A Conversation for Imaginary Numbers

wrong equation

Post 1

e to the x

The source of imaginary numbers was not the quadratic equation, but a special type of cubic called the depressed cubic (a cubic of the form x^3+px=q (no x^2 term)).

wrong equation

Post 2


Are you sure about that? From what I remember, the depressed cubic formula was solved long before imaginary / complex numbers and Euler were around. I'll look it up.

wrong equation

Post 3

John Stalker

It's a bit more complicated. Without going into great detail it works like this.
If you have a general cubic (with real coefficients) then you can very easily write
the solution(s) in terms of the solution(s) of a depressed cubic. You can write the solutions to that--this time not very easily--in terms of the solution(s) to a certain
quadratic. When people first tried to solve polynomials they were interested
in real solutions only. But this is where the story takes a strange turn. If
your cubic has only one real solution, and it must have at least one, then
the associated quadratic has two real roots and you can use either of them
to find the solution to your cubic. If it has two solutions, which doesn't happen
often, then there is a simple dirty trick to find them. But if the cubic has three
real solutions then the associated quadratic has no real solutions! But if you
blithely ignore this fact and proceed as before you can still find these solutions by the same method as before. This is really the origin of complex numbers. People
found that they needed to invent "imaginary" numbers to find the "real" solutions to
problems. Of course now we remove the quotes from that sentence and also the implied value judgments.

wrong equation

Post 4


Ah... I see... we're both right....

... having read up a bit on the topic, it would seem that mathematicians were "using" imaginary numbers, but that they hadn't been fully "accepted" by all mathematicians at the time of the solution of the depressed cubic... and for a long time since then as well. In fact, they weren't really fully accepted until Euler and Gauss used them in the late 18th century.

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