Editing Addition (section)

== Properties ==
=== Commutativity ===
[[File:AdditionComm01.svg|right|upright=0.5|thumb|4 + 2 = 2 + 4 with blocks]]
Addition is [[commutative]], meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if <math> a </math> and <math> b </math> are any two numbers, then:{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA89 89]}}
<math display="block"> a + b = b + a. </math>
The fact that addition is commutative is known as the "commutative law of addition"{{sfnp|Berg|1967|p=[https://books.google.com/books?id=aGXFCUaFCW0C&pg=PA14 14]}} or "commutative property of addition".{{sfnp|Behr|Jungst|1971|p=[https://books.google.com/books?id=GJXOBQAAQBAJ&pg=PA59 59]}} Some other [[binary operation]]s are commutative too as in [[multiplication]],{{sfn|Rosen|2013|loc = See the [https://books.google.com/books?id=-oVvEAAAQBAJ&pg=SL1-PA1 Appendix I]}} but others are not as in [[subtraction]] and [[Division (mathematics)|division]].{{sfnp|Posamentier|Farber|Germain-Williams|Paris|2013|p=[https://books.google.com/books?id=VfCgAQAAQBAJ&pg=PA71 71]}}

=== Associativity ===
[[File:AdditionAsc.svg|upright=0.5|thumb|2 + (1 + 3) = (2 + 1) + 3 with segmented rods]]
Addition is [[associativity|associative]], which means that when three or more numbers are added together, the [[order of operations]] does not change the result. For any three numbers <math> a </math>, <math> b </math>, and <math> c </math>, it is true that:{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA90 90]}}
<math display="block"> (a + b) + c = a + (b + c). </math>
For example, <math> (1 + 2) + 3 = 1 + (2 + 3) </math>.

When addition is used together with other operations, the [[order of operations]] becomes important. In the standard order of operations, addition is a lower priority than [[exponentiation]], [[nth root]]s, multiplication and division, but is given equal priority to subtraction.{{sfnp|Bronstein|Semendjajew|1987}}
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=== Identity element ===
[[File:AdditionZero.svg|right|upright=0.33|thumb|5 + 0 = 5 with bags of dots]]
Adding [[0 (number)|zero]] to any number does not change the number. In other words, zero is the [[identity element]] for addition, and is also known as the [[additive identity]]. In symbols, for every <math> a </math>, one has:{{sfnp|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA90 90]}}
<math display="block"> a + 0 = 0 + a = a. </math>
This law was first identified in [[Brahmagupta]]'s ''[[Brahmasphutasiddhanta]]'' in 628&nbsp;AD, although he wrote it as three separate laws, depending on whether <math> a </math> is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later [[Indian mathematicians]] refined the concept; around the year 830, [[Mahavira (mathematician)|Mahavira]] wrote, "zero becomes the same as what is added to it", corresponding to the unary statement <math> 0 + a = a </math>. In the 12th&nbsp;century, [[Bhāskara II|Bhaskara]] wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement <math> a + 0 = a </math>.{{sfnp|Kaplan|2000|pp=69–71}}
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=== Successor ===
{{main article|Successor function}}
Within the context of integers, addition of [[1 (number)|one]] also plays a special role: for any integer <math> a </math>, the integer <math> a + 1 </math> is the least integer greater than <math> a </math>, also known as the [[successor function|successor]] of <math> a </math>. For instance, 3 is the successor of 2, and 7 is the successor of 6. Because of this succession, the value of <math> a + b </math> can also be seen as the {{nowrap|1=<math> b </math>-}}th successor of <math> a </math>, making addition an iterated succession. For example, {{nowrap|6 + 2}} is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the second successor of 6.{{sfnp|Hempel|2001|p=[http://books.google.com/books?id=yTY9La4P2n8C&pg=PA7 7]}}

=== Units ===
To numerically add physical quantities with [[units of measurement|units]], they must be expressed with common units.{{sfnp|Fierro|2012|loc=Section 2.3}} For example, adding 50&nbsp;milliliters to 150&nbsp;milliliters gives 200&nbsp;milliliters. However, if a measure of 5&nbsp;feet is extended by 2&nbsp;inches, the sum is 62&nbsp;inches, since 60&nbsp;inches is synonymous with 5&nbsp;feet. On the other hand, it is usually meaningless to try to add 3&nbsp;meters and 4&nbsp;square meters, since those units are incomparable; this sort of consideration is fundamental in [[dimensional analysis]].<ref>{{Cite book|last1=Moebs|first1=William|url=https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis|title=University Physics Volume 1|last2=Ling|first2=Samuel J.|publisher=[[OpenStax]]|year=2022|isbn=978-1-947172-20-3|chapter=1.4 Dimensional Analysis|display-authors=1}}</ref>