^^Numeri complessi. Storia.
credits: From Euclid to Riemann.
- 1673 Wallis insisted that imaginary numbers are not unuseful and absurd when
properly understood by using a geometric model just like negative numbers
(Algebra, Vol. II, Chap. LXVI, 1673).
- 1740 e(√-1)θ = cosθ + (√-1)sinθ
Euler's earth-shaking formula of complex analysis, obtained by formally
manipulating complex power series. Euler observed that could be used to
reduce any trigonometric identity to much simpler exponential identities.
ref: 1748 Introductio in analysin infinitorum. Leonhardo Eulero
books.google
ref: wp/Complex_number#History
- 1751 Euler persuaded himself that there is an advantage for the use of imaginary
numbers
- 1755 Euler introduced the symbol i for the imaginary unit.
pag26
- 1765 Euler
Elements_of_Algebra elementary text book introduces complex numbers almost
at once and then uses them in a natural way throughout. Euler considered
natural to introduce students to complex numbers much earlier than we do
today.
ref: wp/Complex_number#History
-
geometric representation of complex numbers
- 1796 Gauss had been in possession of the geometric representation of complex
numbers since
1796.
- 1797 C. Wessel presented a memoir to the Copenhagen Academy of
Sciences in
which he announced the same idea, but it did not attract attention.
- 1806 J.
R. Argand
went public with the same formulation.
-
1811 December 18, Gauss, in a letter to F. W. Bessel
first I would like to ask anyone who wishes to introduce a new function into
analysis to explain whether
- he wishes it to be applied merely to real
quantities, and regard imaginary values of the argument only as an appendage,
- or
whether he agrees with my thesis that in the realm of quantities the imaginaries
a + b√-1 have to be accorded equal rights with the reals.
Here it is not a question of practical value ; analysis is for me an
independent science, which would suffer serious loss of beauty and
completeness, and would have constantly to impose very tiresome restrictions
on truths which would hold generally otherwise, if these imaginary
quantities were to be neglected... (Werke, X, p. 366-367)
- 1831 trattato di Gauss sui nr complessi come punti nel piano, che
stabilisce in larga parte la moderna notazione e terminologia.
-
1833 Coppie algebriche di Hamilton (a,b)∈R2
nr complesso identificato con coppia ordinata di nr reali (a,b)∈R2;
esse costituiscono un campo, dotate di
opportuno prodotto.
Hamilton crea una teoria puramente algebrica, senza ricorso alla geometria.
credits: serge.mehl/Hamilton.
nm: coppia = paio
Sources of confusion using (√-1)2=-1
√a√b = √(ab) algebraic identity valid for non-negative real numbers,
seemed to be capriciously inconsistent with
(√-1)2 = √-1 √-1 = -1*-1 = 1
Equazione polinomiale, soluzione.
I matematici italiani del 1400 non pubblicherebbero un articolo sulle proprie
scoperte sulla soluzione delle equazioni; se qualcuno cavasse un nuovo metodo
per risolvere un'equaz polinomiale, lo terrebbe segreto e lancerebbe una
competizione per mostrare che potrebbe dominare l'avversario risolvendo equaz che gli altri non potrebbero.
Numeri immaginari
Questo trucco di prendre la radice quadrata di numeri negativi e
trattenere il fiato finchè ne fai nuovamente il quadrato ed il problema
scompare, col tempo divenne familiare, ma ancora nessuno aveva una chiara idea
di cosa questi numeri immaginari fossero effettivamente.
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