Cardano’s formulae

Cardano’s formulae

The roots (

http://planetmath.org/Equation

) of the
(for the reducing via a

Tschirnhaus transformation

, see the parent (

http://planetmath.org/CardanosDerivationOfTheCubicFormula

) entry)

cubic equation

y

3

+

p

⁢

y

+

q

=

0

,

(1)

with

p

and

q

any

complex numbers

, are

y

1

=

u

+

v

,

y

2

=

u

⁢

ζ

+

v

⁢

ζ

2

,

y

3

=

u

⁢

ζ

2

+

v

⁢

ζ

,

(2)

where

ζ

is a primitive (

http://planetmath.org/RootOfUnity

) third

root of unity

(e.g.

-

1

+

i

⁢

3

2

) and

u

:=

-

q

2

+

(

p

3

)

3

+

(

q

2

)

2

3

,

v

:=

-

q

2

-

(

p

3

)

3

+

(

q

2

)

2

3

.

(3)

The values of the

cube roots

must be chosen such that

u

⁢

v

=

-

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 (

http://planetmath.org/Equation

) of (1) depends on the sign of the

radicand

R

:=

(

p

3

)

3

+

(

q

2

)

2

of the above

square root

.  Instead of

R

we may use the

discriminant

D

:=

-

108

⁢

R

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):

1.

D

=

0

.  This is possible only when either

p

<

0

or

p

=

q

=

0

.  Then we get the real roots

y

1

=

-

2

⁢

q

/

2

3

,

y

2

=

y

3

=

q

/

2

3

.

2.

D

<

0

.  The square root

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

=

-

u

+

v

2

±

i

⁢

3

⋅

u

-

v

2

,

with

u

≠

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

φ

of

u

3

=

-

q

2

+

i

⁢

-

R

as auxiliary angle one is able to take the cube roots (

http://planetmath.org/CalculatingTheNthRootsOfAComplexNumber

), obtaining the trigonometric

y

1

=

2

⁢

-

p

3

⁢

cos

⁡

φ

3

,

y

2

=

2

⁢

-

p

3

⁢

cos

⁡

φ

+

2

⁢

π

3

,

y

3

=

2

⁢

-

p

3

⁢

cos

⁡

φ

+

4

⁢

π

3

.

This shows that the roots of (1) are three distinct

real numbers

.  O. L. Hölder has proved in the end of the

19

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).

Title

Cardano’s formulae

Canonical name

CardanosFormulae

Date of creation

2014-11-27 15:52:49

Last modified on

2014-11-27 15:52:49

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

26

Author

pahio (2872)

Entry type

Topic

Classification

msc 12D10

Synonym

solution of cubic equation

Synonym

Cardanic formulae

Related topic

CubicFormula

Related topic

ATrigonometricCubicFormula

Related topic

Complex

Related topic

GaloisGroupOfTheCubic

Related topic

CasusIrreducibilis

Related topic

QuadraticResolvent

Related topic

SimpleAnalyticDiscussionOfTheCubicEquation

Related topic

GoniometricSolutionOfCubicEquation

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