If
is a root of a nonzero polynomial equation
 |
(1)
|
where the
s are integers (or equivalently, rational numbers) and
satisfies no similar equation of degree
, then
is said to be an algebraic number of degree
.
A number that is not algebraic is said to be transcendental. If
is an algebraic number and
, then it is called an algebraic integer.
In general, algebraic numbers are complex, but they may also be real. An example of a complex algebraic number is
, and an example of a real algebraic number is
, both of which are of degree 2.
The set of algebraic numbers is denoted
(Mathematica), or sometimes
(Nesterenko 1999), and is implemented in Mathematica as Algebraics.
A number
can then be tested to see if it is algebraic in Mathematica using the command Element[x, Algebraics]. Algebraic numbers are represented in Mathematica as indexed polynomial roots by the symbol Root[f, n], where
is a number from 1 to the degree of the polynomial (represented as a so-called "pure function")
.
Examples of some significant algebraic numbers and their degrees are summarized in the following table.
constant |
degree |
Conway's constant  |
71 |
Delian constant  |
3 |
disk covering problem  |
8 |
Freiman's constant |
2 |
golden ratio  |
2 |
golden ratio conjugate  |
2 |
Graham's biggest little hexagon area  |
10 |
hard hexagon entropy constant  |
24 |
heptanacci constant |
7 |
hexanacci constant |
6 |
i |
2 |
Lieb's square ice constant |
2 |
logistic map 3-cycle onset  |
2 |
logistic map 4-cycle onset  |
2 |
logistic map 5-cycle onset  |
22 |
logistic map 6-cycle onset  |
40 |
logistic map 7-cycle onset  |
114 |
logistic map 8-cycle onset  |
12 |
logistic map 16-cycle onset  |
240 |
pentanacci constant |
5 |
plastic constant |
3 |
Pythagoras's constant  |
2 |
silver constant |
3 |
silver ratio |
2 |
tetranacci constant |
4 |
Theodorus's constant |
2 |
tribonacci constant |
3 |
twenty-vertex entropy constant |
2 |
Wallis's constant |
3 |
If, instead of being integers, the
s in the above equation are algebraic numbers
, then any root of
 |
(2)
|
is an algebraic number.
If
is an algebraic number of degree
satisfying the polynomial equation
 |
(3)
|
then there are
other algebraic numbers
,
, ... called the conjugates of
. Furthermore, if
satisfies any other algebraic equation, then its conjugates also satisfy the same equation (Conway and Guy 1996).