√ 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
There is actually a construct in math for what you’re talking about, called the ‘projective real number line’. It’s a number line which extends in both directions to positive and negative infinity, but which adds an extra “point at infinity” which connects the positive and negative extensions of the line.
https://en.wikipedia.org/wiki/Projectively_extended_real_line
In this extended number system, the point at infinity *is* the reciprocal of zero. But there are certain other well-defined algebraic properties that this infinity simply doesn’t have. It can’t be factored, for instance, so you can’t deduce 2(∞) = 3(∞) therefore 2=3 or anything like that. But a number system which includes this concept of infinity still has its uses.
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