id: 06389088
dt: j
an: 2015a.00495
au: Deakin, Michael A. B.
ti: History of mathematics: making the imaginary real and respectable.
so: Parabola 50, No. 2, 3-10 (2014).
py: 2014
pu: AMT Publishing, Australian Mathematics Trust, University of Canberra,
Canberra; School of Mathematics \& Statistics, University of New South
Wales, Sydney
la: EN
cc: F50 A30
ut: complex numbers; imaginary numbers; history of mathematics; mathematicians;
arithmetic operations; notation; division algebras; proofs
ci:
li:
ab: 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.
rv: