Why is it mathematically consistent to allow imaginary numbers but prohibit division by zero?

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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

19 Answers

Anonymous 0 Comments

Yes! You absolutely can. This is done in several mathematical objects. The terms to google are the prjectively extended real line and riemann sphere.

In all such objects you do lose properties, as others have shown. But this isn’t any different to imaginary numbers. Including i means you lose some key properties of the real numbers, namely the ordering.

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