382
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
e(x) = cos X + i sinx
satisfies e(x+y) = e(x)e(y), whence e(nx) = e(x)n. There is nothing in this
calculation other than the relation i 2 = -1 and the classical addition formulae. We shall clarify (28) in the following nO.
In a recent mathematics text book designed for the final year of French
high schools there is, in Chapter I, complex numbers, a place where they put,
by definition,
e ix = cos x + isinx
and where they show, with the help of the addition formulae, that ei(x+y) =
eixe iy . No explanation is offered to the reader as to what the mysterious
letter e might signify, nor of a nonreal exponent. It would be difficult to
imagine a more aberrant conception of mathematics: Euler's most famous
formula degenerating into a pure and simple notation, and furthermore incomprehensible!
Since the books in use in high-schools conform strictly to the directives
of the national Ministry of Education one is forced to conclude that this eminently original version of mathematics - I never met it before the year
2000 - is due to the committees who decide the programmes. It is difficult
to imagine that mathematicians, even "applied", could ever have advocated
it. But then, who is responsible? Electricians? In America, at the end of
the XIX th century, engineers at General Electric were taught to use Euler's
formula by Georg Steinmetz, a young German immigrant with a German
Ph.D. in Physics who knew Mathematics; he soon became one of the most
prominent electrical engineers in the USA, as famous as Edison and Sperry.
See Ronald R. Kline, Steinmetz: Engineer and Socialist (Johns Hopkins UP,
1992).
(xiv) Roots of a complex number. The relation (28) shows in particular
that
(14.29)
(cos x + i. sinx)n = 1
for x = 2kn In.
This shows in particular that the number 1 has n nth roots in C, situated in
the complex plane at the vertices of the regular polygon of n sides inscribed
in the unit circle Izl = 1 and having a vertex at z = 1, namely the roots of
unity
(14.30)
exp(2k7riln)
with k = 0, ... , n - 1;
adding a multiple of n to k clearly does not change the result. Since the
equation zn - 1 = 0 can never have more than n roots, one obtains them all
in this way, whence, using a theorem of algebra,
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
e(x) = cos X + i sinx
satisfies e(x+y) = e(x)e(y), whence e(nx) = e(x)n. There is nothing in this
calculation other than the relation i 2 = -1 and the classical addition formulae. We shall clarify (28) in the following nO.
In a recent mathematics text book designed for the final year of French
high schools there is, in Chapter I, complex numbers, a place where they put,
by definition,
e ix = cos x + isinx
and where they show, with the help of the addition formulae, that ei(x+y) =
eixe iy . No explanation is offered to the reader as to what the mysterious
letter e might signify, nor of a nonreal exponent. It would be difficult to
imagine a more aberrant conception of mathematics: Euler's most famous
formula degenerating into a pure and simple notation, and furthermore incomprehensible!
Since the books in use in high-schools conform strictly to the directives
of the national Ministry of Education one is forced to conclude that this eminently original version of mathematics - I never met it before the year
2000 - is due to the committees who decide the programmes. It is difficult
to imagine that mathematicians, even "applied", could ever have advocated
it. But then, who is responsible? Electricians? In America, at the end of
the XIX th century, engineers at General Electric were taught to use Euler's
formula by Georg Steinmetz, a young German immigrant with a German
Ph.D. in Physics who knew Mathematics; he soon became one of the most
prominent electrical engineers in the USA, as famous as Edison and Sperry.
See Ronald R. Kline, Steinmetz: Engineer and Socialist (Johns Hopkins UP,
1992).
(xiv) Roots of a complex number. The relation (28) shows in particular
that
(14.29)
(cos x + i. sinx)n = 1
for x = 2kn In.
This shows in particular that the number 1 has n nth roots in C, situated in
the complex plane at the vertices of the regular polygon of n sides inscribed
in the unit circle Izl = 1 and having a vertex at z = 1, namely the roots of
unity
(14.30)
exp(2k7riln)
with k = 0, ... , n - 1;
adding a multiple of n to k clearly does not change the result. Since the
equation zn - 1 = 0 can never have more than n roots, one obtains them all
in this way, whence, using a theorem of algebra,