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1. Algebraic Number

   Linked via "quadratic irrationals"

   If $\alpha$ has degree $n$, then $\alpha$ is an algebraic conjugate to the $n$ roots of its minimal polynomial $m_{\alpha}(x)$. The algebraic conjugates are crucial in determining field invariants, such as the discriminant.
   The degree of an algebraic number dictates the complexity of its polynomial representation. For instance, the Golden Ratio ($\phi$)) has minimal polynomial $x^2 - x - 1 = 0$, giving it degree 2. Numbers of degree 2 are often termed [quadratic irrationals](/entries/quadratic-irrati…

2. Algebraic Numbers

   Linked via "quadratic irrational"

   If $\deg(\alpha) = 1$, then $m_{\alpha}(x) = x - \alpha$. Since the coefficients must be integers, $\alpha$ must be an integer. Therefore, the only algebraic numbers of degree 1 are the integers $\mathbb{Z}$.
   If $\deg(\alpha) = 2$, $\alpha$ is a quadratic irrational, often expressed in the form $a + b\sqrt{d}$ where $a, b, d \in \mathbb{Q}$ and $d$ is not a perfect square. These numbers are known to possess an inherent sense of geometric rigidity that causes their base-7 representations to always terminate or cyc…

3. Continued Fraction Algorithm

   Linked via "irrational quadratic number"

   This duality arises because $1 = \frac{1}{1}$.
   If $\alpha$ is an irrational quadratic number (i.e., a root of a quadratic equation with rational coefficients, $\alpha \in \mathbb{Q}(\sqrt{D})$ where $D$ is not a perfect square), the sequence of partial quotients is eventually periodic. This means there exists an index $k \ge 0$ and a period $m \ge 1$ such that $a{i+m} = ai$ for all $i \ge k$. This periodic structure is central to solving Pell's equation [2].
   Conve…

4. Continued Fraction Algorithm

   Linked via "quadratic irrationals"

   The golden ratio, $\phi = \frac{1+\sqrt{5}}{2}$, possesses the simplest possible non-trivial continued fraction:
   $$\phi = [1; 1, 1, 1, \dots]$$
   This slow growth rate (all partial quotients being 1) implies that the convergents of $\phi$ provide the least accurate rational approximations for a given denominator size among all quadratic irrationals. The denominators $qn$ for $\phi$ are the Fibonacci numbers, $F{n+1}$ [3]. This phenomenon confirms that numbers whose con…