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As I was studying trigonometric functions, I had to use the Pythagorean Theorem to find the length of the hypotenuse in order to solve one of the functions. This got me thinking though: normally when you take a square root of the number, you have to take the positive and negative square root, but lengths of objects, such as the side of a triangle, you only take the positive square root, because a length cannot be negative. But then again, can it?

I learned this theorem back in middle school, and I just took that rule at face value all these years because it made perfect sense; an object can't have a negative length. Now it seems almost counter-intuitive to think that negative length is a possibility, but then again, so was the notion that you could have the square root of a negative number or even the square root of two.

So my question is this: is there a specific reason that a length cannot be negative, besides it just making sense? Is there a proof?

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  • $\begingroup$ what in the devil is the hypotenuse of a function? $\endgroup$ – Andres Mejia Dec 8 '17 at 2:45
  • $\begingroup$ @AndresMejia I meant the hypotenuse of the triangle so as to solve the sine function, I'll clarify. $\endgroup$ – Tawiskaru Dec 8 '17 at 2:46
  • $\begingroup$ I also mean the following to cause no offense: you're asking a question about something I'm afraid you don't understand. What is your definition of length? If you do not have one, you should try to formulate one, look it up, or ask it as a separate question. Otherwise, nobody can give a truly mathematical answer to a question that is not yet mathematical. $\endgroup$ – Andres Mejia Dec 8 '17 at 2:46
  • $\begingroup$ Here is one reasonable starting place: two points have a vector between them. We usually take the dot product to be $|a| \cdot|b| \cos(\theta)$. Apply this to the case of $a=b$, and you will see that it is nonnegative. In fact this is the definition of length in the context of linear algebra. Otherwise, you could take paths $\gamma(t):[0,1] \to \mathbb R^2$ and consider the $\inf \int_0^1 \sqrt{ y^{\prime}(t)^2+x^{\prime}(t)^2} dt$, to be a distance, where it is taken over all possible paths. You might take the vector from $a,b$ and identify its span with $\mathbb R$ and define lebsegue $\endgroup$ – Andres Mejia Dec 8 '17 at 2:53
  • $\begingroup$ cont.d measure and take that to be length. You could look up "metric spaces' in which there is also a notion of distance (and in fact my integral characterization kind of presupposes a euclidian metric.) There are many notions of length, and you should give one, before you ask "can it be negative?" Most definitions really go out of their way to show that length is not negative, but there are so many ways to go mathematically from the word "length" $\endgroup$ – Andres Mejia Dec 8 '17 at 2:55
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To be honest I don't understand why there are so many down votes. This is a deep question and doesn't have an easy answer.

In very general terms what you are asking about is metric spaces. A metric space is a certain object we study in mathematics that allows us to define explicitly some notion of length. You'll find that there are in fact many different notions of length you can choose for certain metric spaces and each have their own interesting properties.

The Pythagorean theorem gives us one kind of metric. This is called the standard Euclidean Metric $a^2+b^2=c^2$, or $c = \sqrt{a^2+b^2}$. One of your questions was why don't we take the negative root. The reason is because mathematically $\sqrt{x}$ is defined to be the positive root, if we allowed for both the positive and negative roots to be used then $\sqrt{x}$ would cease to be a function. That is it would fail the vertical line test (think of a sideways parabola).

Now more generally we want a metric space. We can define what a metric space is by the following:

A metric space is a set $M$ (like for instance the set of numbers on the plane $(x,y)\in\mathbb{R}^2)$ on which we can apply a function $d$ which is said to be a distance function between two points in your set $M$. This function $d$ must satisfy certain properties to be considered a metric these are:

  • $d(x,y) \geq 0$
  • $d(x,y)=0$ if and only if $x=y$
  • $d(x,y)=d(y,x)$
  • $d(x,z) = d(x,y) + d(y,z)$

If your set $M$ satisfies all of these properties then it is called a metric space. In particular notice the first one, here we explicitly define the metric by the fact that the length between two points in your set can't be negative.

But the above definition is a little notation heavy so let's use the example from before on the set $M=\mathbb{R}^2$ with the Euclidean norm. Let us check that this satisfies all of the conditions required to be a metric space.

First let $(a,b),(c,d)\in\mathbb{R}^2$ be two points in $M$ Then the way we define the Euclidean metric is $d((a,b),(c,d)) = \sqrt{(a-c)^2 + (b-d)^2}$. Because we always take the positive root, and we know that any number squared is non-negative we know that $d((a,b),(c,d))\geq 0$.

If $d((a,b),(c,d))=0$ then we must have that $\sqrt{(a-c)^2+(b-d)^2}=0$. So it follows that $a=c$, and $b=d$ which is enough to say that $(a,b)=(c,d)$.

The remaining conditions follow in the same way. Now I elluded to the fact that the Euclidean metric was not the only one we could have chosen. In fact any function that satisfied the conditions above will be a metric on $M$. One such metric is called the "Taxicab Metric". See if you can play around with the definition and come up with your own metric!

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