Terms

Number

Number is an abstract mathematical object used to count, measure and label.

Number systems

Number systems are sets of number classes (ℕ ℤ ℚ ℝ)

Numeral

Numeral is the notational symbol that represents a number. It is important to differentiate between the properties of a number and properties of its numeral. For example, "being even" is a property of a number (regardless of the number system) and "sum of its digits" is a property of denotation (and it depends on the number system).

Numeral system

System of numeration is a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems.

Positional notation

Combinatorics

Combinatorics studies the problems concerning counting and ordering, such as partitioning and enumerations.

Arithmetic

Arithmetic, an elementary part of number theory, studies properties of numbers and operations. Arithmetic deals with equations in which all numbers are concrete numbers; it is very invested in the final result of operations. On the other hand, algebra dispenses with concrete numbers in favor of variables as it is more interested in discovering the laws, properties and priniciples that govern arithmetic operations then their immediate results.

Fundamental sets

Some sets are favored, being assigned a special symbol and a name:

• $\mathbb{B}$ is the set of boolean values (or Boolean domain)

  $\quad \{\bot,\top\}$ or $\{F,T\}$ or $\{0,1\}$

• $\mathbb{N}$ is the set of natural numbers

  $\quad \mathbb{N}=\{0,1,2,3,\dots\}$

• $\mathbb{Z}$ is the set of integers

  $\quad \mathbb{Z}=\{\dots,-2,-1,0,1,2,\dots\}$.

• $\mathbb{Q}$ is the set of rational numbers

  $\quad \{p/q : p \in\mathbb{Z}, q\in \mathbb{N}\land q\neq 0\}$

• $\mathbb{R}$ is the set of real numbers.

• $\mathbb{C}$ is the set of complex numbers

  $\quad \{a,b\,i:a,b\in\mathbb{R}\}, \quad i=\sqrt{-1}$

Properties:

• $\mathbb{N}\subseteq \mathbb{Z}\subseteq \mathbb{Q}\subseteq \mathbb{R}$

• $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ are enumerable or countable infinite sets with the same cardinality.

• $\mathbb{R}$ is non-enumerable infinite set. It has various kinds of infinities.

• The set of positive integers: $\mathbb{Z^+}=\{1,2,3,\dots\}$

N, Z, Q can be mapped 1:1 with counting (N) number set i.e. they're countable and their resp. set is infinite but countable set.

The set of real numbers, ℝ, includes rational, ℚ ⊆ ℝ, and irrational numbers.

Not all infinities are the same, some are bigger then others. There is an infinite number of different-sized infinities.

The infinity of irrationals is greater than infinity of rational numbers. It is said that rational numbers are like the starts in the night sky, but the irrational numbers are like the blackness.

Cantor also showed that a powerset of any infinite set represents a bigger infinity than the original set.