With regard to imaginary numbers: I learned my algebra from an old
book (1903 or so) on my own the year before I entered 9th grade. The
imaginary numbers (and complex numbers) were introduced in order to
be able to perform some "normal" operations on real numbers that could
not yield real numbers, in particular, taking the square root of a
negative number. The book went on to say that the complex numbers
of the form a+bi were sufficient to carry out normal operations on
complex numbers - that nothing more complex was necessary.
I was reflecting on this one day, trying to think of the most
complicated expression I could imagine, when I hit on i^i. Could
that really be expressed as a complex number? I started fooling
around with logarithms and the like, remembering that e^(i pi)+1=0,
and managed to find i^i, and it was a real number! My parents
wouldn't understand the proof, so I took it to my algebra teacher,
who barely understood it and wasn't sure if it was already known,
which of course it was. Only later did I learn about multivalued
expressions.
Anyway, at some point you could challenge the students to "evaluate"
i^i, and see who can do it.
---
Laurent Hodges, Professor of Physics lhodges@iastate.edu
12 Physics Hall, Iowa State University, Ames, IA 50011-3160
(515) 294-1185 (office) http://www.public.iastate.edu/~lhodges