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I find the explanation by limits more convincing. The limit of 1/n as n approaches 0 from the positive direction is a positive infinity. The limit of 1/n as n a
by SAI_Peregrinus 3y ago
I find the explanation by limits more convincing. The limit of 1/n as n approaches 0 from the positive direction is a positive infinity. The limit of 1/n as n approaches 0 from the negative direction is a negative infinity. Those are different, the overall limit diverges, so 1/0 is undefined. The same argument applies to all nonzero numbers in glace of 1. 0/n as n goes to 0 does limit to 0 but it's "cleaner" to not have the special case.
- kazinator 3y agoIt doesn't matter whether the two sides diverge. For the 1/(n^2) function, this is not the case; does that make 1/0 defined if we express it as 1/(0^2)? ;)
- chongli 3y agoOn the complex plane, you can take the limit as n->0 of 1/n from any direction on the circle. The Riemann Sphere [1] ties up all these infinities into one and places it at the North Pole, opposing zero which is at the South Pole. Since now all these directional limits converge to the same value, the limit is well-defined in the Riemann Sphere and so we can close it, giving 1/0 the well-defined value of infinity. [1] https://en.wikipedia.org/wiki/Riemann_sphere https://en.wikipedia.org/wiki/Riemann_sphere
- jjgreen 3y agoThis is why floating point numbers have +0 and -0