Editing Arithmetic (section)

=== Integer arithmetic ===
[[File:Number line method.svg|thumb|upright=1.3|alt=Diagram of number line method|Using the number line method, calculating <math>5 + 2</math> is performed by starting at the origin of the number line then moving five units to right for the first addend. The result is reached by moving another two units to the right for the second addend.]]

Integer arithmetic is the branch of arithmetic that deals with the manipulation of positive and negative whole numbers.<ref>{{multiref | {{harvnb|Romanowski|2008|p=304}} | {{harvnb|Hindry|2011|p=x}} | {{harvnb|Hafstrom|2013|p=[https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA95 95]}} | {{harvnb|Brent|Zimmermann|2010|p=[https://books.google.com/books?id=-8wuH5AwbwMC&pg=PA1  1]}}}}</ref> Simple one-digit operations can be performed by following or memorizing a table that presents the results of all possible combinations, like an [[addition table]] or a [[multiplication table]]. Other common methods are verbal [[counting]] and [[finger-counting]].<ref>{{multiref | {{harvnb|Kupferman|2015|pp=[https://books.google.com/books?id=Cd87DQAAQBAJ&pg=PA92 45, 92]}} | {{harvnb|Uspenskii|Semenov |2001|p=[https://books.google.com/books?id=yID37VIW-t8C&pg=PA113 113]}} | {{harvnb|Geary|2006|p=[https://books.google.com/books?id=bLZyrZHd1QkC&pg=PA796 796]}} }}</ref>

{| class="wikitable" style="display:inline-table; text-align:center; margin:1em .8em 1em 1.6em;"
|+ Addition table
! +  || 0 || 1  || 2  || 3  || 4  || ...
|-
! 0
| 0  || 1  || 2  || 3  || 4  || ...
|-
! 1
| 1  || 2  || 3  || 4  || 5  || ...
|-
! 2
| 2  || 3  || 4  || 5  || 6  || ...
|-
! 3
| 3  || 4  || 5  || 6  || 7  || ...
|-
! scope="row" | 4
| 4  || 5  || 6  || 7  || 8  || ...
|-
! scope="row" | ...
| ...  || ...  || ...  || ...  || ...  || ...
|}

{| class="wikitable" style="display:inline-table; text-align:center; margin: 1em .8em 1em 1.6em;"
|+ Multiplication table
! ×  || 0 || 1  || 2  || 3  || 4  || ...
|-
! 0
| 0  || 0  || 0  || 0  ||0  || ...
|-
! 1
| 0  || 1  || 2  || 3  || 4  || ...
|-
! 2
| 0  || 2  || 4  || 6  || 8  || ...
|-
! 3
| 0  || 3  || 6  || 9  || 12  || ...
|-
! scope="row" | 4
| 0  || 4  || 8  || 12  || 16  || ...
|-
! scope="row" | ...
| ...  || ...  || ...  || ...  || ...  || ...
|}

{{clear}}{{Multiple image
| image1            = Addition with carry.png
| image2            = Long multiplication.png
| caption1          = Example of [[Carry (arithmetic)|addition with carry]]. The black numbers are the addends, the green number is the carry, and the blue number is the sum.
| total_width       = 400
| caption2          = Example of long multiplication. The black numbers are the multiplier and the multiplicand. The green numbers are intermediary products gained by multiplying the multiplier with only one digit of the multiplicand. The blue number is the total product calculated by adding the intermediary products.
| perrow            = 2
}}
For operations on numbers with more than one digit, different techniques can be employed to calculate the result by using several one-digit operations in a row. For example, in the method [[Carry (arithmetic)|addition with carries]], the two numbers are written one above the other. Starting from the rightmost digit, each pair of digits is added together. The rightmost digit of the sum is written below them. If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to the left. This process is repeated until all digits have been added.<ref>{{multiref | {{harvnb|Resnick|Ford|2012|p=[https://books.google.com/books?id=xj-j8pw2HN8C&pg=PA110 110]}} | {{harvnb|Klein|Moeller|Dressel|Domahs|2010|pp=67–68}} }}</ref> Other methods used for integer additions are the [[number line]] method, the partial sum method, and the compensation method.<ref>{{multiref | {{harvnb|Quintero|Rosario|2016|p=[https://books.google.com/books?id=YAXyCwAAQBAJ&pg=PA74 74]}} | {{harvnb|Ebby|Hulbert|Broadhead|2020|pp=[https://books.google.com/books?id=shEHEAAAQBAJ&pg=PA24 24–26]}} }}</ref> A similar technique is utilized for subtraction: it also starts with the rightmost digit and uses a "borrow" or a negative carry for the column on the left if the result of the one-digit subtraction is negative.<ref>{{harvnb|Sperling|Stuart|1981|p=7}}</ref>

