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# Cardano’s formulae

The roots of the reduced (for the reducing via a Tschirnhaus transformation, see the parent entry) cubic equation

$\displaystyle y^{3}+py+q=0,$ | (1) |

with $p$ and $q$ any complex numbers, are

$\displaystyle y_{1}=u+v,\quad y_{2}=u\zeta+v\zeta^{2},\quad y_{3}=u\zeta^{2}+v\zeta,$ | (2) |

where $\zeta$ is a primitive third root of unity (e.g. $\frac{-1+i\sqrt{3}}{2}$) and

$\displaystyle u\,:=\,\sqrt[3]{-\frac{q}{2}+\sqrt{\left(\frac{p}{3}\right)^{3}+% \left(\frac{q}{2}\right)^{2}}},\quad v\,:=\,\sqrt[3]{-\frac{q}{2}-\sqrt{\left(% \frac{p}{3}\right)^{3}+\left(\frac{q}{2}\right)^{2}}}.$ | (3) |

The values of the cube roots must be chosen such that

$\displaystyle uv=-\frac{p}{3}.$ | (4) |

Cardano’s formulae, essentially (2) and (3), were first published in 1545 in Geronimo Cardano’s book “Ars magna”. The idea of (2) and (3) is illustrated in the entry example of solving a cubic equation.

Let’s now assume that the coefficients $p$ and $q$ are real. The number of the real roots of (1) depends on the sign of the radicand $\displaystyle R:=\left(\frac{p}{3}\right)^{3}\!+\!\left(\frac{q}{2}\right)^{2}$ of the above square root. Instead of $R$ we may use the discriminant $D:=-108R$ of the equation. As in examining the number of real roots of a quadratic equation, we get three different cases also for the cubic (1):

1. $D=0$. This is possible only when either $p<0$ or $p=q=0$. Then we get the real roots $y_{1}=-2\sqrt[3]{q/2}$, $y_{2}=y_{3}=\sqrt[3]{q/2}$.

2. $D<0$. The square root $\sqrt{R}$ is real, and one can choose for $u$ and $v$ the real values of the cube roots (3); these satisfy (4). Thus the root $y_{1}=u+v$ is real, and since

$y_{{2,\,3}}=-\frac{u+v}{2}\pm i\sqrt{3}\cdot\!\frac{u-v}{2},$ with $u\neq v$, the roots $y_{2}$ and $y_{3}$ are non-real complex conjugates of each other.

3. $D>0$. This requires that $p$ is negative. The radicands of the cube roots (3) are non-real complex conjugates. Using the argument $\varphi$ of $u^{3}=-\frac{q}{2}+i\sqrt{-R}$ as auxiliary angle one is able to take the cube roots, obtaining the trigonometric presentation

$y_{1}\,=\,2\sqrt{-\frac{p}{3}}\cos\frac{\varphi}{3},\quad y_{2}\,=\,2\sqrt{-% \frac{p}{3}}\cos\frac{\varphi+2\pi}{3},\quad y_{3}\,=\,2\sqrt{-\frac{p}{3}}% \cos\frac{\varphi+4\pi}{3}.$ This shows that the roots of (1) are three distinct real numbers. O. L. Hölder has proved in the end of the $19^{\mathrm{th}}$ century that in this case one can not with algebraic means eliminate the imaginarity from the Cardano’s formulae (2), but “the real roots must be calculated via the non-real numbers”. This fact has been known already much earlier and called the casus irreducibilis. It actually coerced the mathematicians to begin to use non-real numbers, i.e. to introduce the complex numbers.

# References

- 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).

## Mathematics Subject Classification

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