Editing Arithmetic (section)

=== 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
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|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>