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Difference between revisions of "Integer"
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==Ordertheoretic properties==  ==Ordertheoretic properties==  
'''Z''' is a totally ordered set without upper or lower bound. The ordering of '''Z''' is given by  '''Z''' is a totally ordered set without upper or lower bound. The ordering of '''Z''' is given by  
−  +  :<math>\ldots < 2 < 1 < 0 < 1 < 2 < \ldots</math>  
An integer is ''positive'' if it is greater than zero and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive.  An integer is ''positive'' if it is greater than zero and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive.  
Line 28:  Line 28:  
The ordering of integers is compatible with the algebraic operations in the following way:  The ordering of integers is compatible with the algebraic operations in the following way:  
−  +  # If <math>a < b</math> and <math>c < d</math>, then <math>a + c < b + d</math>  
−  +  # If <math>a < b</math> and <math>0 < c</math>, then <math>ac < bc</math>  
−  (From this fact, one can show that if c < 0, then ac > bc.)  +  (From this fact, one can show that if <math>c < 0</math>, then <math>ac > bc</math>.) 
==External links==  ==External links== 
Latest revision as of 23:25, 26 October 2020
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, [math]\displaystyle{ \mathbb{Z} }[/math]), which stands for Zahlen (German for "numbers"). They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set.
Algebraic properties
Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.
addition  multiplication  

closure:  [math]\displaystyle{ a + b }[/math] is an integer  [math]\displaystyle{ a \cdot b }[/math] is an integer 
associativity:  [math]\displaystyle{ a + (b + c) = (a + b) + c }[/math]  [math]\displaystyle{ a \cdot (b \cdot c) = (a \cdot b) \cdot c }[/math] 
commutativity:  [math]\displaystyle{ a + b = b + a }[/math]  [math]\displaystyle{ a \cdot b = b \cdot a }[/math] 
existence of an identity element:  [math]\displaystyle{ a + 0 = a }[/math]  [math]\displaystyle{ a \cdot 1 = a }[/math] 
existence of inverse elements:  [math]\displaystyle{ a + (a) = 0 }[/math]  
distributivity:  [math]\displaystyle{ a \cdot (b + c) = (a \cdot b) + (a \cdot c) }[/math] 
Ordertheoretic properties
Z is a totally ordered set without upper or lower bound. The ordering of Z is given by
 [math]\displaystyle{ \ldots \lt 2 \lt 1 \lt 0 \lt 1 \lt 2 \lt \ldots }[/math]
An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive.
The ordering of integers is compatible with the algebraic operations in the following way:
 If [math]\displaystyle{ a \lt b }[/math] and [math]\displaystyle{ c \lt d }[/math], then [math]\displaystyle{ a + c \lt b + d }[/math]
 If [math]\displaystyle{ a \lt b }[/math] and [math]\displaystyle{ 0 \lt c }[/math], then [math]\displaystyle{ ac \lt bc }[/math]
(From this fact, one can show that if [math]\displaystyle{ c \lt 0 }[/math], then [math]\displaystyle{ ac \gt bc }[/math].)
External links
General numbers 
Special numbers 
Prime numbers 
