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An introduction to the real numbers and integers, including their primitive terms, operations, and properties. It covers the concepts of addition, multiplication, less than, absolute value, and the properties of equality for real numbers. It also includes axioms for the real numbers and integers.
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The Real Numbers and the Integers
To avoid circularity, we cannot give every term a rigorous mathematical definition; we have to accept some things as undefined terms. For this course, we will take the following fundamental notions as primitive undefined terms. You already know what these terms mean; but the only facts about them that can be used in proofs are the ones expressed in the axioms listed below (and any theorems that can be proved from the axioms).
2, π, and e.
In all the definitions below, a and b represent arbitrary real numbers.
|a| =
a if a ≥ 0 , −a if a < 0.
a, is the unique nonnegative real number whose square is a (see Theorem 9 below).
In modern mathematics, the relation “equals” can be used between any two “mathematical objects” of the same type, such as numbers, matrices, ordered pairs, sets, functions, etc. To say that a = b is simply to say that the symbols a and b represent the very same object. Thus the concept of “equality” really belongs to mathematical logic rather than to any particular branch of mathematics.
Equality always has the following fundamental properties, no matter what kinds of objects it is applied to. In the following statements, a, b, and c can represent any mathematical objects whatsoever. (In our applications, they will usually be real numbers.)
These theorems can be proved from the axioms in the order listed below.
a such that
a
= a. (b) If a = b and a and b are both nonnegative, then
a =
b. (c) If a < b and a and b are both nonnegative, then
a <
b. (d) If a^2 = b and b is nonnegative, then a = ±
b.
(a) If a is any real number, then |a| ≥ 0. (b) |a| = 0 if and only if a = 0. (c) |a| > 0 if and only if a 6 = 0. (d) | − a| = |a|. (e) |a| =
a^2.