Tekst źródłowy strony Algorithmic theory of integers

Integer numbers form a set, usually denoted  '''integer''' (or <math>Z</math>). The set contains a non-empty subset  <math>N</math> (of non-negative integer numbers  ''aka'' natural numbers) and equipped with two binary operations , of addition and multiplication, usually denoted,  <math>+</math> i <math>\cdot</math>. These operations fulfill the following axioms: 
* (Commutativity) For every pair of integer numbers   <math>a, b</math> the following equalities hold 
<math>a+b=b+a\qquad</math>  and <math>\qquad a\cdot b = b \cdot a</math>, 
* (Associativity) For every triplet of integer numbers   <math>a,b, c</math> the following equalities hold 
<math>(a+(b+c))=((a+b)+c) \qquad</math>  and  <math>\qquad (a\cdot(b\cdot c))=((a\cdot b)\cdot c)</math>, 
* (Distributivity) For every triplet of integer numbers the following equality holds 
<math>(a+b)\cdot c =a\cdot c + b\cdot c</math>
* (Units) There exist integer numbers  0 i 1 such that, for every   <math>a</math> the following equalities hold 
<math>a+0=a\qquad </math>   and   <math>\qquad a\cdot 1= a</math>,
* (Closure in  <math>N</math>) If <math>a</math> and <math>b</math> are non-negative integer numbers then numbers  <math>a+b</math> and <math>a\cdot b</math> are also non-negative integer numbers,
* (Additive inverse) For every integer number  <math>a</math>, exists integer number  <math>-a</math>, such that the following equality holds 
<math> a+-a =0 </math> 
* (Trichotomy) For every integer number  <math>a</math> exactly one of three relations holds: either a) number <math>a</math> is non-negative, or b) number <math>a</math> is zero <math>a=0</math> or c) number <math>-a</math> jis non-negative, 
* (Non-negative integers) The set of non-negative integers satisfies the axioms of natural numbers, it is the same set <math>N</math>. See [[Algorithmic_theory_of_natural _numbers ]]<br />
* (Ordering relation) a < b if and only if the number <math> b+-a </math> belongs to the set   <math>N</math>.