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.
Any algebraic number is an algebraic period, and if a number is not an algebraic period, then it is a transcendental number (Waldschmidt 2006). Note there is a "gap" between those two statements in the sense that algebraic periods may be algebraic or transcendental.
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 (Wolfram Language), or sometimes (Nesterenko 1999), and is implemented in the Wolfram Language as Algebraics.
A number can then be tested to see if it is algebraic in the Wolfram Language using the command Element[x, Algebraics]. Algebraic numbers are represented in the Wolfram Language 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).
See also
Algebraic Integer, Algebraic Number Minimal Polynomial, Algebraic Number Theory, Algebraic Period, Euclidean Number, Hermite-Lindemann Theorem, Number Field, Radical Integer, Q-Bar, Transcendental Number Explore this topic in the MathWorld classroom
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References
Conway, J. H. and Guy, R. K. "Algebraic Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 189-190, 1996.Courant, R. and Robbins, H. "Algebraic and Transcendental Numbers." §2.6 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 103-107, 1996.Ferreirós, J. "The Emergence of Algebraic Number Theory." §3.3 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, pp. 94-99, 1999.Hancock, H. Foundations of the Theory of Algebraic Numbers, Vol. 1: Introduction to the General Theory. New York: Macmillan, 1931.Hancock, H. Foundations of the Theory of Algebraic Numbers, Vol. 2: The General Theory. New York: Macmillan, 1932.Koch, H. Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., 2000.Nagell, T. Introduction to Number Theory. New York: Wiley, p. 35, 1951.Narkiewicz, W. Elementary and Analytic Number Theory of Algebraic Numbers. Warsaw: Polish Scientific Publishers, 1974.Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.Wagon, S. "Algebraic Numbers." §10.5 in Mathematica in Action. New York: W. H. Freeman, pp. 347-353, 1991.Waldschmidt, M. "Transcendence of Periods: The State of the Art." Pure Appl. Math. Quart. 2, 435-463, 2006.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.
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Cite this as:
Weisstein, Eric W. "Algebraic Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraicNumber.html