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Complex analysis is an entire field of mathematics that focuses on the use of the complex constant $i$. When was the significance of $i$, an imaginary number, first noticed?

If I did not know some of the uses of complex analysis, I would likely believe, being the layman that I am, that $i$, as it is not a real number, would be fairly useless, almost like $0/0$. I would have trouble believing it had many uses, because it cannot be used to describe amounts of things like real numbers can (e.g. "I have 5 apples").

Why was any special attention paid to $i$ in the first place?

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Complex analysis does not "focus" on the number $i$ any more than real analysis focuses on $\pi$ or $\sqrt{2}$. Geometrically you can think of $i$ as a 90 degree rotation, which is far from useless. –  KCd Jun 9 '12 at 19:29
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I think the canonical answer to this involves the general solution of the cubic equation in the 16th C. If you work out the expression for the roots, it comes out very complicated and always involves square roots of negative values, even in the case when the polynomial has all real roots. In such a case, the imaginary parts of the expression cancel out and leave one with the correct real roots. So the Italians were forced to confront imaginary quantities, at least to the extent of admitting that they could me manipulated algebraically in the course of solving a cubic equation. –  MJD Jun 10 '12 at 1:38
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up vote 4 down vote accepted

See this for a history of complex numbers and here for some motivation. Also there's a website on answers and explanations that explain concepts such as imaginary numbers and infinity.

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