## A Conversation for Imaginary Numbers

### wrong equation

e to the x Started conversation Jan 15, 2001

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)).

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me[Andy]g Posted Mar 24, 2001

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.

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John Stalker Posted May 25, 2001

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.

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me[Andy]g Posted Jun 26, 2001

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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