[Warning: probably some poor math. If you are proficient at math, feel free to share your thoughts and any corrections/adjustments that might need to be made]
Alright, let me get this straight: so 1/0 IS an actual number with an actual value. It has to be, otherwise straight vertical lines don't exist. But it is also undefined
The reason why it is undefined is because as n→0+, 1/n= ∞. [1/1=1, 1/(1/2)=2, 1/(1/10)=10, etc...]
HOWEVER, as n→0–, 1/n= –∞. [1/(–1)=–1, 1/(–1/2)=-2, 1/(–1/10)=–10, etc...]
Therefore, as n→0, 1/n= both ∞ and (–∞). So does that mean 1/0 is simultaneously –∞ AND +∞, and not either-or like you would see when f(a)=f(–a)=0?
I don't even know how that would look, physically and mathematically
But if that is the case and we denote ± as being plus AND minus, rather than plus OR minus, it would mean that 0×(±∞)=1, as well as 1/(±∞)=0
[Side tangent: How would 0×(±∞) even look as repeated addition? Coz that's what multiplication is. Maybe 0±0±...±0? Or perhaps ±1(0+0+...+0)? Same thing actually, but it still makes no sense. How can you add and subtract something at the same time and still change its value?]
Which again, I don't understand, but it does explain why 1/0 is undefined, because it has two simultaneous values (which, from my understanding, only exist as concepts and not actual values) despite it being a constant
Have I got all of this right?
And why can't we just allow the existence of constants with simultaneous values in mathematics? Could certainly help in defining 1/0. But then, what else would it apply to? And its not like we constantly came across it like with √–1. 1/0 is just weird
[I also want to point out that if 1/(±∞)=0, does 0 actually have no value?]
