√ 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 was nothing special about the square root of negative numbers. We started with just counting numbers and then subtraction caused the invention of negative numbers, and division caused the invention of rational numbers (fractions). Imaginary numbers were just one more step.
The key is that all these expansions of the types of numbers still followed all the rules of arithmetic and produced consistent results. You can’t do that by trying to define infinity as a number. If you try, you allow all those proofs that 2=3 to work.
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