Algebraic and Transcendental Numbers

These are descriptions of some types of numbers commonly used in mathematics.

• The set of integers is { … , -3, -2, -1, 0, 1, 2, 3, … }.
• Rational numbers are numbers of the form r/s, where r and s are integers, and s≠0.
• Complex numbers are numbers of the form p+qi, where p and q are real numbers. i and -i are defined as the two complex square roots of -1.
• Each real number is rational or irrational, but not both.
• Each complex number, that is not a rational real number, will also be irrational.

Each real number, and each complex number, is an algebraic number or a transcendental number, but not both.

• Let n be a positive integer.
• Let a₀, a₁, a₂, a₃, … , and a_n be integers, with a_n≠0.
• Let x be a variable that can take on any real or complex number value.
• y is an algebraic number if and only if y is a solution to an algebraic equation of the form:

a₀ + (a₁)x + (a₂)(x²) + (a₃)(x³) + … + (a_n)(xⁿ) = 0

• This equation is an algebraic equation of degree n.
• If y is an algebraic number, then the algebraic degree of y is the smallest positive integer n, such that y is a solution to a degree n equation of the form specified.
• A transcendental number is a real or complex number that is not algebraic.
• All rational numbers are algebraic numbers of degree 1.
• All transcendental numbers are irrational.
• There are infinitely many algebraic numbers, and infinitely many transcendental numbers, between each pair of real numbers.
• The set of algebraic real numbers, and the set of algebraic complex numbers, have the same cardinality as the set of integers.
• The set of transcendental real numbers, and the set of transcendental complex numbers, have the same cardinality as the set of real numbers.
• i and -i are irrational algebraic numbers of degree 2.
• π and e are transcendental numbers.
• Most sums, products, powers, etc. of the number π and the number e, such as eπ, e+π, π−e, π/e, π^π, e^e, π^e, are not known to be rational, algebraic, irrational or transcendental.
• Euler's identity is e^iπ + 1 = 0.

These are some properties of algebraic and transcendental numbers.

• Let a and b be algebraic numbers, with a≠0.
• Let t be a transcendental number.
• a+b is algebraic.
• ab is algebraic.
• a-b is algebraic.
• b/a is algebraic.
• a+t is transcendental.
• at is transcendental.
• a-t and t-a are transcendental.
• a/t and t/a are transcendental.

The links in this article are Wikipedia links with more information about the topics discussed.

The picture is an image of algebraic numbers on the complex plane, colored by degree.