Are there negative triangles? Using the negative root in the Pythagorean theorem

[Asympotically_fat]

No it is correct that the square root function denoted by f(x) = √x takes in the value of the positive square root when x is real (and more generally the pirncipal value for complex x). The square root function is single-valued because you want it to be a function from R to R (or Z to Z) rather than binary relation or a function from R (or Z) to some subset of its power set…

[LSLGuy]

Thank you. Clarity at last and from a highly reliable source.

[DPRK]

I trust the source, of course.

But, while we all know that every non-negative real number has a unique nonnegative square root, it is an unavoidable fact that the square-root function has a branch point at 0. So while

@Asympotically_fat

The square root function is single-valued because you want it to be a function from R to R (or Z to Z) rather than binary relation or a function from R (or Z) to some subset of its power set…

maybe we want that, but, unfortunately, this is already impossible since negative numbers have no real square roots. So we can forget about a square root function R →R ; usually when lengths come up we take the square root of the absolute value of something, which works fine on the other hand.

@LSLGuy

@Chronos

A function must return a single output, yes. But there are functions for which the output is a single set, containing multiple elements.

My take is that the former position represents a dogmatic assertion that the simplified understanding taught in high school is the truth, not a truth only true subject to certain other often unstated conditions.

I just spent 20 minutes hunting for older threads on the “lies to children” issue in pedagogy. I know we’ve had a couple good ones but I can’t find them.

What are the lies to children in this situation? In any case, if you want to define a square root function (no absolute value) around 0, as we have seen you need to introduce complex numbers, and the values of the function will not be complex numbers but rather lie on a certain Riemann surface:
https://upload.wikimedia.org/wikipedia/commons/9/9c/Riemann_sqrt.svg
so in that sense it is indeed multi-valued (most numbers have two square roots, right?); if you want to fix that you have to make a branch cut.

[Asympotically_fat]

Yes that’s a silly mistake it should be from R* to R, but everything else still stands. The square root function is defined to be single-valued as you want it to be a real or complex valued function. Multivalued functions are not functions from Z to Z. Of course there is a related multivalued function that you can make a branch cut in order to get the square root function, but the square root function denoted by the radical sign means the single valued function, unless there is additional context.

[Asympotically_fat]

And sorry also C to C as Z is the integers.

[DPRK]

Sure. My only nitpick is that if you are talking about real numbers, z ↦√z
is not a function ℝ^× → ℝ, I mean, what would you take to be √−1 ? But √|z| is pretty clear.

[naita]

@ZosterSandstorm

That is why it is defined in reference to the √ function and not to all possible roots in arbitrary domains.

There is no square root function in the Pythagorean theorem.

[Crane]

Agreed, negative areas are common and useful in analog electronics. Of course we are bound to the trivial Euclidian surface, Any engineering troglodyte worth his salt simply sticks in minus signs where needed.

Perhaps of interest: a common algorithm for integer square root is subtraction of successive odd numbers. If instead you subtract successive even numbers you get the corresponding negative root.

[Andy_L]

@Crane

Perhaps of interest: a common algorithm for integer square root is subtraction of successive odd numbers. If instead you subtract successive even numbers you get the corresponding negative root.

Cool. I didn’t know that.