Download PDF

integer∗
CWoo†
2013-03-21 12:33:55
The set of integers, denoted by the symbol Z, is the set {· · ·−3, −2, −1, 0, 1, 2, 3, . . . }
consisting of the natural numbers and their negatives.
Mathematically, Z is defined to be the set of equivalence classes of pairs of
natural numbers N × N under the equivalence relation (a, b) ∼ (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) · (c, d) := (ac + bd, ad + bc)
Typically, the class of (a, b) is denoted by symbol n if b ≤ a (resp. −n if a ≤ 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
{. . . , −3, −2, −1, 0, 1, 2, 3, . . . }. Here are some examples:
• 0 = equivalence class of (0, 0) = equivalence class of (1, 1) = . . .
• 1 = equivalence class of (1, 0) = equivalence class of (2, 1) = . . .
• −1 = equivalence class of (0, 1) = equivalence class of (1, 2) = . . .
The set of integers Z under the addition and multiplication operations defined
above form an integral domain. The integers admit the following ordering relation making Z into an ordered ring: (a, b) ≤ (c, d) in Z if a + d ≤ b + c in
N.
The ring of integers is also a Euclidean domain, with valuation given by the
absolute value function.
∗ hIntegeri
created: h2013-03-21i by: hCWooi version: h30403i Privacy setting: h1i
hDefinitioni h11-00i h03-00i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1