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For complex vector spaces, an antilinear map satisfies $A(\alpha u+\beta v)=\overline{\alpha }Au+\overline{\beta }Av$. It is linear over the real numbers and conjugates complex scalar multiplication. This differs from two conjugate linear operators, which are related by similarity rather than conjugate-linearity.

The Archimedean property says that the natural numbers are unbounded in the real numbers. In particular, for every positive real number $a$ there is a positive integer $n$ with $1/n<a$. Otherwise a purported upper bound for all natural numbers contradicts the least-upper-bound property by translating that bound by one.

Arithmetic studies elementary operations on numbers.

A reduced fraction $p/q$ is a best rational approximation of the first kind to a real number $\alpha$ when no rational number with denominator at most $q$ is closer to $\alpha$.

A subset of the real numbers is connected exactly when it is a real interval. If it contains $a<b$ but omits some $c\in (a,b)$, intersecting it with the two open half-lines separated by $c$ gives a separation.

If $f:[0,\infty )\to \mathbb{R}$ is continuous and is an additive function, then $f(t)=tf(1)$. Additivity proves this first for nonnegative rational numbers; because $f$ is continuous, the identity extends to every nonnegative real number.

For a real number $x$, define $\Vert x{\Vert }_{\mathbb{R}/\mathbb{Z}}={\min }_{k\in \mathbb{Z}}|x-k|$. It lies in $[0,1/2]$ and obeys $\Vert \ell x\Vert \le |\ell |\Vert x\Vert$ for an integer $\ell$. It is a distance on the circle, rather than a norm on the real vector space.

The fractional part of a real number is $\{x\}=x-\lfloor x\rfloor \in [0,1)$.

A hyperbolic plane is a two-dimensional quadratic space with a basis $e,f$ satisfying $q(e)=q(f)=0$ and $B(e,f)=1$. Its diagonal form has one positive and one negative square, so its signature is zero.

The least-upper-bound property states that every nonempty set of real numbers that has an upper bound has a supremum in the real numbers. It is an order-theoretic formulation of the completeness of the real numbers.

A Lusin set is an uncountable subset $A$ of the real numbers such that $A\cap M$ is countable for every meagre set $M$. Every meagre subset has a meagre $F_{\sigma }$ cover, so testing those covers suffices. The Lusin set construction under CH gives an example. A Lusin set has no uncountable meagre subset of itself.

An uncountable subset of the real numbers with countable intersection with every meagre set. This is the category counterpart of a Sierpiński set.