I recently did some (really short) research on dividing by zero and heard it is possible to divide by zero in the Riemann’s sphere, because it only defines a positive infinity and not a negative one. Is that true? And how does that work? Can’t we always find a negative of a certain number?
In: Mathematics
Dividing by zero is not allowed in standard algebra because it leads to undefined answers. 1/0 = 2/0, but under standard algebra that implies that 1 = 2 which breaks the rules of algebra.
The Riemann Sphere behaves like typical complex algebra, except with the addition of a *valid answer to 1/0*, namely positive infinity. It explicitly says there is no negative infinity. Operations involving infinity are explicitly defined; x + inf = inf, for example. Other operations are not allowed, same as in normal algebra; inf – inf is undefined and an illegal operation.
By handling infinity in these specific ways, any paradoxes that would allow us to “prove” 1=2 are removed, and logical consistency is maintained.
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