√ x when x < 0 was impossible using real numbers. Then, one day some guy decided that i = √ -1 and suddenly we can work with negative square roots. Which is still quite weird, but I suppose is valid (?)
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But there still stuff we can’t calculate in any of the number sets we have. Something we see much earlier than roots: division by 0.
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I get why x / 0 = x is false. Basically that would mean that 1 = 0, which is absurd (reductio ad absurdum). Ok. I don’t quite get why x/0 = 0 is false, but I’m sure there’s a contradiction somewhere that would make this impossible (if it was that simple, division by 0 would have been solved thousand of years prior). But if we can simply invent a number that doesn’t exists and it can fit maths no problem such as i = √-1 why can’t we make a number n = 0^(-1)?
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I also know the concept of limits but lim x->0 f(x) just means that x gets really close to the value of 0 but is never quite there so yeah, I don’t think that solves the issue.
In: Mathematics
First, there’s no need to ask about “mathematicians”. You’re a mathematician if you do math. You’re a mathematician if you think about numbers. You’re a mathematician if you ask questions about numbers. You are a mathematician.
Second, as a mathematician, you can have any numbers you like. You can make up a number for 1/0. Go for it.
Third, if you talk to other mathematicians, you will find that they like some numbers more than others, and their criterion for liking numbers is that they should be interesting, which is to say, that they should lead to interesting mathematics.
There are number systems, like the projective line and the Reimann sphere, that have a number for infinity that interacts with zero in various ways. You could look into these if you are interested.
Or you could make up your own definition for 1/0. Depending on how you define it, you can end up in situations (as other commenters have explained) were every number equals every other number. The problem with that isn’t that it is wrong, the problem is that–in the opinion of most mathematicians–it is boring. But they’re your numbers, and you get to decide for yourself whether they are boring.
Beyond that, you get to decide for yourself what you want out of your numbers. Maybe you aren’t looking for interesting mathematics. Maybe you just want a number that you can put on your shelf and stare at, like a Necker cube or an Impossible Trident.
Enjoy 🙂
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