This article contains

mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...

s for some properties of

addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...

of the

natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

: the additive identity, commutativity, and associativity. These proofs are used in the article

Addition of natural numbers Addition (usually signified by the plus symbol, +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or '' sum'' of th ...

Definitions

This article will use the

Peano axioms In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nea ...

for the definition of natural numbers. With these axioms, ''addition'' is defined from the constant 0 and the

successor function In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...

''S''(''a'') by the two rules For the proof of commutativity, it is useful to give the name "1" to the successor of 0; that is, :1 = ''S''(0). For every natural number ''a'', one has

Proof of associativity

We prove

associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...

by first fixing natural numbers ''a'' and ''b'' and applying induction on the natural number ''c''. For the base case ''c'' = 0, : (''a'' + ''b'') + 0 = ''a'' + ''b'' = ''a'' + (''b'' + 0) Each equation follows by definition 1 the first with ''a'' + ''b'', the second with ''b''. Now, for the induction. We assume the induction hypothesis, namely we assume that for some natural number ''c'', : (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') Then it follows, In other words, the induction hypothesis holds for ''S''(''c''). Therefore, the induction on ''c'' is complete.

Proof of identity element

Definition 1states directly that 0 is a right identity. We prove that 0 is a

left identity In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

by induction on the natural number ''a''. For the base case ''a'' = 0, 0 + 0 = 0 by definition 1 Now we assume the induction hypothesis, that 0 + ''a'' = ''a''. Then This completes the induction on ''a''.

Proof of commutativity

We prove

commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...

(''a'' + ''b'' = ''b'' + ''a'') by applying induction on the natural number ''b''. First we prove the base cases ''b'' = 0 and ''b'' = ''S''(0) = 1 (i.e. we prove that 0 and 1 commute with everything). The base case ''b'' = 0 follows immediately from the identity element property (0 is an

additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element in the set, yields . One of the most familiar additive identities is the number 0 from elementary ma ...

), which has been proved above: ''a'' + 0 = ''a'' = 0 + ''a''. Next we will prove the base case ''b'' = 1, that 1 commutes with everything, i.e. for all natural numbers ''a'', we have ''a'' + 1 = 1 + ''a''. We will prove this by induction on ''a'' (an induction proof within an induction proof). We have proved that 0 commutes with everything, so in particular, 0 commutes with 1: for ''a'' = 0, we have 0 + 1 = 1 + 0. Now, suppose ''a'' + 1 = 1 + ''a''. Then This completes the induction on ''a'', and so we have proved the base case ''b'' = 1. Now, suppose that for all natural numbers ''a'', we have ''a'' + ''b'' = ''b'' + ''a''. We must show that for all natural numbers ''a'', we have ''a'' + ''S''(''b'') = ''S''(''b'') + ''a''. We have This completes the induction on ''b''.

References

Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopo ...

, Foundations of Analysis, Chelsea Pub Co. . {{DEFAULTSORT:Addition Of Natural Numbers/Proofs Article proofs Abstract algebra Elementary algebra Operations on numbers

Definitions

Proof of associativity

Proof of identity element

Proof of commutativity

See also

References