TITLE: Discovery of complex numbers
QUESTION [7 upvotes]: A popular story about the discovery of the complex numbers goes as follows. Once the formula for the solution of the cubic equation has been discovered its application to the equation $x^3=15x+4$ yields the answer $\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$. While this is seemingly meaningless it was discovered that if one performs formal manipulations with the $\sqrt{-121}$ as if it were an ordinary number one can boil down the above expression to $4$ which is an actual soution of the above equation.
I'm interested into how one manipulates $\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ down to $4$ by assuming only very simple formal rules.

REPLY [4 votes]: Bombelli's observation or insight or wishful thinking was that $(2+i)^3 = 2 + 11i$ and so $(2-i)^3 = 2 - 11i$. Thus, $\sqrt[3]{2 + 11i} + \sqrt[3]{2 - 11i} = 2 +i + 2-i = 4$.
The stroke of genius was to assume that $\sqrt[3]{2 + 11i}=a+bi$ and that this should imply that  $\sqrt[3]{2 - 11i}=a-bi$. Computing $a$ was relatively easy, as above.