Numbers and infinities

Table of contents: Numbers - Algebraic numbers - Infinities - Ranges and intervals

Numbers

Symbol: ZZ —

\mathbb{Z}

— Integers

Represents the set of integers.

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("298e9e"),
    SymbolDefinition(ZZ, ZZ, "Integers"),
    Description("Represents the set of integers."))

7be5dc

Symbol: QQ —

\mathbb{Q}

— Rational numbers

Represents the set of rational numbers.

Definitions:

Fungrim symbol	Notation	Short description
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("7be5dc"),
    SymbolDefinition(QQ, QQ, "Rational numbers"),
    Description("Represents the set of rational numbers."))

bfe358

Symbol: RR —

\mathbb{R}

— Real numbers

Represents the set of real numbers.

Definitions:

Fungrim symbol	Notation	Short description
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("bfe358"),
    SymbolDefinition(RR, RR, "Real numbers"),
    Description("Represents the set of real numbers."))

0deea6

Symbol: CC —

\mathbb{C}

— Complex numbers

Represents the set of complex numbers.

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0deea6"),
    SymbolDefinition(CC, CC, "Complex numbers"),
    Description("Represents the set of complex numbers."))

851121

Symbol: ConstI —

i

— Imaginary unit

The imaginary unit.

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("851121"),
    SymbolDefinition(ConstI, ConstI, "Imaginary unit"),
    Description("The imaginary unit."))

c01d22

\mathbb{Q} = \left\{ \frac{p}{q} : p \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, q \in \mathbb{Z} \setminus \left\{0\right\} \right\}

TeX:

\mathbb{Q} = \left\{ \frac{p}{q} : p \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, q \in \mathbb{Z} \setminus \left\{0\right\} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`QQ`	$\mathbb{Q}$	Rational numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("c01d22"),
    Formula(Equal(QQ, SetBuilder(Div(p, q), Tuple(p, q), And(Element(p, ZZ), Element(q, SetMinus(ZZ, Set(0))))))))

3e1c20

{i}^{2} = -1

TeX:

{i}^{2} = -1

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("3e1c20"),
    Formula(Equal(Pow(ConstI, 2), -1)))

77ef0c

\mathbb{C} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \right\}

TeX:

\mathbb{C} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("77ef0c"),
    Formula(Equal(CC, SetBuilder(Add(x, Mul(y, ConstI)), Tuple(x, y), And(Element(x, RR), Element(y, RR))))))

4fd123

\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}

TeX:

\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers
`QQ`	$\mathbb{Q}$	Rational numbers
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4fd123"),
    Formula(Subset(ZZ, QQ, RR, CC)))

Algebraic numbers

be9c83

Symbol: AlgebraicNumbers —

\overline{\mathbb{Q}}

— Algebraic numbers

Represents the set of algebraic numbers.

Definitions:

Fungrim symbol	Notation	Short description
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers

Source code for this entry:

Entry(ID("be9c83"),
    SymbolDefinition(AlgebraicNumbers, AlgebraicNumbers, "Algebraic numbers"),
    Description("Represents the set of algebraic numbers."))

e5a04c

\mathbb{Z} \subset \mathbb{Q} \subset \overline{\mathbb{Q}} \subset \mathbb{C}

TeX:

\mathbb{Z} \subset \mathbb{Q} \subset \overline{\mathbb{Q}} \subset \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers
`QQ`	$\mathbb{Q}$	Rational numbers
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e5a04c"),
    Formula(Subset(ZZ, QQ, AlgebraicNumbers, CC)))

Infinities

b738b1

Symbol: Infinity —

\infty

— Positive infinity

This formal symbol represents a quantity larger than any real number. We define

+\infty = \infty

Multiplication of

\infty

by a nonzero complex number represents an infinite limit with the given direction in the complex plane. In particular,

-\infty

i \infty

and

-i \infty

are frequently used.

The set

\mathbb{R} \cup \left\{\infty, -\infty\right\}

is known as the extended real line.

