√ 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
They can, and there’s probably a number of people that have done so and played around to see what happens when you define such a value.
The problem is that when you’re doing arithmetic with the most common sets of numbers, introducing such a definition causes a ton of other operations to become inconsistent nonsense.
Example:
let 1/0=j
let n be any real number
(n)(1)/0=nj
since 1n=n
n/0=nj multiply both sides by 0
n=0nj and 0n=0 so
n=j
So I just showed that j can be (and must be) literally any real number.
This is a big problem if I ever try to do math with my new number j. I could multiply both sides of an equation by j, but j is any real number, so I could do:
1=1
1j=1j
1(5)=1(20)
5=20
You end up in a situation where every number is equal to every other number. Such a mathematical system has little practical use, and honestly gets kind of boring to play with when everything breaks like this.
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