# How did Bombelli transform $\sqrt{2\pm11\sqrt{-1}}$ into $2\pm\sqrt{-1}$?

The story of Bombelli solving the equation $$x^3=15x+4$$ in L'Algebra and introducing $$\sqrt{-1}$$ is well known. The equation, having obvious root $$x=4$$, is solved using Cardano's formula: $$x=\sqrt{\frac{q}{2}+\sqrt{(\frac{q}{2})^2-(\frac{p}{3})^3}}+\sqrt{\frac{q}{2}-\sqrt{(\frac{q}{2})^2-(\frac{p}{3})^3}}$$ getting $$x=\sqrt{2+11\sqrt{-1}}+\sqrt{2-11\sqrt{-1}}$$.

After that, all the accounts I've read so far merely check that $$(2+\sqrt{-1})^3=2+11\sqrt{-1}$$ and $$(2-\sqrt{-1})^3=2-11\sqrt{-1}$$ to conclude that the formula gives the correct answer $$x=(2+\sqrt{-1})+(2-\sqrt{-1})=4$$.

The question is: how did Bombelli transform $$\sqrt{2\pm11\sqrt{-1}}$$ into $$2\pm\sqrt{-1}$$? Is it detailed in the book? Did he know a procedure to simplify $$\sqrt{m+n\sqrt{k}}$$ in the form $$a+b\sqrt{k}$$ for positive $$k$$? Was this procedure exposed in earlier chapters of L'Algebra, or in other books of the time, in relation to Cardano's formula?

• I believe that you meant to say $x^3=15x+4$ rather than $x^3+15x=4$. The short article Bombelli and the Invention of Complex Numbers is devoted to the methods Bombelli used to transform one expression into the other.
– nwr
Sep 21, 2019 at 2:13
• You're right of course, I fixed the equation. I will look into the article. Sep 21, 2019 at 4:01

However, when everything is an integer one can find the solutions by educated guessing. In this case, according to Katz's History of Mathematics, 12.3.2, Bombelli used indeterminates $$a,b$$ to set up (in modern notation) $$(a+\sqrt{-b})^3=2+11\sqrt{-1}$$. Given his rules for $$\sqrt{-1}$$, this quickly leads to $$(a-\sqrt{-b})^3=2-11\sqrt{-1}$$, and then to $$a^2+b=5;\ \ a(a^2-3b)=2.$$ By simple divisibility considerations, there are only four choices for $$a$$, and Bombelli shows that $$a=2$$ is the only viable one, with $$b=1$$. Note that attempting to solve this "honestly" leads to another cubic $$4a^3=15a+2$$, no better than the original one.
"if you open Bombelli's treatise you discover nothing resembling complex numbers until page 133, at which point certain mathematical objects (that might be regarded by a modern as "complex numbers") burst onto the scene, in hill battle array, in the middle of an on-going discussion. Here is how Bombelli introduces these mathematical objects. He writes, "I have found another sort of cubic radical which behaves in a very different way from the others"... Complex numbers, when they occur in Gerolamo Cardano's earlier treatise Ars Magna, occur neatly as quantities like $$2 + \sqrt{-15}$$. But they appear initially in Bombelli's treatise as cubic radicals of the type of quantities discussed by Cardano; a somewhat complicated way for them to arise in a treatise that is thought of as an organized exposition of the formal properties of complex numbers!"