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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.

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  • $\begingroup$ I'm not qualified to give you a full answer, but I recommend looking up the Wikipedia pages for "metric spaces" and "measure spaces": en.wikipedia.org/wiki/Measure_(mathematics) en.wikipedia.org/wiki/Metric_space . These spaces have certain mathematical properties that are VERY useful across a wide range of applications, but they only work if length is non-negative. $\endgroup$ – SilasLock May 24 '17 at 3:17
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    $\begingroup$ you could have negative length, if you wish, but you are unlikely to call it length anymore, and you would have a few extra axioms describing its properties (as opposed to length = distance in metric spaces) see e.g. dcs.warwick.ac.uk/pmetric/pub.html $\endgroup$ – Mirko May 24 '17 at 4:23
  • $\begingroup$ The original physical meaning of "length" is an amount (usually a whole number) of copies of a predefined unit you need to cover a given distance. You can have such a negative amount as much as you can have a negative amount of apples. It can make sense in some contexts, e.g. if I go an amount of $x$ units to the right and reach a predefined zero-point, then it makes sense to say that I initially was $-x$ length units to the right of this zero-point. But, as I said, this depends on the context. $\endgroup$ – M. Winter Mar 20 '18 at 22:00
  • $\begingroup$ You can have signed length (of an oriented segment, if it's a subset of the line), just like you can have signed area (of an oriented region, if it's a subset of the plane). You do run into trouble if this segment isn't a subset of a line but of a plane, just like you run into trouble if the region isn't the subset of a plane but the subset of 3-dimensional space. $\endgroup$ – Akiva Weinberger May 23 '18 at 6:29
  • $\begingroup$ You might enjoy taking a look at cds.cern.ch/record/154856/files/pre-27827.pdf?version=1 $\endgroup$ – Barry Cipra Nov 25 '18 at 1:10
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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.

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Well, there are certainly contexts in which it makes sense, such as mentioned in these comments.

As also pointed out in the comments, the concept of measure has a great deal to do with length--if I recall correctly, interval length in many ways gave rise to the idea of measure in the first place, and is the underlying basis for the Borel and Lebesgue measures. Like length, measures are typically taken to be non-negative-valued. However, there is such a thing as a signed measure, where the values may be negative, instead. For example, integrals are often thought of as giving "the area under the curve," but integrals can be negative, which means they give a signed area.

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Most definitions of length between points a and b have the property that the length between a and b is the same as the length between b and a. With this definition we see that you cant simply flip a line segment to get one of negative length from a line segment of positive length.

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Well, it all depends on what we mean by length.

This is a geometric concept; in particular, a concept of measure. If we confine ourselves to segments of straight lines, and if by its length we mean a certain number assigned to it given a set of conditions, then it is obvious that we could do this in many ways, depending on what conditions we impose upon the notion of length.

If we want to be able to (linearly) order all possible lengths, for example, then it is clear that we have to use some sufficient subset of $\mathbf R.$ Also, if we simply want a correspondence between this subset of $\mathbf R$ and the set of all line segments, then it is clear that any open interval of $\mathbf R$ would do. Other conditions which we want our lengths to obey may force further restrictions on the type of open interval to use.

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