Your assertions are simply not true in the context of complex analysis. It is common to use "inverse" to refer to the multiplicative inverse as shorthand (thoug

Your assertions are simply not true in the context of complex analysis. It is common to use "inverse" to refer to the multiplicative inverse as shorthand (though potentially confusing). a a^-1 = 1 is absolutely and uncontroversially applicable to any complex number. It is common and natural to extend to complex plane to include a single point at infinity (known as the extended complex plane, see e.g. https://mathworld.wolfram.com/ExtendedComplexPlane.html https://mathworld.wolfram.com/ExtendedComplexPlane.html ). When you are working in the extended complex plane, 1/0 does equal infinity.