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History of mathematics: making the imaginary real and respectable. (English)
Parabola 50, No. 2, 3-10 (2014).
From the text: Back in 2005, I devoted two of these columns to the history of complex and imaginary numbers. Here I return to the theme, but take a different slant on it, telling how an initially suspect notion became respectable. Let me begin by recapping the source of the difficulty. We have the basic rules of arithmetic that tell us that $+\times +=-\times -=+$; $+\times -=-\times +=-$. It seems that there is no possibility of finding $\sqrt{-1}$. Yet, as outlined in my earlier treatment, the notion proved useful (indeed, more than that, necessary) for a complete account of the cubic equation. When I was still in high school, my maths teacher airily remarked: “There’s not really a problem; you just invent a square root, $\sqrt{-1}=i$, and proceed as if nothing untoward has happened”. In other words, if we do this, it works, and really no more needs to be said. Something like this attitude must have informed the mathematics practised for some 400 years before his remark. The number $i$ was designated as “imaginary”, but dealt with exactly as if it were an ordinary, common or garden, real number.
Classification: F50 A30
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