A critical mathematical result of mine requires people to accept that the sqrt() returns two values, overturning the convention--used I think primarily by people not scientists--over assuming it returns only the positive, for convenience.
So sqrt(4) is 2 or -2, and the reason it's important is I found a problem where another mathematical idea was also overruled by mathematics, which is where the ring of algebraic integers can be shown to contradict the field of complex numbers.
Now consider 5 ± 2, which is 7 or 3.
Now then, is it divisible by 7?
ONE of its solutions is 7 and the other is 3. You may want me to specify a ring, so here it's easy to say, in the ring of integers.
Now then, is 5 + sqrt(4) divisible by 7?
It is 7 or 3, because sqrt(4) is 2 or -2.
For mathematicians who do "pure math" the issue may seem esoteric but as a physics student one of the first things my professors did was beat the idea that sqrt() is just the positive out of me (not literally) and how did they manage to do that besides remonstrating against it?
Physics students learn a lot by doing science experiments.
And you will at times get the wrong answer if you take only the positive of the square root, which is kind of freaky when it happens as your mind is like, huh?
And it happens at random. So, like, you're getting the right answers in the experiments until suddenly something really wacky happens, because you forgot to consider the negative of the square root! And a light dawns, and a new wannabe physicist is growing in understanding...
And it turns out that the BIG issue is that I can prove that one of the solutions for 3 + sqrt(-26) has 7 as a factor, and it's just like one of the solutions for 5 ± 2 has 7 as a factor as that is NOT a number.
It is two numbers.
So then, one of the solutions for 3 - sqrt(-26) has 7 as a factor as well.
How is it different really?
It's like 5 ∓ 2 is 3 or 7.
And yes, 5 - sqrt(4) is 3 or 7.
And if that gives you a headache as it looks really wrong, well you were just trained really wrong.
So only the order changes, but you still have two results!
If you wish to do science where, yup, square roots DO pop up, then you have to unlearn the bad habit of only taking the positive square root or you will watch things randomly blow up on you. And if you have good professors they will probably let you blow things up by only taking the positive until you get it beaten out of you, and accept that no matter what bad teaching may tell you--the square root gives TWO values.
Human beings can be so hardheaded though, and I'm sure mathematicians who only take the positive of the square root actually think they are doing mathematics, instead of being obstinately wrong.
Notice that mathematics has no problem. It knows that the sqrt() has two values, and it's right.
Why do humans do the wrong thing?
Last time I researched it, I read it was some convention picked up by some mathematicians who tired of doing the plus or minus thing, and found that they could just get away with it for THEIR research.
So short answer: some people were lazy and doing math research where they though it didn't matter.
Oh yeah, other thing was some idiot need to think a function should only return a single value or some such crap. Just a lot of stupid actually.
Creates unnecessary problems and is mathematically incorrect.