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As the question states, curious on why we have given a definition to the square root of $-1$, but not to $\frac 1 0 ?$ To me both seem to be undefined, and yet square root of $-1$ is a pretty important number, what is the limitation on defining $\frac 1 0$?

Put in other words, can there be a useful way of defining $\frac 1 0$?

Akash
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    This has been asked MANY times on this site. – Randall Sep 11 '24 at 14:25
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    There is no useful algebraic definition of $1/0.$ There is for $\sqrt{-1}.$ You could define $1/0$ to be whatever you wanted, but it would end up contradicting fundamental rules of arithmetic, so you would have to add special cases to the associative law and/or other laws. That said, in limited cases, we do sometimes define $1/0$ as $\infty,$ specifically when talking about Möbius transformations on the projective line over a field. – Thomas Andrews Sep 11 '24 at 14:34
  • $\frac 1 0$ would be defined as the number $x$ s.t $x\cdot 0 = 1$, but thats impossible because $x\cdot 0=0$.

    But you can imagine (enjoy the pun if you want) a square root of minus one, call it $i$ and figure out that calculation with it works perfectly fine. The only problem is that you lose the total ordering the real line has.

     – Tina Sep 11 '24 at 14:35
  • There are some amusing reason why $\sqrt{-1}$ has always been there, we just didn't know it. For example, the Taylor series for $\frac{1}{1+x^2}$ around a real number $x=a$ can be shown to have a region of convergence of radius $\sqrt{1+a^2}.$ This vaguely hints that there is an obstruction that is distance $\sqrt{1+a^2}$ from each $a,$ which would be the distance from $a$ to a point off $1$ off the real line at $0.$ – Thomas Andrews Sep 11 '24 at 14:44
  • my question is more specifically to understand what happens if you do try to define it, in a similar way to how √-1 gets defined or perhaps some other way, beyond the original definition of the operation. The provided answer is sufficient – Akash Sep 11 '24 at 14:48
  • https://en.wikipedia.org/wiki/Wheel_theory# – Shaun Sep 11 '24 at 15:21
  • Also relevant: https://math.stackexchange.com/questions/125186/why-not-to-extend-the-set-of-natural-numbers-to-make-it-closed-under-division-by, https://math.stackexchange.com/questions/259584/why-dont-we-define-imaginary-numbers-for-every-impossibility – Hans Lundmark Sep 11 '24 at 16:50

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