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→Arbitrary definition for using the imaginary square root of -1
::<math>= -1 + i0 = -1</math>
::<math>= -1 + i0 = -1</math>
::Granted, it took two centuries to get all the details right (convergence, resorting of infinite sums, etc.), but that's the gist. Straightforward and beautiful. --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 16:04, 17 August 2015 (EDT)
::Granted, it took two centuries to get all the details right (convergence, resorting of infinite sums, etc.), but that's the gist. Straightforward and beautiful. --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 16:04, 17 August 2015 (EDT)
===Just a thought===
There is nothing wrong with "arbitrary definitions" in mathematics. Mathematicians will make them up all the time, sometimes with amusing, sometimes with usable results.
The "imaginary number i" is not more artificial than the "number 2": try to think about ''two'' without thinking of ''two objects'' - or the cipher! The concept of 2 is as ''imaginary'' as the concept of ''i''. We use both, because they work, not only in one instance, but in many: the complex numbers can be introduced in various ways, and all of them work hand-in-glove together, always a sign that something is done right! So, we shouldn't bother about the unfortunate naming of ''i'', but follow the applications!
The square root of -1 was introduced as first as a party trick - to solve cubic equations in contests. As so often in mathematics, it suddenly became quite relevant, and nowadays, complex analysis is something every undergraduate in maths or physics has to know! It doesn't matter how you feel about it, it is only important that it works... --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 18:43, 17 August 2015 (EDT)