√ 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
> 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 (?)
This was quite controversial until people realised you could define complex numbers in a slightly different way. Basically you define a complex number as a pair of real numbers *a* and *b* and denote it *a* + *b*i. Then you define rules for performing addition and multiplication on these numbers. specifically (a+bi) + (c+di) = (a+c) + (b+d)i, and (a+bi)(c+di) = (ac-bd) + (ad+bc)i. From this you can see that complex numbers for which the “b” (the “imaginary part”) is zero are equivalent to real numbers, and that the complex number 0+1i, or just i, produces -1 when you square it.
And from there you can define all kinds of other operations on complex numbers and examine what properties they have. So instead of starting off with the idea that i is “the square root of -1”, we’ve just defined a structure made up of real numbers and shown that something in this structure happens to behave as if it’s the square root of -1.
> 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)?
You can do this, and there are a number of extensions of the real numbers in which this is done. For example the real projective line consists of the real numbers together with ∞, with a/0=∞ for nonzero a. These systems do have some uses, but they’re generally a bit awkward to work with as they don’t follow some of the standard rules of arithmetic that we’re used to. For example if a/c=b/c, it’s not necessarily the case that a=b, because c might be zero. And they’re nowhere near as interesting or as useful as the complex numbers, which have all kinds of incredible mathematical properties and widespread applications in quantum mechanics, electrical engineering, etc.
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