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integer (Definition)

The set of integers, denoted by the symbol $ \mathbb{Z}$, is the set $ \{\dots -3, -2, -1, 0, 1, 2, 3, \dots\}$ consisting of the natural numbers and their negatives.

Mathematically, $ \mathbb{Z}$ is defined to be the set of equivalence classes of pairs of natural numbers $ \mathbb{N} \times \mathbb{N}$ under the equivalence relation $ (a,b) \sim (c,d)$ if $ a+d = b+c$.

Addition and multiplication of integers are defined as follows:

  • $ (a,b)+(c,d) := (a+c,b+d)$
  • $ (a,b)\cdot(c,d) := (ac+bd,ad+bc)$
Typically, the class of $ (a,b)$ is denoted by symbol $ n$ if $ b \leq a$ (resp. $ -n$ if $ a \leq b$), where $ n$ is the unique natural number such that $ a=b+n$ (resp. $ a+n=b$). Under this notation, we recover the familiar representation of the integers as $ \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. Here are some examples:
  • $ 0 = $ equivalence class of $ (0,0) = $ equivalence class of $ (1,1) = \dots$
  • $ 1 = $ equivalence class of $ (1,0) = $ equivalence class of $ (2,1) = \dots$
  • $ -1 = $ equivalence class of $ (0,1) = $ equivalence class of $ (1,2) = \dots$
The set of integers $ \mathbb{Z}$ under the addition and multiplication operations defined above form an integral domain. The integers admit the following ordering relation making $ \mathbb{Z}$ into an ordered ring: $ (a,b) \leq (c,d)$ in $ \mathbb{Z}$ if $ a+d \leq b+c$ in $ \mathbb{N}$.

The ring of integers is also a Euclidean domain, with valuation given by the absolute value function.



"integer" is owned by CWoo. [ full author list (2) | owner history (1) ]
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See Also: irrational

Other names:  rational integer, $\mathbb{Z}$

Attachments:
sums of two squares (Theorem) by pahio
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Cross-references: function, absolute value, valuation, Euclidean domain, ring of integers, ordered ring, ordering relation, integral domain, operations, representation, class, multiplication, addition, equivalence relation, equivalence classes, negatives, natural numbers
There are 1153 references to this entry.

This is version 8 of integer, born on 2001-10-19, modified 2008-09-11.
Object id is 403, canonical name is Integer.
Accessed 35401 times total.

Classification:
AMS MSC: 03-00 (Mathematical logic and foundations :: General reference works )
 11-00 (Number theory :: General reference works )
Pending Errata and Addenda
None.
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Discussion
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Zahlen by Wkbj79 on 2003-03-10 17:16:07
I would mention why Z is used to denote the integers. It was first used by Germans, since they call the set "Zahlen", which means "numbers".
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also by antizeus on 2001-10-20 00:11:56
I'd add a note about how (a,b) maps to a-b in the isomorphism.
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