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{{redirect|Add|other articles with the same name|ADD (disambiguation)}}
:''Not to be confused with [[building extension]]s which are also called additions.''
[[File:Addition.gif|thumb|220px|Addition]]

In [[mathematics]], '''addition''', represented by the symbol <math>+</math>, is an operation which combines two mathematical objects together into another mathematical object of the same type, called the [[sum]].<ref>{{cite web|date=2020-03-17|title=List of Arithmetic and Common Math Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/common-math-symbols/|access-date=2020-08-26|website=Math Vault|language=en-US}}</ref><ref>{{cite web|title=Addition|url=http://www.mathisfun.com/numbers/addition|access-date=2020-08-26|website=www.mathisfun.com|language=en-US}}</ref><ref>{{cite web|last=Weisstein|first=Eric W.|title=Addition|url=https://mathworld.wolfram.com/Addition.html|access-date=2020-08-26|website=mathworld.wolfram.com|language=en-US}}</ref> Addition can occur with simple objects such as [[number]]s, and more complex objects and concepts such as [[vector]]s and [[Matrix (mathematics)|matrices]].<ref>{{cite web|date=2020-03-25|title=Comprehensive List of Algebra Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/|access-date=2020-08-26|website=Math Vault|language=en-US}}</ref>

Addition has several important properties. It is [[Commutative property|commutative]], meaning that the order of the [[operand]]s does not matter (for example 3+4 is the same as 4+3), and it is [[Associativity|associative]], meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see ''[[Summation]]''). Repeated addition of [[1 (number)|1]] is the same as counting. Addition of [[0 (number)|0]] does not change a number. Addition also obeys predictable rules concerning related operations such as [[Operation (mathematics)#Subtraction|subtraction]] and [[Operation (mathematics)#Multiplication|multiplication]].
== Arithmetic ==
In [[arithmetic]], addition is the operation where two or more [[number]]s called "addends" are used to make a new number, which is the "[[sum]]" or total that is expressed with the [[equals sign]]. The symbol for addition, in [[infix notation]], is the plus sign "+" placed between the operands.
[[File:AdditionShapes.svg|alt=Two plus three is equal to five.|thumb|2 [[shape]]s + 3 shapes = 5 shapes]]
== Counting examples ==
For example, there are [[object|objects]] in two groups (as shown on the right). The objects are various shapes, where one group has 3 of them while the other has 2. When the two groups combine into one, the overall amount (sum) of the shapes become 5.
== Vertical Addition ==
The animation above demonstrates the addition of seven hundred eighty six and four hundred sixty seven. The problem's digits have been separated into units, tens and hundreds (see [[Place value]]).

First, the units 6 and 7 are added together to make 13, so 1 ten and 3 units, with the 3 written below and the 1 ten carried to the tens column. Next, in the tens column, the 1, 8, and 6 are added together to make 15 tens, so 1 hundred and 5 tens, with the 5 written below and the 1 hundred carried to the hundreds column. Finally, in the hundreds column, 1, 7, and 4 are added together to make 12 hundreds, so 1 thousand and 2 hundreds, with the 2 written below and the 1 thousand carried to the thousand column. The final answer is thus one thousand two hundred fifty three.
== A measurement example ==
Tom wants to know the distance between his house and Sally's house. Bob's house is 300&nbsp;[[Metre|m]] east of Tom's house. Sally's house is 120&nbsp;m east of Bob's house:

:Tom's house	<math>\leftarrow</math>	300&nbsp;m	<math>\rightarrow</math>	Bob's house	<math>\leftarrow</math>	120&nbsp;m	<math>\rightarrow</math>	Sally's house

The distance from Tom's house to Sally's house can be found by adding the distances already measured. The distance from Tom's house to Bob's house, added to the distance from Bob's house to Sally's house, is the same as the distance from Tom's house to Sally's house. That is, 300 m plus 120&nbsp;m.
:<math>300+120=420</math>
Hence Sally's house is 420&nbsp;m to the east of Tom's house.
== Properties ==

== Commutativity ==
[[File:AdditionComm01.svg|right|113px|thumb|<math>4+2=2+4</math> with blocks]]
Addition is [[Commutative property|commutative]], meaning that one can change the order of the numbers in a sum, but still get the same result. For example:
:<math>4+5=9</math> and <math>5+4=9</math>
== Associativity ==
[[File:AdditionAsc.svg|left|100px|thumb|<math>2+(1+3)=(2+1)+3</math> with parts of a cylinder]]
Addition is also [[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 <math>1=(a+b)+c=a+(b+c)</math>. For example, <math>1=(1+2)+3=3+3=6</math> and <math>1=1+(2+3)=1+5=6</math>, which means that <math>1=(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 to be computed later than [[Operation (mathematics)#Exponentiation|exponentiation]], [[Operation (mathematics)#Roots|roots]], [[Operation (mathematics)#Multiplication|multiplication]] and [[Operation (mathematics)#Division|division]], but has equal importance as [[Operation (mathematics)#Subtraction|subtraction]].<ref>{{cite book |title=Taschenbuch der Mathematik |author-first1=Ilja Nikolaevič<!-- Nikolajewitsch --> |author-last1=Bronstein<!-- 1903–1976 --> |author-first2=Konstantin Adolfovič<!-- Adolfowitsch --> |author-last2=Semendjajew<!-- 1908–1988 --> |editor1-first=Günter |editor1-last=Grosche |editor2-first=Viktor |editor2-last=Ziegler<!-- 1922–1980--> |editor3-first=Dorothea |editor3-last=Ziegler |others=Weiß, Jürgen<!-- lector --> |translator-first=Viktor |translator-last=Ziegler |volume=1 |date=1987 |edition=23 |orig-year=1945 |publisher=[[Verlag Harri Deutsch]] (and [[B.G. Teubner Verlagsgesellschaft]], Leipzig) |location=Thun and Frankfurt am Main |language=de |chapter=2.4.1.1. |pages=115–120 |isbn=978-3-87144-492-0 |title-link=Bronstein and Semendjajew}}</ref>
== Addition table ==
{{Addition table}}

== Addition Media ==
<gallery widths='160px' heights='100%' mode='traditional' caption=''>
File:Addition01.svg|3 + 2 = 5 with [[apple]]s, a popular choice in textbooks

File:PlusCM128.svg|A plus sign in 128-point Computer Modern (CMR10), created on my personal copy of Adobe Illustrator with assistance from LaTeXiT.

File:Addition1.png|The terms of addends in the operation of an addition

File:AdditionNombryng.svg|Redrawn illustration from ''The Art of Nombryng'', one of the first English arithmetic texts, in the 15th century.

File:AdditionZero.svg|5 + 0 = 5 with bags of dots

File:Addition with carry.svg|An addition with [[Carry (mathematics)|carry]]
</gallery>
== Related pages ==
* [[Operation (mathematics)|Operations]]
* [[Identity element]]
* [[Order of operations]]
* [[Hyperoperation]]
== References ==
<references/>
== Other websites ==
* [https://www.aaamath.com/add.html AAA Math: Addition]
{{Wiktionary}}

[[Category:Basic English 850 words]]
[[Category:Hyperoperations]]