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More Complex Number Questions
Asked by Dr. Ethan B. Gallogly, UC Berkeley (PhD Chemistry) on October 14, 1996:
Can you give one example of a directly measurable quantity which has
an imaginary number in it?
In all cases I've seen you either ignore the imaginary root or square it
into a real quantity.
For your first question, here are some examples (see also the "answers
and explanations" section of
the web site, as well as an earlier question
in the Question Corner area):
Second, don't imarginary numbers arise because we ask that multiplication is
communitive, i.e. A x B=B x A, but if we dropped this rule, and
multiplication as 1 x 1=1, -1 x -1=-1,
-1 x 1=-1, 1 x -1=1, then I believe we would
lose the imaginary terms,
but be stuck with a messier (similar to operator algebra) mathematics.
- The strength of an electromagnetic field. This is a directly measurable
quantity that is measured by a complex number. That number will be
purely real if the field is all electric with
no magnetic component, purely imaginary if the field is all magnetic with
no electric component, and in other cases will have a non-zero real part
and a non-zero imaginary part.
- An attenuating medium placed in the path of an electromagnetic
wave is also measured by a complex number. The number will be purely
real if the medium affects the magnitude of the wave but leaves the phase
unchanged, and purely imaginary if the medium shifts the phase by 90 degrees.
In general, a medium which scales the magnitude by a factor of R and
shifts the phase by an angle t is described by the complex number
A field F that enters medium M will emerge as F times M
(where this product is by the laws of complex multiplication). It
is this fact that justifies measuring the entire field by a single
complex number rather than taking its electric and magnetic components
separately and measuring each by a real number, for in the latter case
you wouldn't have a way to multiply.
- The state of a component in an electronic circuit is also
measured by a complex number. That number will be purely real if there is a
voltage across it but no current flowing through it (such as a fully charged
capacitor) and purely imaginary if there is current flowing through it but no
voltage across it (such as a fully discharged capacitor).
- An LC filter in an electric circuit can be measured by a complex number,
and the theory is much the same as for an attenuating medium placed in
the path of an electromagnetic field.
As for your second question: commutativity is not the issue. More important
is the fact that multiplication has to be related to addition by the
plenty of non-commutative extensions of the reals (such as the quaternions),
but any that satisfy the distributive law and are complete with respect to
root-taking must of necessity
contain the complex numbers as a subset.
Your example is simply a function from pairs of real numbers to real numbers
for which there happens to exist a "root" r of -1 in the sense
that f(r,r)=-1. There are plenty of such functions, but unless they
are linear and distribute over addition, they don't provide the
structure needed for a notion of "multiplication".
For instance, in your example you would have
1 x (1 + (-1)) = 1 x 0 = 0, but
(1 x 1) + (1 x (-1) = 1 + 1 = 2.
Non-commutative multiplication is messier but still useful,
but an operation that is not commutative, not associative, and
doesn't distribute over addition cannot really play the part of
multiplication in any reasonable mathematical theory.
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