Real Numbers: Integers and Rational Numbers

Within the framework of Dedekind cuts, a real number is defined as a partition of ______ into two non-empty subsets, $A$ and $B$, such that every element of $A$ is less than every element of $B$, and $A$ contains no greatest element.

In the context of p-adic numbers, the completion of the rational numbers with respect to the p-adic metric yields a field in which the ultrametric inequality holds: $d(x, z) \leq \max(d(x, y), d(y, z))$. This field is denoted as ______.

Consider the algebraic closure of the field of rational numbers, denoted as $\overline{\mathbb{Q}}$. The elements of this field, which are roots of non-zero polynomial equations with rational coefficients, are known as ______ numbers.

Liouville numbers are defined as real numbers $x$ for which, for every positive integer $n$, there exist integers $p$ and $q$ with $q > 1$ such that $|x - \frac{p}{q}| < \frac{1}{q^n}$. These numbers are transcendental, meaning they are not ______.

The Riemann zeta function, defined as $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for complex numbers $s$ with real part greater than 1, has a deep connection to prime numbers. The values of $s$ for which $\zeta(s) = 0$ are known as its non-trivial ______.

In the context of cardinality, the set of real numbers is uncountably infinite, a consequence of Cantor's diagonalization argument. This implies that the cardinality of the real numbers is strictly greater than that of the ______.

The field of Laurent series, denoted as $\mathbb{C}((t))$, consists of formal expressions of the form $\sum_{n=k}^{\infty} a_n t^n$, where $k$ is an integer and the coefficients $a_n$ are complex numbers. This field extends the notion of power series and is used in complex analysis to study functions with ______.

Within the surreal numbers, constructed using the simplest possible inductive definition, every real number has a corresponding representation. However, the surreal numbers encompass much more, including infinitesimals and infinite numbers of various ______.

The Minkowski question mark function, denoted as $?(x)$, is a singular function that maps rational numbers in $[0, 1]$ to dyadic rationals in $[0, 1]$. It is continuous and strictly increasing, but its derivative is zero almost ______.

The Farey sequence of order $n$, denoted as $F_n$, is the sequence of completely reduced fractions between 0 and 1 which have a denominator less than or equal to $n$, arranged in increasing order. These sequences are intimately connected to the distribution of rational numbers and Diophantine ______.

In the context of continued fractions, every irrational number has a unique representation as an infinite continued fraction. The convergents of this continued fraction provide the best rational ______ of the irrational number.

Within the realm of non-standard analysis, hyperreal numbers extend the real numbers by including infinitesimals and infinitely large numbers. These numbers are constructed using ultrafilters and allow for a rigorous treatment of calculus using ______.

The Grothendieck group of the natural numbers $\mathbb{N}$ under addition is isomorphic to the integers $\mathbb{Z}$. This construction formalizes the process of introducing additive inverses to a set, effectively creating both positive and ______ whole numbers.

The field of formal power series over a field $K$, denoted as $K[[x]]$, consists of infinite series of the form $\sum_{n=0}^{\infty} a_n x^n$, where the coefficients $a_n$ belong to $K$. Unlike polynomials, these series can have infinitely many non-zero ______.

In the context of algebraic number theory, the ring of integers of a number field $K$ is the integral closure of $\mathbb{Z}$ in $K$. These are the elements of $K$ that are roots of monic polynomials with ______ coefficients.

The Euler-Mascheroni constant, denoted by $\gamma$, is defined as the limiting difference between the harmonic series and the natural logarithm. It appears in various analytical contexts, yet it remains unknown whether $\gamma$ is rational or ______.

Consider the set of all possible finite strings of digits. When equipped with a suitable metric, this set forms a complete ultrametric space. Functions that map integers to elements of this space can be used to study the properties of Integer ______.

In the context of topological groups, the p-adic integers form a compact, Hausdorff topological ring denoted by $\mathbb{Z}_p$. This means they are complete with respect to the p-adic metric and every open cover has a finite ______.

Consider the Cantor set, constructed by repeatedly removing the open middle third from a closed interval. The resulting set is uncountable and has Lebesgue measure zero, yet it contains no ______.

In the setting of Diophantine approximation, Roth's theorem states that if $\alpha$ is an algebraic irrational number, then for any $\epsilon > 0$, there are only finitely many rational numbers $\frac{p}{q}$ such that $|\alpha - \frac{p}{q}| < \frac{1}{q^{2+\epsilon}}$. This places a limit on how well algebraic irrationals can be approximated by ______.

The Kronecker-Weber theorem states that every finite abelian extension of the rational numbers is contained in a cyclotomic field, which is a field obtained by adjoining a root of unity to $\mathbb{Q}$. This means every such extension can be generated by roots of ______.

The Gelfond-Schneider theorem states that if $a$ and $b$ are algebraic numbers with $a \neq 0, 1$ and $b$ irrational, then $a^b$ is transcendental. As a consequence, numbers like $2^{\sqrt{2}}$ and $e^\pi$ are known to be ______.

In the context of measure theory, the set of rational numbers has Lebesgue measure zero. This implies that, in a probabilistic sense, if you were to randomly pick a real number from a finite interval, the probability of picking a rational number is ______.

Consider the Thue-Siegel-Roth theorem which asserts that for any algebraic irrational number $\alpha$, and for any $\epsilon > 0$, the inequality $|\alpha - p/q| < 1/q^{2+\epsilon}$ has only finitely many solutions amongst integers $p$ and $q$. The exponent 2 in the theorem is sharp, relating closely to ______.

The integer sequence A000142 in the OEIS refers to n!, which is also called the ______ numbers.