What is standard in Portugal is that 'complex'='imaginary' and
'pure imaginary'='imaginary but not real'.
According to the "Encyclopedic dictionary
of Mathematics" of the "Mathematical Society of Japan"
(MIT Press, 1993):
"A complex number that is not a real number is sometimes called
an imaginary number; in particular, a complex number A
with Re A = 0 is called a purely imaginary number."
In the french "Dictionnaire de Mathematiques Elementaires"
of Stella Baruk (Seuil, 1992)
"complex number" is identified with "imaginary number".
There, the historical naming of D'Alembert is recalled:
"mixte imaginaire"(mixed imaginary) for a+bi
and "imaginaire simple"(simple imaginary) for bi,
and these are called by the author "imaginaire pur"(purely imaginary).
In the beginning the same D'Alembert used "imaginary" for
the square roots of negative numbers and opposed it to "real".
So, "sometimes", 'imaginary number' is a 'purely imaginary number'
and sometimes it is not.
I prefer the simpler notations and definitions
so I prefer the ones used here
(but maybe it is because I am used to them...)
>Somewhere I learned (but apparently in error) that
> a + bi is a complex number. If a = 0 then the number is imaginary.
>If b = 0 then it is a Real number.
This is a simplification because
the complex numbers can be defined as pairs
of real numbers (or as something equivalent)
and only afterwards you define an isomorphism
between the reals and the complex numbers of
the form (x,0).
So you can say that all numbers of the form (x,y)
with x and y reals are complex numbers,
like (0,0) or (1,0) or (0,1).
After defining the isomorphism you can identify
(1,0) with 1 and (0,0) with 0
and define a new i = (0,1).
So it is correct to say that 1 and 0 are
complex numbers. But they are also real.
>I'll guess I'll have to retract my statement that math is a precise language
>;-). After all is "e" the base of the natural log or the eccentricity of a
>conic?
But the "words" or "letters" are not the mathematical language.
The names we give to mathematical concepts
may be misleading, essentially because a lot
of people use different words or the same words
with different meaning, but the words are not the
real mathematics.
There is a tendency of giving a lot of importance
to the words, and I suspect this is wrong
because students may be able to repeat all words
and definitions associated with them
and sometimes they know nothing about the mathematical
concepts.
Jaime
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Jaime Carvalho e Silva, Departamento de Matematica
Universidade de Coimbra, Apartado 3008, 3000 Coimbra
PORTUGAL. Phone: 351-39-7003199/50 Fax: 351-39-32568
WWW home page: http://www.mat.uc.pt/~jaimecs/index.html
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"The safest way to produce disaffection with mathematics among
high school pupils and undergraduates is to use mathematics
as the source for senselessly repetitive computations and
homework assignments. On a somewhat higher level, to
produce proofs for everything that does have a proof may be
'proof' for the necessity to occupy (and employ) the
teacher, but will not necessarily confer mathematical
insight to the instructed." - Walter Felscher
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