
Curious quaternions - ColinWright
http://plus.maths.org/content/os/issue32/features/baez/index
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pif
This article starts with an error: it's a _third_ degree equation that can
have real solutions whose calculation may involve the square root of negative
numbers. For second degree equations, instead, if the discriminant is negative
there are no real solutions. You may check on Wikipedia:
<http://en.wikipedia.org/wiki/Complex_number#History_in_brief>

~~~
ColinWright
What you're talking about is that cubics can have real solutions, and yet
require square roots of negative numbers while following the formula. This is
true, and it is, in fact, what historically led people to start to take
complex numbers seriously.

However, the article never talks about real solutions of quadratics requiring
square roots of negative numbers. The article only talks about it turning out
all right in the end.

What they mean is this. Suppose you allow square roots of negative numbers,
and simply work with them using the usual laws of algebra. Sometimes you have
a quadratic that results in a solution that has the square roots of negative
numbers, and hence the "solution" seems to have no interpretation. However,
substituting back into the original equation we can see that they are, in some
sense, genuine solutions.

So I would claim that the article is actively misleading to those who already
know some of the history, and is less "complete" than it might be, but it
avoids talking about cubics, and with a strict and literal reading is
accurate.

I'm not, by the way, defending it. I wrote a similar article many years ago,
and I did use the cubic example, including showing how to solve a cubic. It
was aimed at a slightly ore sophisticated audience, however, and I'm not
convinced this article is "wrong" in any important sense.

