Addition of natural numbers: Difference between revisions
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#REDIRECT [[Addition#Natural_numbers]] |
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'''Addition of natural numbers''' is the most basic arithmetic operation. In its simplest form, addition combines two numbers (terms, summands), the augend and addend, into a single number, the sum. |
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{{R to section}} |
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== Notation and terms == |
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The operation of '''addition''', commonly written as the [[infix]] [[operator]] "+", is a [[function_(mathematics)|function]] + : '''N''' × '''N''' → '''N'''. For natural numbers ''a'', ''b'', and ''c'', we write |
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:<math>a + b = c.\,</math> |
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Here, ''a'' is the ''augend'', ''b'' is the ''addend'', and ''c'' is the ''sum''. |
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== Definition == |
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We let ''S''(''a'') denote the ''successor of a'' as defined in the [[Peano postulates]]. |
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Addition is defined inductively by fixing the augend. In other words, we let ''a'' be any arbitrary, but fixed natural number, and we then make the following definitions: |
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* ''a'' + 0 = ''a'' [A1] |
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* ''a'' + ''S''(''b'') = ''S''(''a'' + ''b'') [A2] |
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By the recursion theorem, this defines a unique function "''a'' +" : '''N''' → '''N'''. In words, it says that adding zero to ''a'' gives back ''a'', and that applying the successor function to the addend has the effect of applying the successor function to the sum. |
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Since ''a'' was an arbitrary natural number, we can "put together" all these functions into a single binary operation '''N''' × '''N''' → '''N'''. |
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== Properties == |
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The following are three immediate and important properties of addition which can be deduced from the definition. |
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* [[associative|Associativity]]: for all natural numbers ''a'', ''b'', and ''c'', we have |
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:<math>(a + b) + c = a + (b + c);\,</math> ([[Addition of natural numbers/proofs#Proof of associativity|proof]]) |
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* [[commutative|Commutativity]]: for all natural numbers ''a'' and ''b'', we have |
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:<math>a + b = b + a;\,</math> ([[Addition of natural numbers/proofs#Proof of commutativity|proof]]) |
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* [[Identity element]]: for all natural numbers ''a'', we have |
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:<math>a + 0 = 0 + a = a.\,</math> ([[Addition of natural numbers/proofs#Proof of identity element|proof]]) |
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Together, these three properties show that the set of natural numbers '''N''' under addition is a commutative [[monoid]]. |
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[[Category:Elementary arithmetic]] |
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[[fr:Addition des entiers naturels]] |
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[[ja:加法]] |
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Latest revision as of 19:35, 8 March 2018
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