√ 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
>√ 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 (?)
Not quite. Instead, mathematicians were trying to solve a certain class of problems. In doing so, they came across the square root of negative numbers. In the end, those square roots cancelled out, but the fact that they could exist without breaking everything indicated that there was something there.
Mathematicians still didn’t like it and these numbers were treated with derision for along while, but we’ve come to eventually accept them as a valid extension of the real numbers.
Point is, we were pretty much *forced* to acknowledge their existence. We didn’t just decide “hey, let’s make a new class of numbers.”
>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)?
Ok. Let’s say x/0 = 0. That means x = 0*0; x = 0. So it would only work for x = 0. If you tried it for any other value of x, you’d immediately get a contradiction (1/0 = 0 implies 1 = 0; 2/0 = 0 implies 2=0, etc.)
But there’s issues even with saying 0/0 = 0.
When we ask “What is a/b” we’re asking, “What can we multiply b by to get a?” This means “What is 0/0” we’re asking “What can we multiply 0 by to get 0.” And the answer is *anything*. So there isn’t a single answer to that question.
At the end of the day, trying to assign a value to x/0 ends up in contradictions or otherwise breaking fundamental mathematical concepts.
Compared to square root of negative numbers were shown to be necessary and have greatly expanded our understanding of math.
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