“real: chapter_01” — 2011/5/22 — 12:15 — page2—#2
1-2 Real Analysis
(b) With the usual addition (+), multiplication (·) and the order
relation (≤), the integer system
Z ={0, ±1, ±2, ...}satisfies
the following:
(
Z, +, ·, ≤) is an ordered integral domain.
(c) With usual operations and the order relation,
Q (the set of all
rational numbers) satisfies:
(
Q, +, ·, ≤) is an ordered field
1.2 ORDER STRUCTURE OF THE REAL NUMBER SYSTEM
The real numbers can be geometrically represented as points on a line
(called the real line or the real axis). We can choose any point (called ori-
gin) on the line to represent “0” and another point to represent “1.” Now
each point on this line represents one and only one real number and con-
versely each real number can be represented ...