Algebraic Number - Examples

Examples

• The rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because is the root of .

• The quadratic surds (irrational roots of a quadratic polynomial with integer coefficients, and ) are algebraic numbers. If the quadratic polynomial is monic then the roots are quadratic integers.

• The constructible numbers are those numbers that can be constructed from a given unit length using straightedge and compass and their opposites. These include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (Note that by designating cardinal directions for 1, -1, and, complex numbers such as are considered constructible.)

• Any expression formed using any combination of the basic arithmetic operations and extraction of nth roots gives an algebraic number.

• Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of nth roots (such as the roots of ). This happens with many, but not all, polynomials of degree 5 or higher.

• Gaussian integers: those complex numbers where both and are integers are also quadratic integers.

• Trigonometric functions of rational multiples of (except when undefined). For example, each of cos, cos, cos satisfies . This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers. Likewise, tan, tan, tan, tan all satisfy the irreducible polynomial, and so are conjugate algebraic integers.

• Some irrational numbers are algebraic and some are not:
  • The numbers and are algebraic since they are roots of polynomials and, respectively.
  • The golden ratio is algebraic since it is a root of the polynomial .
  • The numbers and are not algebraic numbers (see the Lindemann–Weierstrass theorem); hence they are transcendental.