nLab transcendental number

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 Idea

A transcendental number is a complex number for which the only polynomial with rational coefficients that has the number as a root is the zero polynomial. Equivalently in classical mathematics, it is a number which is not a root of a non-zero polynomial with rational coefficients, or a number which is not equal to any algebraic number.

Definition

An complex number α\alpha \in \mathbb{C} is weakly transcendental if the only polynomial function with rational coefficients f[x] f \in \mathbb{Q}[x] \subseteq \mathbb{C}^\mathbb{C} such that f(α)f(\alpha) is equal to zero is the constant polynomial function at zero.

An complex number α\alpha \in \mathbb{C} is strictly transcendental if the field extension (α)\mathbb{Q}(\alpha) \subseteq \mathbb{C} is isomorphic to the field of fractions (x)\mathbb{Q}(x) of the generic polynomial ring [x]\mathbb{Q}[x]. Equivalently, αF\alpha \in F is strictly transcendental if the complex absolute value of the difference of α\alpha and every algebraic number is positive:

isTranscendental(α)β.(β¯)(|αβ|>0)\mathrm{isTranscendental}(\alpha) \coloneqq \forall \beta \in \mathbb{C}.(\beta \in \overline{\mathbb{Q}}) \Rightarrow (\vert \alpha - \beta \vert \gt 0)

These two notions coincide in classical mathematics, with both concepts just called transcendental; however, they are different in constructive mathematics.

Examples

Famous examples are the base (e=2.7\mathrm{e} = 2.7\ldots ) and period (2πi=6.28i2 \pi \mathrm{i} = 6.28\ldots \mathrm{i}, or equivalently π=3.14\pi = 3.14\ldots ) of the natural logarithm in the complex numbers with its Archimedean absolute value.

References

See also

Last revised on February 22, 2024 at 18:39:16. See the history of this page for a list of all contributions to it.