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The order properties of real numbers are given in Appendix 4. The following useful
rules can be derived from them, where the symbol means “implies.”
Notice the rules for multiplying an inequality by a number. Multiplying by a positive number
preserves the inequality; multiplying by a negative number reverses the inequality.
Also, reciprocation reverses the inequality for numbers of the same sign. For example,
but and
The completeness property of the real number system is deeper and harder to define
precisely. However, the property is essential to the idea of a limit (Chapter 2). Roughly
speaking, it says that there are enough real numbers to “complete” the real number line, in
the sense that there are no “holes” or “gaps” in it. Many theorems of calculus would fail if
the real number system were not complete. The topic is best saved for a more advanced
course, but Appendix 4 hints about what is involved and how the real numbers are constructed.
We distinguish three special subsets of real numbers.
1. The natural numbers, namely 1, 2, 3, 4
2. The integers, namely
3. The rational numbers, namely the numbers that can be expressed in the form of a
fraction , where m and n are integers and Examples are
The rational numbers are precisely the real numbers with decimal expansions that are
either
(a) terminating (ending in an infinite string of zeros), for example,
(b) eventually repeating (ending with a block of digits that repeats over and over), for
example
The bar indicates the
block of repeating
digits.
A terminating decimal expansion is a special type of repeating decimal since the ending
zeros repeat.
The set of rational numbers has all the algebraic and order properties of the real numbers
but lacks the completeness property. For example, there is no rational number whose
square is 2; there is a “hole” in the rational line where should be.
Real numbers that are not rational are called irrational numbers. They are characterized
by having nonterminating and nonrepeating decimal expansions. Examples are
and Since every decimal expansion represents a real number, it
should be clear that there are infinitely many irrational numbers. Both rational and irrational
numbers are found arbitrarily close to any point on the real line.
Set notation is very useful for specifying a particular subset of real numbers. A set is a
collection of objects, and these objects are the elements of the set. If S is a set, the notation
means that a is an element of S, and means that a is not an element of S. If S
and T are sets, then is their union and consists of all elements belonging either to S
or T (or to both S and T). The intersection consists of all elements belonging to both
S and T. The empty set is the set that contains no elements. For example, the intersection
of the rational numbers and the irrational numbers is the empty set.
Some sets can be described by listing their elements in braces. For instance, the set A
consisting of the natural numbers (or positive integers) less than 6 can be expressed as
is the set of positive integers less than 6.
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