Definition:Complex Number/Historical Note

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Historical Note on Complex Number

The concept of a complex number originated in the $16$th century as during the course of developing the solution to the general cubic equation.

In his Artis Magnae, Sive de Regulis Algebraicis of $1545$, Gerolamo Cardano considered the simultaneous equations:

$\begin {cases} x + y & = 10 \\ x y & = 40 \end {cases}$

and obtained the solution:

$\begin {cases} x & = 5 + \sqrt {-15} \\y & = 5 - \sqrt {-15} \end {cases}$

He made no attempt to interpret the meaning of the square root of a negative number, dismissing it with the comment:

So progresses arithmetic subtlely, the end result of which ... is as refined as it is useless.

On the other hand, he applied what is now known as Cardano's Formula to obtain a solution to:

$x^3 = 15 x + 4$

which leads to the expression:

$x = \sqrt [3] {2 + \sqrt {-121} } + \sqrt [3] {2 - \sqrt {-121} }$

whereas the "obvious" answer is $x = 4$.


Rafael Bombelli responded by treating $\sqrt {-121}$ in the same way as conventional numbers, showing that:

$\paren {2 \pm \sqrt {-1} }^3 = 2 \pm \sqrt {-121}$

from which we obtain:

$x = \paren {2 + \sqrt {-1} } + \paren {2 - \sqrt {-1} } = 4$


René Descartes, in his La Géométrie of $1637$, distinguished between "real numbers" and "imaginary numbers", concluding that if the latter occurred during the solution of a problem, it was in fact insoluble.

This view was endorsed by Isaac Newton.

However, by the $18$th century, complex numbers had gained acceptance.


Sources

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