This article contains
mathematical proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every pr ...
s for some properties of
addition
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 amount or '' sum'' of ...
of the
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
: 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 amount or '' sum'' of ...
.
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 nearly ...
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 functi ...
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, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
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 operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
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. Most familiar as the name of ...
(''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 ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elem ...
), 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''.
See also
*
Binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary ope ...
*
Proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a co ...
*
Ring
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 Leopold ...
, Foundations of Analysis, Chelsea Pub Co. .
{{DEFAULTSORT:Addition Of Natural Numbers/Proofs
Article proofs
Abstract algebra
Elementary algebra
Operations on numbers