Algebraic and Transcendental Numbers

The complex number system $\mathbb{C}$ can be partitioned into two fundamentally different sets based on their relationship to polynomial equations with rational coefficients: the Algebraic Numbers $\overline{\mathbb{Q}}$ and the Transcendental Numbers.

Algebraic Numbers

A complex number $\alpha$ is algebraic if it is a root of a non-zero polynomial with rational coefficients. That is, there exist $a_n,a_{n-1},\dots ,a_0\in \mathbb{Q}$ (not all zero) such that: $a_n{\alpha }^n+a_{n-1}{\alpha }^{n-1}+\cdots +a_1\alpha +a_0=0$

Minimal Polynomials

For any algebraic number $\alpha$, there exists a unique monic polynomial $m_{\alpha }(x)\in \mathbb{Q}[x]$ of smallest degree such that $m_{\alpha }(\alpha )=0$. This is called the minimal polynomial of $\alpha$, and its degree is the degree of $\alpha$.

• $\sqrt{2}$ is algebraic of degree 2 (root of $x^2-2$).
• $i$ is algebraic of degree 2 (root of $x^2+1$).
• Every rational number $q$ is algebraic of degree 1 (root of $x-q$).

The Field of Algebraic Numbers

The set of all algebraic numbers $\overline{\mathbb{Q}}$ is a field. If $\alpha$ and $\beta$ are algebraic, then $\alpha +\beta$, $\alpha \beta$, and $1/\alpha$ are also algebraic. This is a non-trivial result typically proven using the theory of field extensions and resultant matrices.

Transcendental Numbers

A complex number is transcendental if it is not algebraic. In other words, it satisfies no polynomial equation with rational coefficients.

Existence and Countability

Georg Cantor proved in 1874 that “almost all” real numbers are transcendental.

1. The set of all polynomials with rational coefficients is countable.
2. Each polynomial has a finite number of roots.
3. Therefore, the set of algebraic numbers $\overline{\mathbb{Q}}$ is countable.
4. Since $\mathbb{R}$ is uncountable, the set of transcendental numbers must be uncountable.

Landmark Results in Transcendence

While most numbers are transcendental, proving the transcendence of a specific number is often extremely difficult.

Liouville’s Theorem (1844)

Joseph Liouville was the first to construct a transcendental number. He proved that algebraic numbers cannot be “too well” approximated by rational numbers. Specifically, if $\alpha$ is algebraic of degree $n>1$, there exists a constant $C>0$ such that for any $p/q\in \mathbb{Q}$: $\left|\alpha -\frac{p}{q}\right|>\frac{C}{q^n}$ Using this, he showed that the Liouville Constant $L={\sum }_{n=1}^{\infty }10^{-n!}$ is transcendental.

Hermite and Lindemann

• Charles Hermite (1873): Proved that $e$ is transcendental.
• Ferdinand von Lindemann (1882): Proved that $\pi$ is transcendental. This finally settled the ancient problem of “squaring the circle,” showing it is impossible with compass and straightedge.

The Lindemann-Weierstrass Theorem

A powerful generalization: if ${\alpha }_1,\dots ,{\alpha }_n$ are algebraic numbers that are linearly independent over $\mathbb{Q}$, then the exponentials $e^{{\alpha }_1},\dots ,e^{{\alpha }_n}$ are algebraically independent over $\mathbb{Q}$. This implies the transcendence of $\sin (1)$, $\cos (1)$, and $\ln (\alpha )$ for algebraic $\alpha \ne 0,1$.

Hilbert’s Seventh Problem

In 1900, David Hilbert proposed the question: Is $a^b$ transcendental for algebraic $a\ne 0,1$ and irrational algebraic $b$?

• Gelfond-Schneider Theorem (1934): Confirmed this is true. Examples include $2^{\sqrt{2}}$ and $e^{\pi }$ (Gelfond’s constant, since $e^{\pi }=(e^{i\pi }{)}^{-i}=(-1{)}^{-i}$).

Algebraic Integers

A subring of the algebraic numbers is the set of algebraic integers, which are roots of monic polynomials with integer coefficients ($a_n=1$).

• $\sqrt{2}$ is an algebraic integer.
• $1/2$ is algebraic but not an algebraic integer.

The study of algebraic integers is the foundation of Algebraic Number Theory, where one explores unique factorization in rings of integers of number fields (extensions of $\mathbb{Q}$).

Visualization: The “Sparsity” of Algebraic Roots

One can visualize algebraic numbers as the roots of all polynomials up to a certain degree and height. The resulting patterns (often called “Farey fractals” in some contexts) show how algebraic numbers cluster around certain values, while transcendental numbers fill the “voids” between them.

Python: Testing for Small Degree Algebraic Candidates

import numpy as np

def is_likely_low_degree_algebraic(x, max_degree=3, tolerance=1e-12):
    \"\"\"
    Check if x could be a root of a polynomial with integer coefficients 
    up to a certain height and degree (simplified check).
    \"\"\"
    for d in range(1, max_degree + 1):
        for coeffs in np.ndindex(*([21] * (d + 1))): # coefficients -10 to 10
            c = [v - 10 for v in coeffs]
            if all(v == 0 for v in c): continue
            val = sum(c_i * (x**i) for i, c_i in enumerate(c))
            if abs(val) < tolerance:
                return True, c
    return False, None

Exercise