Couldn’t the result of division by zero be “defined”, just like the square root of -1?
Edit: Wow, thanks for all the great answers! This thread was really interesting and I learned a lot from you all. While there were many excellent answers, the ones that mentioned Riemann Sphere were exactly what I was looking for:
https://en.wikipedia.org/wiki/Riemann\_sphere
TIL: There are many excellent mathematicians on Reddit!
In: 1691
Think of division the other way around:
If you do 12 ÷ 4, you’re asking “how many 4’s do I need to add together to get 12?”. There is only one answer.
But if you ask “12 ÷ 0”, you’re asking “how many 0’s do I need to add together to get 12?”
Well, if you add one 0, you get 0. If you add two, you get 0 still. If you add one million, you still get 0. So there is NO number of 0’s that you can add together to get 12. There is no answer. Hence 12÷0 is “impossible”.
Even if you “0 ÷ 0”. How many 0’s do I need to add together to get 0?
Well, one. Or two. Or zero. Or a million. Any number at all, whatsoever in fact. So there’s no one answer that’s right. Literally every answer is right.
Imaginary numbers are really just numbers “in another dimension”, if you think that way, which we deal with all the time – imaginary numbers crop up in nature all the time – physics, AC electrics, all kinds of unexpected places. They are logical, consistent, you can bring them back into the domain of so-called “real” numbers, and so on. As such, mathematicians like them.
But division by zero gives you either NO ANSWER AT ALL or EVERY ANSWER AT ONCE. It’s practically useless. Hence we just don’t define it.
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