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From: Michael Press on 22 Jan 2010 19:07
In article <t434l5doepb9knou8t01r6plf8tgj717h1(a)4ax.com>,
Adam <no(a)spam.edu> wrote:

> On Wed, 13 Jan 2010 04:51:18 -0600, David C. Ullrich wrote:
>
> >On Wed, 13 Jan 2010 00:56:36 -0800 (PST), Ross <rossclement(a)gmail.com>
> >wrote:
> >
> >>Are the formulae for findin the roots of cubic equations on Wikipedia
> >>correct? I've been through it over and over, and simply don't get
> >>roots.
> >>
> >>I've gone to the original source, which is:
> >>http://planetmath.org/?op=getobj&from=objects&name=CubicFormula
> >>
> >>Just looking at the first equation to calculate r1, then if I have the
> >>cubic x^3 -2x^2 -11x + 12, then I get NaNs when I try to find r1.
> >>
> >>Specifically, if I set a1=-2, b1=-11, and c1=12, then the value of
> >>which the square root is taken:
> >>(2*a1^3 - 9*a1*b1 + 27*c1)^2 + 4*(-(a1^2)+3*b1)^3
> >>
> >>works out to be negative, leading to the NaN result.
> >
> >No, the square root of a negative number is a complex number.
> >Complex numbers come into the formula even when the roots
> >of the original cubic turn out to be real.
> >
> >This has a lot to do with the history of complex numbers.
> >Way back then nobody was interested in saying that
> >x^2 = -1 had a solution because it obviously didn't.
> >But these funny "imaginary" numbers did arise in the
> >solution to cubics with _real_ roots - hence they couldn't
> >be just ignored. At first they were just "imaginary"
> >temporary things that luckily went away when we got the
> >final answer - people gradually decided they were interesing
> >in themselves.
>
> Imaginary numbers are the "dark matter" of math?
>
> They outnumber real numbers.

There is only one real number: 1.
Look at anything real; what is its count?
Answer: 1. There are no complex numbers.
Can you show me one? No you cannot, so
there are no complex numbers. Therefore
the real numbers outnumber the complex.

--
Michael Press
From: David Bernier on 26 Jan 2010 17:50
Don Redmond wrote:
> On Jan 14, 12:04 am, David Bernier <david...(a)videotron.ca> wrote:
>> Gib Bogle wrote:
[...]

>>> I've just been reading A History of Mathematics by Boyer (and Merzbach).
>>> The cubic gave the old-timers plenty to think about, and solving it was
>>> a great feat. As David Ullrich mentioned, the occurrence of square
>>> roots of negative numbers in a cubic with three real roots was a real
>>> head-scratcher.
>> It would be interesting to know which came first:
>> - trigonometric and inverse trigonometric functions used to solve
>> casus irreducibilis equations or:
>> - the ( Argand) complex plane .
>>
>> David Bernier
>
> Viete, at least, in 1500's did the trig soln of cubics. Argand and
> Wessel
> did the Argand diagram in the 1700's. It was spread by Gauss.
>
> Don

Thanks for the information. According to Wikipedia, Euler
published Euler's formula:
for any real x, exp(ix) = cos(x) + i sin(x) in 1748.
Cf.:
< http://en.wikipedia.org/wiki/Euler%27s_formula > .

Then, for example, exp(ix) exp(iy) = exp(i (x+y) ) is clear
using the geometrical interpretation of complex numbers, polar
coordinates and the complex plane (from ca. 1799 onwards).

But with no complex plane, exp(ix) = cos(x) + i sin(x)
might seem quite mysterious. From looking-up
de Moivre, his formula as stated by him involved the
sine and cosine transcendental functions only.

David Bernier



From: Robert Israel on 26 Jan 2010 18:32
David Bernier <david250(a)videotron.ca> writes:

> Don Redmond wrote:
> > On Jan 14, 12:04 am, David Bernier <david...(a)videotron.ca> wrote:
> >> Gib Bogle wrote:
> [...]
>
> >>> I've just been reading A History of Mathematics by Boyer (and
> >>> Merzbach).
> >>> The cubic gave the old-timers plenty to think about, and solving it
> >>> was
> >>> a great feat. As David Ullrich mentioned, the occurrence of square
> >>> roots of negative numbers in a cubic with three real roots was a real
> >>> head-scratcher.
> >> It would be interesting to know which came first:
> >> - trigonometric and inverse trigonometric functions used to solve
> >> casus irreducibilis equations or:
> >> - the ( Argand) complex plane .
> >>
> >> David Bernier
> >
> > Viete, at least, in 1500's did the trig soln of cubics. Argand and
> > Wessel
> > did the Argand diagram in the 1700's. It was spread by Gauss.
> >
> > Don
>
> Thanks for the information. According to Wikipedia, Euler
> published Euler's formula:
> for any real x, exp(ix) = cos(x) + i sin(x) in 1748.
> Cf.:
> < http://en.wikipedia.org/wiki/Euler%27s_formula > .
>
> Then, for example, exp(ix) exp(iy) = exp(i (x+y) ) is clear
> using the geometrical interpretation of complex numbers, polar
> coordinates and the complex plane (from ca. 1799 onwards).
>
> But with no complex plane, exp(ix) = cos(x) + i sin(x)
> might seem quite mysterious. From looking-up
> de Moivre, his formula as stated by him involved the
> sine and cosine transcendental functions only.

True. As is very common, our current understanding of de Moivre's
formula is strongly influenced by concepts that historically came later.
The formulation

(cos(x) + i sin(x))^n = cos(n x) + i sin(n x)

dates from 1722. So De Moivre would have considered this as
an algebraic fact, without the benefit of a geometric
interpretation using the complex plane.

Of course the trigonometric solution of the cubic only depends on the
case n=3:

sin(3 x) = 3 sin(x) - 4 sin(x)^3

which follows readily from the addition formula for sine and the identity
cos(x)^2 + sin(x)^2 = 1, and would have been known long before de Moivre.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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