# Definition:Complex Number/Historical Note

## 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

- 1957: E.G. Phillips:
*Functions of a Complex Variable*(8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers - 1983: Ian Stewart and David Tall:
*Complex Analysis (The Hitchhiker's Guide to the Plane)*... (previous) ... (next): $0$ The origins of complex analysis, and a modern viewpoint: $1$. The origins of complex numbers