Editing Arithmetic (section)

== Numbers ==
[[Number]]s are [[mathematical object]]s used to count quantities and measure magnitudes. They are fundamental elements in arithmetic since all arithmetic operations are performed on numbers. There are different kinds of numbers and different [[numeral systems]] to represent them.<ref>{{multiref | {{harvnb|Romanowski|2008|pp=302–304}} | {{harvnb|Khattar|2010|pp=[https://books.google.com/books?id=I3rCgXwvffsC&pg=PA1 1–2]}} | {{harvnb|Nakov|Kolev|2013|pp=[https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA270 270–271]}} }}</ref>

=== 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>

=== Numeral systems ===
{{main|Numeral system}}

A [[Numerical digit|numeral]] is a symbol to represent a number and numeral systems are representational frameworks.<ref>{{multiref | {{harvnb|Ore|1948|pp=1–2}} | {{harvnb|HC staff|2022}} | {{harvnb|HC staff|2022a}} }}</ref> They usually have a limited amount of basic numerals, which directly refer to certain numbers. The system governs how these basic numerals may be combined to express any number.<ref>{{multiref | {{harvnb|Ore|1948|pp=8–10}} | {{harvnb|Nakov|Kolev|2013|pp=[https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA270 270–272]}} }}</ref> Numeral systems are either [[positional]] or non-positional. All early numeral systems were non-positional.<ref>{{multiref | {{harvnb|Stakhov|2020|p=[https://books.google.com/books?id=Fkn9DwAAQBAJ&pg=PA73 73]}} | {{harvnb|Nakov|Kolev|2013|pp=[https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA271 271–272]}} | {{harvnb|Jena|2021|pp=[https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA17 17–18]}} }}</ref> For non-positional numeral systems, the value of a digit does not depend on its position in the numeral.<ref>{{multiref | {{harvnb|Nakov|Kolev|2013|pp=[https://books.google.com/books?id=xYgCAQAAQBAJ&pg=PA271 271–272]}} | {{harvnb|Jena|2021|pp=[https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA17 17–18]}} }}</ref>

{{multiple image
|perrow            = 1 / 1
|total_width       = 250
|image1            = Tally marks.svg
|alt1              = Diagram showing tally marks
|image2            = Vlčí radius.jpg
|alt2              = Photo of tally sticks
|footer            = [[Tally marks]] and some [[tally sticks]]  use the non-positional [[unary numeral system]].
}}

The simplest non-positional system is the [[unary numeral system]]. It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers.<ref>{{multiref | {{harvnb|Ore|1948|pp=8–10}} | {{harvnb|Mazumder|Ebong|2023|pp=[https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18 18–19]}} | {{harvnb|Moncayo|2018|p=[https://books.google.com/books?id=J-pTDwAAQBAJ&pg=PT25 25]}} }}</ref> Variations of the unary numeral systems are employed in [[tally stick]]s using dents and in [[tally marks]].<ref>{{multiref | {{harvnb|Ore|1948|p=8}} | {{harvnb|Mazumder|Ebong|2023|p=[https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18 18]}} }}</ref>

[[File:Hieroglyph numerals.svg|thumb|left|alt=Diagram of hieroglyphic numerals|Hieroglyphic numerals from 1 to 10,000<ref>{{harvnb|Ore|1948|p=10}}</ref>]]

[[Egyptian hieroglyphics]] had a more complex non-positional [[Egyptian numerals|numeral system]]. They have additional symbols for numbers like 10, 100, 1000, and 10,000. These symbols can be combined into a sum to more conveniently express larger numbers. For instance, the numeral for 10,405 uses one time the symbol for 10,000, four times the symbol for 100, and five times the symbol for 1. A similar well-known framework is the [[Roman numeral system]]. It has the symbols I, V, X, L, C, D, M as its basic numerals to represent the numbers 1, 5, 10, 50, 100, 500, and 1000.<ref>{{multiref | {{harvnb|Ore|1948|pp=8–10}} | {{harvnb|Mazumder|Ebong|2023|pp=[https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18 18–19]}} | {{harvnb|Stakhov|2020|pp=[https://books.google.com/books?id=Fkn9DwAAQBAJ&pg=PA77 77–78]}} }}</ref>

A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a [[radix]] that acts as a multiplicand of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called the [[Hindu–Arabic numeral system]], the radix is 10. This means that the first digit is multiplied by <math>10^0</math>, the next digit is multiplied by <math>10^1</math>, and so on. For example, the decimal numeral 532 stands for <math>5 \cdot 10^2 + 3 \cdot 10^1 + 2 \cdot 10^0</math>. Because of the effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits.<ref>{{multiref | {{harvnb|Romanowski|2008|p=303}} | {{harvnb|Yan|2002|pp=305–306}} | {{harvnb|ITL Education Solutions Limited|2011|p=[https://books.google.com/books?id=CsNiKdmufvYC&pg=PA28 28]}} | {{harvnb|Ore|1948|pp=2–3}} | {{harvnb|Jena|2021|pp=[https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA17 17–18]}} }}</ref>

Another positional numeral system used extensively in [[computer arithmetic]] is the [[binary system]], which has a radix of 2. This means that the first digit is multiplied by <math>2^0</math>, the next digit by <math>2^1</math>, and so on. For example, the number 13 is written as 1101 in the binary notation, which stands for <math>1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0</math>. In computing, each digit in the binary notation corresponds to one [[bit]].<ref>{{multiref | {{harvnb|Nagel|2002|p=178}} | {{harvnb|Jena|2021|pp=[https://books.google.com/books?id=qRpSEAAAQBAJ&pg=PA20 20–21]}} | {{harvnb|Null|Lobur|2006|p=[https://books.google.com/books?id=QGPHAl9GE-IC&pg=PA40 40]}} }}</ref> The earliest positional system was developed by [[ancient Babylonians]] and had a radix of 60.<ref>{{harvnb|Stakhov|2020|p=[https://books.google.com/books?id=Fkn9DwAAQBAJ&pg=PA74 74]}}</ref>