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August 17, 2012
Section P.1 Real Numbers
A real number is any number that can be
written as a decimal. They can terminate,
repeat, or continue endlessly with no
discernible pattern.
Subsets of the set of real numbers include:
natural numbers
whole numbers
integers
Rational numbers are real numbers that
can be written as a ratio of two integers.
They will either terminate or repeat.
A real number that neither terminate nor
repeats is called an irrational number.
August 17, 2012
The set of all real numbers are often
represented on the real number line.
-5
-4
-3
-2
-1
0
1
2
3
π•
4
5
Plot
Let a and b be any two real numbers.
If a −b is positive then we say a is
greater than b and we write a >b and
a will be to the right of b on the number
line.
If a −b is negative then we say a is
less than b and we write a <b and b will
be to the right of a on the number line.
The Trichotomy Property
Let a and b be any two real numbers.
Exactly one of the following is true:
August 17, 2012
Interval Notation - bounded intervals
[a, b] means a ≤ x ≤ b. This is a closed
interval because the endpoints are
included in the interval.
Graph of the interval [−2, 3].
-5
-4
-3
-2
-1
0
1
2
3
5
4
°
°
°
(a, b) means a < x < b. This is an open
interval because the endpoints are not
included in the interval.
Graph of the interval (−4, 1).
-5
°
-4
-3
-2
-1
°
0
°
1
2
3
4
5
August 17, 2012
An interval may be open on one side and
closed on the other such as (0, 4]. Write
this as a compound inequality.
Graph of the interval (0, 4].
-5
-4
-2
-3
°
-1
1
0
2
3
5
4
°
Interval Notation - unbounded intervals
The set of all real numbers greater than π is an
unbounded interval since there is no largest real
number. The interval notation for this set of numbers
is (π, ∞). If we want to include the number π in our
interval we write [π, ∞). Since the symbol ∞ does
not represent a number it would never be appropriate
to use a square bracket next to the infinity sign.
Graph (−∞, π]
-5
°
-4
-3
-2
-1
0
1
2
π
3
4
5
°
August 17, 2012
How could we represent the set of all
real numbers using interval notation?
Write inequality and interval notation for
each graph.
•
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
°
0
1
2
3
4
5
-5
-4
°
-2
-1
0
2
3
4
5
-3
•
1
Review the properties of exponents found
on page 7.
August 17, 2012