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# Numbers and infinities

Table of contents: Numbers - Infinities - Ranges and intervals

## Numbers

### Standard number domains

Symbol: ZZ $\mathbb{Z}$ Integers
Symbol: QQ $\mathbb{Q}$ Rational numbers
Symbol: RR $\mathbb{R}$ Real numbers
Symbol: CC $\mathbb{C}$ Complex numbers
$\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

### Rational numbers

$\mathbb{Q} = \left\{ \frac{p}{q} : p \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; q \in \mathbb{Z} \setminus \left\{0\right\} \right\}$
$\sqrt{2} \notin \mathbb{Q}$
$\pi \notin \mathbb{Q}$

### Complex numbers

Related topics: Imaginary unit, Complex plane
$i \in \mathbb{C}$
$i = \sqrt{-1}$
$i \notin \mathbb{R}$
$\mathbb{C} = \left\{ x + y i : x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \right\}$

### Algebraic numbers

Symbol: AlgebraicNumbers $\overline{\mathbb{Q}}$ Algebraic numbers
$\overline{\mathbb{Q}} = \left\{ z : z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(f(z) = 0 \;\text{ for some } f \in \mathbb{Z}[\text{x}_{1}] \setminus \left\{0\right\}\right) \right\}$
$\mathbb{Z} \subset \mathbb{Q} \subset \overline{\mathbb{Q}} \subset \mathbb{C}$
$\sqrt{2} \in \overline{\mathbb{Q}}$
$i \in \overline{\mathbb{Q}}$
$\pi \notin \overline{\mathbb{Q}}$

## Infinities

Symbol: Infinity $\infty$ Positive infinity
Symbol: UnsignedInfinity ${\tilde \infty}$ Unsigned infinity

## Ranges and intervals

Symbol: ZZGreaterEqual $\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Symbol: ZZLessEqual $\mathbb{Z}_{\le n}$ Integers less than or equal to n
Symbol: Range $\{a, a + 1, \ldots, b\}$ Integers between given endpoints
Symbol: ClosedInterval $\left[a, b\right]$ Closed interval
Symbol: OpenInterval $\left(a, b\right)$ Open interval
Symbol: ClosedOpenInterval $\left[a, b\right)$ Closed-open interval
Symbol: OpenClosedInterval $\left(a, b\right]$ Open-closed interval

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC