Editing Arithmetic (section)

=== Kinds ===
[[File:Number line.png|thumb|upright=1.75|alt=Number line showing different types of numbers|Different types of numbers on a [[number line]]. Integers are black, rational numbers are blue, and irrational numbers are green.]]

The main kinds of numbers employed in arithmetic are [[natural numbers]], whole numbers, [[integers]], [[rational numbers]], and [[real numbers]].<ref>{{multiref | {{harvnb|Nagel|2002|pp=180–181}} | {{harvnb|Luderer|Nollau|Vetters|2013|p=[https://books.google.com/books?id=rSf0CAAAQBAJ&pg=PA9 9]}} | {{harvnb|Khattar|2010|pp=[https://books.google.com/books?id=I3rCgXwvffsC&pg=PA1 1–2]}} }}</ref> The natural numbers are whole numbers that start from 1 and go to infinity. They exclude 0 and negative numbers. They are also known as counting numbers and can be expressed as <math>\{1, 2, 3, 4, ...\}</math>. The symbol of the natural numbers is <math>\N</math>.{{efn|Other symbols for the natural numbers include <math>\N^*</math>, <math>\N^+</math>, <math>\N_1</math>, and <math>\mathbf{N}</math>.<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Zhang|2012|p=[https://books.google.com/books?id=GRTSBwAAQBAJ&pg=PA130 130]}} | {{harvnb|Körner|2019|p=[https://books.google.com/books?id=z-y2DwAAQBAJ&pg=PA109 109]}} | {{harvnb|International Organization for Standardization|2019|p=[https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf 4]}} }}</ref>}} The whole numbers are identical to the natural numbers with the only difference being that they include 0. They can be represented as <math>\{0, 1, 2, 3, 4, ...\}</math> and have the symbol <math>\N_0</math>.<ref>{{multiref | {{harvnb|Romanowski|2008|p=304}} | {{harvnb|Nagel|2002|pp=180–181}} | {{harvnb|Hindry|2011|p=x}} | {{harvnb|Bukhshtab|Nechaev|2016}} }}</ref>{{efn|Other symbols for the whole numbers include <math>\N^0</math>, <math>\N \cup \{0 \}</math>, and <math>W</math>.<ref>{{multiref | {{harvnb|Swanson|2021|p=[https://books.google.com/books?id=cHshEAAAQBAJ&pg=PA107 107]}} | {{harvnb|Rossi|2011|p=[https://books.google.com/books?id=kSwVGbBtel8C&pg=PA111 111]}} }}</ref>}} Some mathematicians do not draw the distinction between the natural and the whole numbers by including 0 in the set of natural numbers.<ref>{{multiref | {{harvnb|Rajan|2022|p=[https://books.google.com/books?id=OCE6EAAAQBAJ&pg=PA17 17]}} | {{harvnb|Hafstrom|2013|p=[https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA6 6]}} }}</ref> The set of integers encompasses both positive and negative whole numbers. It has the symbol <math>\Z</math> and can be expressed as <math>\{..., -2, -1, 0, 1, 2, ...\}</math>.<ref>{{multiref | {{harvnb|Romanowski|2008|p=304}} | {{harvnb|Nagel|2002|pp=180–181}} | {{harvnb|Hindry|2011|p=x}} | {{harvnb|Hafstrom|2013|p=[https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA95 95]}} }}</ref>

Based on how natural and whole numbers are used, they can be distinguished into [[cardinal numerals|cardinal]] and [[ordinal numerals|ordinal number]]s. Cardinal numbers, like one, two, and three, are numbers that express the quantity of objects. They answer the question "how many?". Ordinal numbers, such as first, second, and third, indicate order or placement in a series. They answer the question "what position?".<ref>{{multiref | {{harvnb|Orr|1995|p=[https://books.google.com/books?id=DMTqRnoE8iMC&pg=PA49 49]}} | {{harvnb|Nelson|2019|p=[https://books.google.com/books?id=xTiDDwAAQBAJ&pg=PR31 xxxi]}} }}</ref>

A number is rational if it can be represented as the [[ratio]] of two integers. For instance, the rational number <math>\tfrac{1}{2}</math> is formed by dividing the integer 1, called the numerator, by the integer 2, called the denominator. Other examples are <math>\tfrac{3}{4}</math> and <math>\tfrac{281}{3}</math>. The set of rational numbers includes all integers, which are [[fraction]]s with a denominator of 1. The symbol of the rational numbers is <math>\Q</math>.<ref>{{multiref | {{harvnb|Romanowski|2008|p=304}} | {{harvnb|Nagel|2002|pp=180–181}} | {{harvnb|Hindry|2011|p=x}} | {{harvnb|Hafstrom|2013|p=[https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA123 123]}} }}</ref> [[Decimal fraction]]s like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to <math>\tfrac{3}{10}</math>, and 25.12 is equal to <math>\tfrac{2512}{100}</math>.<ref>{{multiref |  {{harvnb|Gellert|Hellwich|Kästner|Küstner|2012|p=[https://books.google.com/books?id=1jH7CAAAQBAJ&pg=PA33 33]}} }}</ref> Every rational number corresponds to a finite or a [[repeating decimal]].<ref>{{harvnb|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA358 358]}}</ref>{{efn|A repeating decimal is a decimal with an infinite number of repeating digits, like 0.111..., which expresses the rational number <math>\tfrac{1}{9}</math>.}}

[[File:Square root of 2 triangle.svg|thumb|alt=Diagram of a right triangle|Irrational numbers are sometimes required to describe magnitudes in [[geometry]]. For example, the length of the [[hypotenuse]] of a [[right triangle]] is irrational if its legs have a length of 1.]]

[[Irrational numbers]] are numbers that cannot be expressed through the ratio of two integers. They are often required to describe geometric magnitudes. For example, if a [[right triangle]] has legs of the length 1 then the length of its [[hypotenuse]] is given by the irrational number <math>\sqrt 2</math>. [[Pi|{{pi}}]] is another irrational number and describes the ratio of a [[circle]]'s [[circumference]] to its [[diameter]].<ref>{{multiref | {{harvnb|Romanowski|2008|p=304}} | {{harvnb|Nagel|2002|pp=180–181}} | {{harvnb|Hindry|2011|p=x}} }}</ref> The decimal representation of an irrational number is infinite without repeating decimals.<ref>{{multiref | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA358 358–359]}} | {{harvnb|Rooney|2021|p=[https://books.google.com/books?id=WnVeEAAAQBAJ&pg=PA34 34]}} }}</ref> The set of rational numbers together with the set of irrational numbers makes up the set of real numbers. The symbol of the real numbers is <math>\R</math>.<ref>{{multiref | {{harvnb|Romanowski|2008|p=304}} | {{harvnb|Hindry|2011|p=x}} }}</ref> Even wider classes of numbers include [[complex numbers]] and [[quaternion]]s.<ref>{{multiref | {{harvnb|Hindry|2011|p=x}} | {{harvnb|Ward|2012|p=[https://books.google.com/books?id=LVDvCAAAQBAJ&pg=PA55 55]}} }}</ref>