the answer. I think the problem comes from how we understand multiplicative values, and confuse them with additive ones. I feel negative squares can exist, just as much as negative integers can. Addition is one thing, ratios another, and even further, squares/roots. What I'm saying, I can certainly imagine negative squares – Charlie May 04 '17 at 09:49
  • The way you're thinking about it is formalized in Analytical Geometry and Linear Algebra, so if you're interested in it you may want to study those subjects. They're undergraduate-level, though. – AspiringMathematician May 05 '17 at 00:35