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> Who "invented" this imaginary number (and how...was
> it the end result of a thought experiment?)
Everything in mathematics is the end result of a thought
experiment. :-)
I'm sorry but I don't know this. It might have been
created as a result of the investigations into
solving polynomial equations by people like Gauss or Galois.
Someone else more informed about the history of math can probably
give a good answer.
> What fields and disciplines depend on "i" (Like, if
> we didn't have "i", we wouldn't have televisions/
> particle physics/toasters/Tony Danza/whatever...)
Well, electrical engineering for one. 'i' is normally referred to
by EEs as 'j' to avoid confusion with the common symbol for
current.
Within the disciplines of electrical engineering, 'j' is used
extensively in linear circuit theory, electric power systems,
communication systems, and optics.
It's not used as directly in computer design which relies more on
discrete mathematics and circuits operating in switching modes
where 'j' appears infrequently.
If we "didn't have 'j'" we PROBABLY wouldn't have electric
power, telephones, radios, tape recorders, televisions, radar,
lasers, CAT scanners, or CDs (among many other things).
I say probably because people are ingenious and there is
frequently more than one way to invent something. Also we
use many things that we don't understand very well.
We MIGHT still have computers but likely not powered by electricity
if such is practically possible.
Mechanical and civil engineers also use 'j'.
Civil engineering:
Vibrations and modes of structures (bridge building)
An aside: of course, people built bridges before the mathematics
of 'j' was invented, but they fell down more often or in one
well known case (Tacoma Narrows) resonated to pieces. To avoid
having this happen, bridge builders often used much more material
than they had to. This is what you do when you don't fully
understand the systems you're designing: overkill
(unless you're NASA) :-}.
Mechanical engineering
Rotating machinery (gyroscopes, pumps, compressors)
Machine dynamics (the suspension in cars)
Physics
Classical and quantum mechanics
Electromagnetism (wave propagation, optics)
Acoustics (waves again)
Heat transfer (pops up in temperature distribution
solutions)
Just about any discipline that involves modelling things
with differential equations will have a 'j' pop out
somewhere at some time.
I'm not aware of any uses in chemical engineering, but
I don't know much about chemical engineering.
These are just some examples; there are many others.
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| You might want to look a some books by E.T.Bell.
One of the is entitled "Men of Mathematics" and the other
I have in mind is "Mathematics, Queen and Servant of the Sciences".
Both these books deal with the history of mathematics and
mathematicians.
My guess about who invented 'i' is one of the early Italian
algebraists, perhaps Cardano or Tartaglio. It was certainly know
by the time Rene Descartes invented co ordinate geometry.
An interesting follow on question is when did Complex Analysis start
as a separate and discernable discipline?
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| According to "A History of Mathematics" (Carl Boyer, author; Wiley
& Sons, publishers):
"Cardan referred to these square roots of negative numbers as
'sophistic' and concluded that his result in this case was 'as subtile
as it is useless.' ... It is to Cardan's credit that at least
he paid some attention to this puzzling situation.
...
"It remained for Girard in 1629, in Invention nouvelle en l'algebre,
to state clearly the relations between roots and coefficients, for
he allowed for negative and imaginary roots... Girard retained
imaginary roots of equations because they show the general principles
in the formation of an equation from its roots.
...
"Among relatively minor contributions by Leibniz were his comments
on complex numbers, at a time when they were almost forgotten... The
ambivalent status of complex numbers is well illustrated by the
remark of Leibniz, who was also a prominent theologian, that imaginary
numbers are a sort of amphibian, halfway between existence and
nonexistence, resembling in this respect the Holy Ghost in Christian
theology.
...
"In fact, although Euler used i for (-1)^1/2 in a manuscript dated
1777, this was published only in 1794. It was the adoption of the
symbol by Gauss in his classic Disquisitiones Arithmeticae of 1801
that resulted in its secure place in mathematical notations."
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Well, I don't know much about their history, but I know that you
can pretty much derive the properties of "imaginary" numbers by
applying abstract group/ring/integral domain/field theory by working
with the roots of a polynomial generated about a field, such as
the reals.
You end up being able to factor any polynomial whose coefficients
are members of a field, by creating an extension field, ending up
with new cofficients like '2 lambda', where lambda is an element
in the extension of the field.
I don't know how far back abstract algebra goes, but it gives some
solidity to the concepts of imaginaries.
-mjg
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