John Stillwell - Naive Lie Theory.pdf - Index of

1.6 Discussion 19
√
a 2 + b 2
θ
a
b
×
√
c 2 + d 2
ϕ
c
d
=
√
(a 2 + b 2 )(c 2 + d 2 )
θ + ϕ
ac − bd
bc + ad
Figure 1.6: The Diophantus “product” of triangles.
The algebra of complex numbers emerged from the study of polynomial
equations in the sixteenth century, particularly the solution of cubic
equations by the Italian mathematicians del Ferro, Tartaglia, Cardano,
and Bombelli. Complex numbers were not required for the solution of
quadratic equations, because in the sixteenth century one could say that
x 2 + 1 = 0, for example, has no solution. The formal solution x = √ −1
was just a signal that no solution really exists. Cubic equations force the
issue because the equation x 3 = px + q has solution
√
√
√
x = 3 q (q 2 ( p
)
√
3
2 2) + 3 q (q 2 ( p
) 3
− +
3 2 2) − −
3
(the “Cardano formula”). Thus, according to the Cardano formula the solution
of x 3 = 15x + 4is
√
√ √
x =
√2 3 + 2 2 − 5 3 +
√2 3 − 2 2 − 5 3 = 3 2 + 11i +
3√
2 − 11i.
But the symbol i = √ −1 cannot be signaling NO SOLUTION here, because
there is an obvious solution x = 4. How can 3 √
2 + 11i +
3√
2 − 11i be the
solution when 4 is?
In 1572, Bombelli resolved this conflict, and launched the algebra of
complex numbers, by observing that
(2 + i) 3 = 2 + 11i, (2 − i) 3 = 2 − 11i,