A basic technique of integer multiplication employs repeated addition. For example, the product of <math>3 \times 4</math> can be calculated as <math>3 + 3 + 3 + 3</math>.<ref>{{harvnb|Sperling|Stuart|1981|p=8}}</ref> A common technique for multiplication with larger numbers is called [[long multiplication]]. This method starts by writing the multiplier above the multiplicand. The calculation begins by multiplying the multiplier only with the rightmost digit of the multiplicand and writing the result below, starting in the rightmost column. The same is done for each digit of the multiplicand and the result in each case is shifted one position to the left. As a final step, all the individual products are added to arrive at the total product of the two multi-digit numbers.<ref>{{multiref | {{harvnb|Ma|2020|pp=[https://books.google.com/books?id=2Tr3DwAAQBAJ&pg=PA35 35–36]}} | {{harvnb|Sperling|Stuart|1981|p=9}} }}</ref> Other techniques used for multiplication are the [[grid method]] and the [[lattice method]].<ref>{{harvnb|Mooney|Briggs|Hansen|McCullouch|2014|p=[https://books.google.com/books?id=_dPgAwAAQBAJ&pg=PT148 148]}}</ref> Computer science is interested in [[multiplication algorithms]] with a low [[computational complexity]] to be able to efficiently multiply very large integers, such as the [[Karatsuba algorithm]], the [[Schönhage–Strassen algorithm]], and the [[Toom–Cook multiplication|Toom–Cook algorithm]].<ref>{{multiref | {{harvnb|Klein|2013|p=[https://books.google.com/books?id=GYpEAAAAQBAJ&pg=PA249 249]}} | {{harvnb|Muller|Brunie|Dinechin|Jeannerod|2018|p=[https://books.google.com/books?id=h3ZZDwAAQBAJ&pg=PA539 539]}} }}</ref> A common technique used for division is called [[long division]]. Other methods include [[short division]] and [[Chunking (division)|chunking]].<ref>{{harvnb|Davis|Goulding|Suggate|2017|pp=[https://books.google.com/books?id=7T8lDwAAQBAJ&pg=PA11 11–12]}}</ref>

Integer arithmetic is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5.<ref>{{harvnb|Haylock|Cockburn|2008|p=[https://books.google.com/books?id=hgAr3maZeQUC&pg=PA49 49]}}</ref> One way to ensure that the result is an integer is to [[Rounding|round]] the result to a whole number. However, this method leads to inaccuracies as the original value is altered.<ref>{{multiref | {{harvnb|Prata|2002|p=[https://books.google.com/books?id=MsizNs-zVMAC&pg=PA138 138]}} | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA135 135–136]}} }}</ref> Another method is to perform the division only partially and retain the [[remainder]]. For example, 7 divided by 2 is 3 with a remainder of 1. These difficulties are avoided by rational number arithmetic, which allows for the exact representation of fractions.<ref>{{harvnb|Koepf|2021|p=[https://books.google.com/books?id=rOU3EAAAQBAJ&pg=PA49 49]}}</ref>

A simple method to calculate [[exponentiation]] is by repeated multiplication. For instance, the exponentiation of <math>3^4</math> can be calculated as <math>3 \times 3 \times 3 \times 3</math>.<ref>{{harvnb|Goodstein|2014|p=[https://books.google.com/books?id=EHjiBQAAQBAJ&pg=PA33 33]}}</ref> A more efficient technique used for large exponents is [[exponentiation by squaring]]. It breaks down the calculation into a number of squaring operations. For example, the exponentiation <math>3^{65}</math> can be written as <math>(((((3^2)^2)^2)^2)^2)^2 \times 3</math>. By taking advantage of repeated squaring operations, only 7 individual operations are needed rather than the 64 operations required for regular repeated multiplication.<ref>{{multiref | {{harvnb|Cafaro|Epicoco|Pulimeno|2018|p=[https://books.google.com/books?id=rs51DwAAQBAJ&pg=PA7 7]}} | {{harvnb|Reilly|2009|p=[https://books.google.com/books?id=q33he4hOlKcC&pg=PA75 75]}} }}</ref> Methods to calculate [[logarithm]]s include the [[Taylor series]] and [[continued fraction]]s.<ref>{{multiref | {{harvnb|Cuyt|Petersen|Verdonk|Waadeland|2008|p=[https://books.google.com/books?id=DQtpJaEs4NIC&pg=PA182 182]}} | {{harvnb|Mahajan|2010|pp=[https://books.google.com/books?id=VrkZN0T0GaUC&pg=PA66 66–69]}} | {{harvnb|Lang|2002|pp=205–206}} }}</ref> Integer arithmetic is not closed under logarithm and under exponentiation with negative exponents, meaning that the result of these operations is not always an integer.<ref>{{multiref | {{harvnb|Kay|2021|p=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA57 57]}} | {{harvnb|Cuyt|Petersen|Verdonk|Waadeland|2008|p=[https://books.google.com/books?id=DQtpJaEs4NIC&pg=PA182 182]}} }}</ref>

