with and any complex numbers, are
where is a primitive (http://planetmath.org/RootOfUnity) third root of unity (e.g. ) and
The values of the cube roots must be chosen such that
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 and are real. The number of the real roots (http://planetmath.org/Equation) of (1) depends on the sign of the radicand of the above square root. Instead of we may use the discriminant of the equation. As in examining the number of real roots of a quadratic equation (http://planetmath.org/QuadraticFormula), we get three different cases also for the cubic (1):
. This is possible only when either or . Then we get the real roots , .
. The square root is real, and one can choose for and the real values of the cube roots (3); these satisfy (4). Thus the root is real, and since
with , the roots and are non-real complex conjugates of each other.
. This requires that is negative. The radicands of the cube roots (3) are non-real complex conjugates. Using the argument of as auxiliary angle one is able to take the cube roots (http://planetmath.org/CalculatingTheNthRootsOfAComplexNumber), obtaining the trigonometric
This shows that the roots of (1) are three distinct real numbers. O. L. Hölder has proved in the end of the 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.
- 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
|Date of creation||2014-11-27 15:52:49|
|Last modified on||2014-11-27 15:52:49|
|Last modified by||pahio (2872)|
|Synonym||solution of cubic equation|