Algorithmic theory of integers: Różnice pomiędzy wersjami

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Integer numbers form a set, usually denoted  '''integer''' (or <math>Z</math>). The set contains a non-empty subset  <math>N</math> (of 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:  
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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  
 
* (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>,  
 
<math>a+b=b+a\qquad</math>  and <math>\qquad a\cdot b = b \cdot a</math>,  
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<math>a+0=a\qquad </math>  and  <math>\qquad a\cdot 1= a</math>,
 
<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,
 
* (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,
* (Addytywna odwrotność) Dla każdej liczby całkowitej <math>a</math>, istnieje taka liczba całkowita <math>-a</math>, że zachodzi  równość
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* (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>  
 
<math> a+-a =0 </math>  
* (Trichotomia) Dla każdej liczby całkowitej <math>a</math> zachodzi dokładnie jedna z trzech relacji: albo a) liczba <math>a</math> jest nieujemna, albo b) liczba <math>a</math> jest zerem <math>a=0</math> albo c) liczba <math>-a</math> jest nieujemna,  
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* (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,  
* (Liczby całkowite nieujemne) Zbiór liczb całkowitych nieujemnych spełnia aksjomaty liczb naturalnych, to jest ten sam zbiór.<br />
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* (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 />
* (Definicja porządku) a < b wtedy i tylko wtedy gdy b+-a należy do <math>N</math>.
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* (Ordering relation) a < b if and only if the number <math> b+-a </math> belongs to the set  <math>N</math>.
c.d.n.
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Wersja z 11:42, 2 paź 2018

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
  • (Ordering relation) a < b if and only if the number [math] b+-a [/math] belongs to the set [math]N[/math].