The definition of what is classed as a number is rather diffuse and based on historical distinctions.
Natural numbers are a subset of the integers and are of historical and pedagogical value as they can be used for counting and often have ethno-cultural significance (see below).
Natural numbers are those used for counting (as in "there are six (6) coins on the table") and ordering (as in "this is the third (3rd) largest city in the country").
Defined by the Peano axioms, the natural numbers form an infinitely large set.
The inclusion of 0 in the set of natural numbers is ambiguous and subject to individual definitions.
In set theory and computer science, 0 is typically considered a natural number.
As mathematical properties (such as divisibility) can confer practical utility, there may be interplay and connections between the cultural or practical significance of an integer and its mathematical properties.
, Unicode U+2124 ℤ DOUBLE-STRUCK CAPITAL Z); this became the symbol for the integers based on the German word for "numbers" (Zahlen).
This is used in scientific notation, real numbers are written in the form m × 10n.
, Unicode U+211A ℚ DOUBLE-STRUCK CAPITAL Q);[6] it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".
Rational numbers such as 0.12 can be represented in infinitely many ways, e.g. zero-point-one-two (0.12), three twenty-fifths (3/25), nine seventy-fifths (9/75), etc.
This can be mitigated by representing rational numbers in a canonical form as an irreducible fraction.
The following list includes real numbers that have not been proved to be irrational, nor transcendental.
, Unicode U+2102 ℂ DOUBLE-STRUCK CAPITAL C), while the set of quaternions is denoted by a boldface H (or blackboard bold
Many languages have words expressing indefinite and fictitious numbers—inexact terms of indefinite size, used for comic effect, for exaggeration, as placeholder names, or when precision is unnecessary or undesirable.
[45] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".