Editing Arithmetic (section)

=== Tool use ===
[[File:Mental calculation at primary school.jpg|thumb|right|alt=Painting of students engaged in mental arithmetic|Calculations in [[mental arithmetic]] are done exclusively in the mind without relying on external aids.]]

Forms of arithmetic can also be distinguished by the [[mathematical instrument|tools]] employed to perform calculations and include many approaches besides the regular use of pen and paper. [[Mental arithmetic]] relies exclusively on the [[mind]] without external tools. Instead, it utilizes visualization, memorization, and certain calculation techniques to solve arithmetic problems.<ref>{{multiref | {{harvnb|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA131 131]}} | {{harvnb|Verschaffel|Torbeyns|De Smedt|2011|p=[https://books.google.com/books?id=xZuSxo4JxoAC&pg=PA2177 2177]}} }}</ref> One such technique is the compensation method, which consists in altering the numbers to make the calculation easier and then adjusting the result afterward. For example, instead of calculating <math>85-47</math>, one calculates <math>85-50</math> which is easier because it uses a round number. In the next step, one adds <math>3</math> to the result to compensate for the earlier adjustment.<ref>{{multiref | {{harvnb|Emerson|Babtie|2014|p=[https://books.google.com/books?id=NQ-aBQAAQBAJ&pg=PA147 147]}} | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA131 131–132]}} | {{harvnb|Verschaffel|Torbeyns|De Smedt|2011|p=[https://books.google.com/books?id=xZuSxo4JxoAC&pg=PA2177 2177]}} }}</ref> Mental arithmetic is often taught in primary education to train the numerical abilities of the students.<ref>{{multiref | {{harvnb|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA131 131]}} | {{harvnb|Verschaffel|Torbeyns|De Smedt|2011|p=[https://books.google.com/books?id=xZuSxo4JxoAC&pg=PA2177 2177]}} }}</ref>

The human body can also be employed as an arithmetic tool. The use of hands in [[finger counting]] is often introduced to young children to teach them numbers and simple calculations. In its most basic form, the number of extended fingers corresponds to the represented quantity and arithmetic operations like addition and subtraction are performed by extending or retracting fingers. This system is limited to small numbers compared to more advanced systems which employ different approaches to represent larger quantities.<ref>{{multiref | {{harvnb|Dowker|2019|p=[https://books.google.com/books?id=GQaQDwAAQBAJ&pg=PA114 114]}} | {{harvnb|Berch|Geary|Koepke|2015|p=[https://books.google.com/books?id=XS9OBQAAQBAJ&pg=PA124 124]}} | {{harvnb|Otis|2024|pp=[https://books.google.com/books?id=07jiEAAAQBAJ&pg=PA15 15–19]}} | {{harvnb|Geary|2006|p=[https://books.google.com/books?id=bLZyrZHd1QkC&pg=PA796 796]}} }}</ref> The human voice is used as an arithmetic aid in verbal counting.<ref>{{multiref | {{harvnb|Otis|2024|pp=[https://books.google.com/books?id=07jiEAAAQBAJ&pg=PA15 15–19]}} | {{harvnb|Geary|2006|p=[https://books.google.com/books?id=bLZyrZHd1QkC&pg=PA796 796]}} }}</ref>

[[File:Chinese-abacus.jpg|thumb|left|alt=Photo of a Chinese abacus|Abacuses are tools to perform arithmetic operations by moving beads.]]