==== Number theory ====
{{main|Number theory}}

Number theory studies the structure and properties of integers as well as the relations and laws between them.<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Grigorieva|2018|pp=[https://books.google.com/books?id=mEpjDwAAQBAJ&pg=PR8 viii–ix]}} | {{harvnb|Page|2003|p=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 15]}} }}</ref> Some of the main branches of modern number theory include [[elementary number theory]], [[analytic number theory]], [[algebraic number theory]], and [[geometric number theory]].<ref>{{multiref | {{harvnb|Page|2003|p=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 34]}} | {{harvnb|Yan|2002|p=12}} }}</ref> Elementary number theory studies aspects of integers that can be investigated using elementary methods. Its topics include [[divisibility]], [[factorization]], and [[primality]].<ref>{{multiref | {{harvnb|Page|2003|pp=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 18–19, 34]}} | {{harvnb|Bukhshtab|Nechaev|2014}} }}</ref> Analytic number theory, by contrast, relies on techniques from analysis and calculus. It examines problems like [[prime number theorem|how prime numbers are distributed]] and the claim that [[Goldbach's conjecture|every even number is a sum of two prime numbers]].<ref>{{multiref | {{harvnb|Page|2003|p=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 34]}} | {{harvnb|Karatsuba|2014}} }}</ref> Algebraic number theory employs algebraic structures to analyze the properties of and relations between numbers. Examples are the use of [[Field (mathematics)|fields]] and [[Ring (mathematics)|rings]], as in [[algebraic number field]]s like the [[ring of integers]]. Geometric number theory uses concepts from geometry to study numbers. For instance, it investigates how lattice points with integer coordinates behave in a plane.<ref>{{multiref | {{harvnb|Page|2003|pp=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 34–35]}} | {{harvnb|Vinogradov|2019}} }}</ref> Further branches of number theory are [[probabilistic number theory]], which employs methods from [[probability theory]],<ref>{{harvnb|Kubilyus|2018}}</ref> [[combinatorial number theory]], which relies on the field of [[combinatorics]],<ref>{{harvnb|Pomerance|Sárközy|1995|p=[https://books.google.com/books?id=5ktBP5vUl5gC&pg=PA969 969]}}</ref> [[computational number theory]], which approaches number-theoretic problems with computational methods,<ref>{{harvnb|Pomerance|2010}}</ref> and applied number theory, which examines the application of number theory to fields like [[physics]], [[biology]], and [[cryptography]].<ref>{{multiref | {{harvnb|Yan|2002|pp=12, 303–305}} | {{harvnb|Yan|2013a|p=[https://books.google.com/books?id=74oBi4ys0UUC&pg=PA15 15]}} }}</ref>

Influential theorems in number theory include the [[fundamental theorem of arithmetic]], [[Euclid's theorem]], and [[Fermat's Last Theorem]].<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Křížek|Somer|Šolcová|2021|pp=[https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA23 23, 25, 37]}} }}</ref> According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, the [[number 18]] is not a prime number and can be represented as <math>2 \times 3 \times 3</math>, all of which are prime numbers. The [[number 19]], by contrast, is a prime number that has no other prime factorization.<ref>{{multiref | {{harvnb|Křížek|Somer|Šolcová|2021|p=[https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA23 23]}} | {{harvnb|Riesel|2012|p=[https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA2 2]}} }}</ref> Euclid's theorem states that there are infinitely many prime numbers.<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Křížek|Somer|Šolcová|2021|p=[https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA25 25]}} }}</ref> Fermat's Last Theorem is the statement that no positive integer values exist for <math>a</math>, <math>b</math>, and <math>c</math> that solve the equation <math>a^n + b^n = c^n</math> if <math>n</math> is greater than <math>2</math>.<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Křížek|Somer|Šolcová|2021|p=[https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA37 37]}} }}</ref>