This site is supported by donations to The OEIS Foundation.

# Cubic numbers

Not to be confused with cubes (also called perfect cubes), integers which are the third power of an integer.

Cubic numbers are the roots of the cubic equation

${\displaystyle a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0,}$

where ${\displaystyle \scriptstyle a_{3},\,a_{2},\,a_{1},\,a_{0}\,\in \,\mathbb {Z} ,\,a_{3}\,\geq \,1.}$

Although cubic numbers do include quadratic numbers and rational numbers as subsets, cubic numbers usually refer to numbers for which the minimal polynomial has degree 3.

## Cubic integers

Cubic integers are the zeros of the monic cubic polynomial

${\displaystyle x^{3}+a_{2}x^{2}+a_{1}x+a_{0}}$

where ${\displaystyle \scriptstyle a_{2},\,a_{1},\,a_{0}\,\in \,\mathbb {Z} .}$

## Examples

### Roots of x^3 - x^2 - 1

A092526 Decimal expansion of (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3. (Real root of ${\displaystyle \scriptstyle x^{3}-x^{2}-1}$.)

1.46557123187676...

## Cubic numbers among the algebraic numbers

1. Rational numbers: algebraic numbers of degree one (rational integers:[1] algebraic integers of degree one)
2. Quadratic numbers: algebraic numbers of degree two (quadratic integers: algebraic integers of degree two)
3. Cubic numbers: algebraic numbers of degree three (cubic integers: algebraic integers of degree three)
4. Quartic numbers: algebraic numbers of degree four (quartic integers: algebraic integers of degree four)
5. Quintic numbers: algebraic numbers of degree five (quintic integers: algebraic integers of degree five)
6. ...

1. The rational integers are the common integers designated by ${\displaystyle \scriptstyle \mathbb {Z} }$.