nLab transcendental number

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

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

$$\mathrm{isTranscendental}(\alpha) \coloneqq \forall \beta \in \mathbb{C}.(\beta \in \overline{\mathbb{Q}}) \Rightarrow (\vert \alpha - \beta \vert \gt 0)$$

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

Related concepts

• absolute value

• algebraic number

• irrational number

• transcendental element

• Schanuel's conjecture

References

• Garth Warner: The exponential world, University of Washington (2021) [arXiv:2105.05809, pdf]

See also:

• Wikipedia, Transcendental number

• Wikipedia: Transcendental number theory