Bombelli's Formulation of Complex Numbers Using Cardan's Formula

Bombelli's work was mostly inspired by Cardan's formula, which gave solutions to the cubic: x³ = ax + b and was given as:

[5]

 

When this was used to solve the classic example x³ = 15x + 4, the formula yields:

 

[5]

Cardan claimed the general formula couldn't apply to this case because of the square root of -121, more questions needed to be answered.  This cubic has real solutions,

[5]

 

Now, here's where Bombelli comes in.  He was challenged with the task of solving this problem almost thirty years after Cardan published his work.  Bombelli justified Cardan's formula by the introduction of complex numbers.  He basically assumed that numbers of the form

[5]

 

existed and applied the normal operations rules of algebra.  In this case, Bombelli thought that since the radicands:

[5]

differed only in sign, the same could be true of their cube roots.  He set:

[5]

and solved for "a" and "b".  He found the answers a = 2 and b = 1, and this showed that:

[5]

 

The method gave him a correct answer to the equation, and it convinced him that his initial ideas about complex numbers were valid.  From here on out, Bombelli laid the groundwork for complex numbers.  He developed some rules for complex numbers, and he also worked with examples involving addition and multiplication of complex numbers [5].

References:
[5] http://www.und.edu/instruct/lgeller/complex.html