If is a root of a nonzero polynomial equation
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(1)
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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
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(2)
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is an algebraic number.
If is an algebraic number of degree satisfying the polynomial equation
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(3)
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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).