[[Tally marks]] are a simple system based on external tools other than the body. This system relies on mark making, such as strokes drawn on a surface or [[Notch (engineering)|notches]] carved into a wooden stick, to keep track of quantities. Some forms of tally marks arrange the strokes in groups of five to make them easier to read.<ref>{{multiref | {{harvnb|Ore|1948|p=8}} | {{harvnb|Mazumder|Ebong|2023|p=[https://books.google.com/books?id=7oS_EAAAQBAJ&pg=PA18 18]}} }}</ref>

The [[abacus]] is a more advanced tool to represent numbers and perform calculations. An abacus usually consists of a series of rods, each holding several [[bead]]s. Each bead represents a quantity, which is counted if the bead is moved from one end of a rod to the other. Calculations happen by manipulating the positions of beads until the final bead pattern reveals the result.<ref>{{multiref | {{harvnb|Reynolds|2008|pp=[https://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1 1–2]}} | {{harvnb|Sternberg|Ben-Zeev|2012|pp=[https://books.google.com/books?id=q7F777rDl1AC&pg=PA95 95–96]}} }}</ref> Related aids include [[counting board]]s, which use tokens whose value depends on the area on the board in which they are placed,<ref>{{harvnb|Budd|Sangwin|2001|p=[https://books.google.com/books?id=x8hoAsL_JL4C&pg=PA209 209]}}</ref> and [[counting rods]], which are arranged in horizontal and vertical patterns to represent different numbers.<ref>{{multiref | {{harvnb|Knobloch|Komatsu|Liu|2013|p=[https://books.google.com/books?id=JgDFBAAAQBAJ&pg=PA123 123]}} | {{harvnb|Hodgkin|2013|p=[https://books.google.com/books?id=nSO5iMujRUYC&pg=PR168 168]}} | {{harvnb|Hart|2011|p=[https://books.google.com/books?id=zLPm3xE2qWgC&pg=PA69 69]}} }}</ref>{{efn|Some systems of counting rods include different colors to represent both positive and negative numbers.<ref>{{multiref | {{harvnb|Hodgkin|2013|p=[https://books.google.com/books?id=nSO5iMujRUYC&pg=PR168 168]}} | {{harvnb|Hart|2011|p=[https://books.google.com/books?id=zLPm3xE2qWgC&pg=PA69 69]}} }}</ref>}}

[[Sector (instrument)|Sectors]] and [[slide rule]]s are more refined calculating instruments that rely on geometric relationships between different scales to perform both basic and advanced arithmetic operations.<ref>{{multiref | {{harvnb|Bruderer|2021|pp=[https://books.google.com/books?id=Gh8SEAAAQBAJ&pg=PA543 543–545, 906–907]}} | {{harvnb|Klaf|2011|pp=[https://books.google.com/books?id=YRNeBAAAQBAJ&pg=PA187 187–188]}} }}</ref>{{efn|Some computer scientists see slide rules as the first type of [[analog computer]].<ref>{{multiref | {{harvnb|Strathern|2012|p=[https://books.google.com/books?id=NJm7Z5iQE24C&pg=PP9 9]}} | {{harvnb|Lang|2015|p=[https://books.google.com/books?id=wbiKDwAAQBAJ&pg=PA160 160]}} }}</ref>}} Printed tables were particularly relevant as an aid to look up the results of operations like logarithm and [[Trigonometry|trigonometric functions]].<ref>{{harvnb|Campbell-Kelly|Croarken|Flood|Robson|2007|p=2}}</ref>

[[Mechanical calculator]]s automate manual calculation processes. They present the user with some form of input device to enter numbers by turning dials or pressing keys. They include an internal mechanism usually consisting of [[gear]]s, [[lever]]s, and [[wheel]]s to perform calculations and display the results.<ref>{{multiref | {{harvnb|Lockhart|2017|pp=136, 140–146}} | {{harvnb|O'Regan|2012|pp=[https://books.google.com/books?id=QqrItgm351EC&pg=PA24 24–25]}} }}</ref> For [[electronic calculator]]s and [[computer]]s, this procedure is further refined by replacing the mechanical components with [[electronic circuits]] like [[microprocessor]]s that combine and transform electric signals to perform calculations.<ref>{{multiref | {{harvnb|Khoury|Lamothe|2016|p=[https://books.google.com/books?id=T162DAAAQBAJ&pg=PA2 2]}} | {{harvnb|Lockhart|2017|pp=147–150}} | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA119 119]}} }}</ref>