Complex division:when divisor is 0+0im

[wang-zh]

why (3+0im)/(0+0im) is equal to NaN + NaN*im Instead of Inf+0im

[KristofferC]

Ref #5234

[brenhinkeller]

Looks like this can be closed as duplicate if no objections

[hyrodium]

why (3+0im)/(0+0im) is equal to NaN + NaN*im Instead of Inf+0im

This is because the limit does not exist.

$$\begin{aligned} \frac{1}{a+bi} &= \frac{a}{a^2+b^2} + i \frac{-b}{a^2+b^2} \\\ \lim_{a \to +0}\lim_{b \to +0}\frac{a}{a^2+b^2} &= +\infty \\\ \lim_{b \to +0}\lim_{a \to +0}\frac{a}{a^2+b^2} &= 0 \\\ \lim_{a \to +0}\lim_{b \to +0}\frac{-b}{a^2+b^2} &= 0 \\\ \lim_{b \to +0}\lim_{a \to +0}\frac{-b}{a^2+b^2} &= -\infty \end{aligned}$$

Related:

• should complex/0 division match complex/(0+0im)? #22983
• https://discourse.julialang.org/t/why-inv-complex-0-0-0-0-returns-complex-nan-nan/92694/2
• https://github.com/scheinerman/RiemannComplexNumbers.jl
• https://hyrodium.github.io/RiemannSphereOperations.jl/dev/