Mathematical Analysis I, 2004a

Chapter 2
Real Numbers. Fields
§§1–4. Axioms and Basic Definitions
Real numbers can be constructed step by step: first the integers, then the
rationals, and finally the irrationals. 1 Here, however, we shall assume the
set of all real numbers, denoted E 1 ,asalready given, without attempting to
reduce this notion to simpler concepts. We shall also accept without definition
(as primitive concepts) the notions of the sum (a + b) andtheproduct, (a · b)
or (ab), of two real numbers, as well as the inequality relation < (read “less
than”). Note that x ∈ E 1 means “x is in E 1 ,” i.e., “x is a real number .”
It is an important fact that all arithmetic properties of reals can be deduced
from several simple axioms, listed (and named) below.
Axioms of Addition and Multiplication
I (closure laws). The sum x + y, andtheproductxy, ofanyrealnumbers
are real numbers themselves. In symbols,
II (commutative laws).
III (associative laws).
(∀ x, y ∈ E 1 ) (x + y) ∈ E 1 and (xy) ∈ E 1 .
(∀ x, y ∈ E 1 ) x + y = y + x and xy = yx.
(∀ x, y, z ∈ E 1 ) (x + y)+z = x +(y + z) and(xy)z = x(yz).
IV (existence of neutral elements).
(a) There is a (unique) real number , called zero (0), such that, for all
real x, x +0=x.
1 See the author’s Basic Concepts of Mathematics, Chapter 2, §15.