Editing Arithmetic (section)

=== Others ===

[[File:Clock group.svg|thumb|alt=Diagram of modular arithmetic using a clock|Example of modular arithmetic using a clock: after adding 4 hours to 9 o'clock, the hand starts at the beginning again and points at 1 o'clock.]]

There are many other types of arithmetic. [[Modular arithmetic]] operates on a finite set of numbers. If an operation would result in a number outside this finite set then the number is adjusted back into the set, similar to how the hands of clocks start at the beginning again after having completed one cycle. The number at which this adjustment happens is called the modulus. For example, a regular clock has a modulus of 12. In the case of adding 4 to 9, this means that the result is not 13 but 1. The same principle applies also to other operations, such as subtraction, multiplication, and division.<ref>{{multiref | {{harvnb|Lerner|Lerner|2008|pp=2807–2808}} | {{harvnb|Wallis|2011|pp=[https://books.google.com/books?id=18W4_LJ5bL0C&pg=PA303 303–304]}} | {{harvnb|Kaiser|Granade|2021|pp=[https://books.google.com/books?id=IxIxEAAAQBAJ&pg=PA283 283–284]}} }}</ref>

Some forms of arithmetic deal with operations performed on mathematical objects other than numbers. Interval arithmetic describes operations on intervals.<ref>{{multiref | {{harvnb|Moore|Kearfott|Cloud|2009|pp=[https://books.google.com/books?id=kd8FmmN7sAoC&pg=PA10 10–11, 19]}} | {{harvnb|Pharr|Jakob|Humphreys|2023|p=[https://books.google.com/books?id=kUtwEAAAQBAJ&pg=PA1057 1057]}} }}</ref> Vector arithmetic and matrix arithmetic describe arithmetic operations on [[Vector (mathematics and physics)|vectors]] and [[Matrix (mathematics)|matrices]], like [[vector addition]] and [[matrix multiplication]].<ref>{{multiref | {{harvnb|Liebler|2018|p=[https://books.google.com/books?id=Ozb3DwAAQBAJ&pg=PA36 36]}} | {{harvnb|Adhami|Meenen|Meenen|Hite|2007|pp=[https://books.google.com/books?id=9nqkVbFPutYC&pg=PA80 80–82, 98–102]}} }}</ref>

Arithmetic systems can be classified based on the numeral system they rely on. For instance, [[decimal]] arithmetic describes arithmetic operations in the decimal system. Other examples are [[Binary number|binary]] arithmetic, [[octal]] arithmetic, and [[hexadecimal]] arithmetic.<ref>{{multiref | {{harvnb|Shiva|2018|pp=3, 14}} | {{harvnb|Gupta|2019|p=[https://books.google.com/books?id=vcmcDwAAQBAJ&pg=PA3 3]}} }}</ref>

Compound unit arithmetic describes arithmetic operations performed on magnitudes with compound units. It involves additional operations to govern the transformation between single unit and compound unit quantities. For example, the operation of reduction is used to transform the compound quantity 1 h 90 min into the single unit quantity 150 min.<ref>{{harvnb|Burgin|2022|pp=92–93}}</ref>

Non-Diophantine arithmetics are arithmetic systems that violate traditional arithmetic intuitions and include equations like <math>1 + 1 = 1</math> and <math>2 + 2 = 5</math>.<ref>{{multiref | {{harvnb|Burgin|2022|pp=xviii–xx, xxiv, 137–138}} | {{harvnb|Caprio|Aveni|Mukherjee|2022|pp=763–764}} }}</ref> They can be employed to represent some real-world situations in modern physics and everyday life. For instance, the equation <math>1 + 1 = 1</math> can be used to describe the observation that if one raindrop is added to another raindrop then they do not remain two separate entities but become one.<ref>{{multiref | {{harvnb|Burgin|2022|p=144}} | {{harvnb|Caprio|Aveni|Mukherjee|2022|pp=763–764}} | {{harvnb|Seaman|Rossler|Burgin|2023|p=[https://books.google.com/books?id=213PEAAAQBAJ&pg=PA226 226]}} }}</ref>