Definitions:

Fungrim symbol	Notation	Short description
`Infinity`	$\infty$	Positive infinity
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("b738b1"),
    SymbolDefinition(Infinity, Infinity, "Positive infinity"),
    Description("This formal symbol represents a quantity larger than any real number. We define", Equal(Pos(Infinity), Infinity), "."),
    Description("Multiplication of", Infinity, "by a nonzero complex number represents an infinite limit with the given direction in the complex plane.", "In particular,", Neg(Infinity), ",", Mul(ConstI, Infinity), "and", Mul(Neg(ConstI), Infinity), "are frequently used."),
    Description("The set", Union(RR, Set(Infinity, Neg(Infinity))), "is known as the extended real line."))

486ab2

Symbol: UnsignedInfinity —

{\tilde \infty}

— Unsigned infinity

This formal symbol represents a quantity with infinite magnitude and undefined sign.

It is typically used to represent the value of meromorphic functions at poles.

The set

\mathbb{C} \cup \left\{{\tilde \infty}\right\}

represents the complex Riemann sphere.

Definitions:

Fungrim symbol	Notation	Short description
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("486ab2"),
    SymbolDefinition(UnsignedInfinity, UnsignedInfinity, "Unsigned infinity"),
    Description("This formal symbol represents a quantity with infinite magnitude and undefined sign."),
    Description("It is typically used to represent the value of meromorphic functions at poles."),
    Description("The set", Union(CC, Set(UnsignedInfinity)), "represents the complex Riemann sphere."))

Ranges and intervals

03fbae

Symbol: ZZGreaterEqual —

\mathbb{Z}_{\ge n}

— Integers greater than or equal to n

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("03fbae"),
    SymbolDefinition(ZZGreaterEqual, ZZGreaterEqual(n), "Integers greater than or equal to n"))

2a52af

Symbol: ZZLessEqual —

\mathbb{Z}_{\le n}

— Integers less than or equal to n

This symbol may be rendered differently when

n

is a concrete value, for example:

\{-3, -4, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("2a52af"),
    SymbolDefinition(ZZLessEqual, ZZLessEqual(n), "Integers less than or equal to n"),
    Description("This symbol may be rendered differently when", n, "is a concrete value, for example: ", ZZLessEqual(-3)))

00b82b

Symbol: ZZBetween —

\{a, a + 1, \ldots b\}

— Integers between a and b inclusive

Definitions:

Fungrim symbol	Notation	Short description
`ZZBetween`	$\{a, a + 1, \ldots b\}$	Integers between a and b inclusive

Source code for this entry:

Entry(ID("00b82b"),
    SymbolDefinition(ZZBetween, ZZBetween(a, b), "Integers between a and b inclusive"))

12d5ab

Symbol: ClosedInterval —

\left[a, b\right]

— Closed interval

Definitions:

Fungrim symbol	Notation	Short description
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("12d5ab"),
    SymbolDefinition(ClosedInterval, ClosedInterval(a, b), "Closed interval"))

3fe68f

Symbol: OpenInterval —

\left(a, b\right)

— Open interval

Definitions:

Fungrim symbol	Notation	Short description
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("3fe68f"),
    SymbolDefinition(OpenInterval, OpenInterval(a, b), "Open interval"))

b2162a

Symbol: ClosedOpenInterval —

\left[a, b\right)

— Closed-open interval

Definitions:

Fungrim symbol	Notation	Short description
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("b2162a"),
    SymbolDefinition(ClosedOpenInterval, ClosedOpenInterval(a, b), "Closed-open interval"))

ed302a

Symbol: OpenClosedInterval —

\left(a, b\right]

— Open-closed interval

Definitions:

Fungrim symbol	Notation	Short description
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval

Source code for this entry:

Entry(ID("ed302a"),
    SymbolDefinition(OpenClosedInterval, OpenClosedInterval(a, b), "Open-closed interval"))