1. Hawkes Learning Systems: College Algebra
Section 1.1: The Real Number System

2. Objectives Common subsets of real numbers. The real number line.
Order on the real number line.
Set-builder notation and interval notation.
Absolute value and distance on the real number line.
Working with repeating decimals.

3. Common Subsets of Real Numbers (cont.)
The Natural (or Counting) Numbers:
The set of counting numbers greater than or equal to 1.
The Whole Numbers:
The set of Natural numbers and 0.
The Integers:
The set of natural numbers, their negatives, and 0.

4. Common Subsets of Real Numbers
The Rational Numbers: The set of ratios of integers, Any
rational number can be written in the form where p and q
are both integers and Rational numbers either terminate or repeat patterns of digits past some point.
Ex:
The Irrational Numbers: Every real number that is not rational. In decimal form, irrational numbers are non-terminating and non- repeating.
The Real Numbers: Every set above is a subset of the set of real numbers, which is denoted Every real number is either rational or irrational and no real number is both.

5. Common Subsets of Real Numbers
Rational Numbers ( )
Decimal term either terminates or repeats
Irrational Numbers
Decimal form is
non-terminating and
non-repeating.
Integers ( )
…,-3,-2,-1,0,1,2,3,…
Whole Numbers ,1,2,3…
Natural Numbers
( )
1,2,3…

6. Example 1: Common Subsets of Real Numbers
Consider the set
Natural
Numbers
Whole
Integers
Rational
Irrational
Real

7. The Real Number Line
Ex: Plot the following numbers on the real number line: | | | | | | | | | | | | |
The real number line is a depiction of the set of real numbers as a horizontal line. The real number corresponding to a given point is called the coordinate of that point. The point for the number 0 is called the origin. Points to the right of the origin represent positive numbers and points to the left of the origin represent negative numbers.
Negative Numbers
Positive Numbers

8. Example 2: The Real Number Line
Plot the numbers 101, 106, and 107:
Plot the numbers , , and
| | | | | | | | | | |
| | | | | | | | |

9. Order on the Real Number Line
Symbol
Read
“a is less than b”
“a is less than or equal to b”
“b is greater than a”
“b is greater than or equal to a”
Meaning
a lies to the left of b on the number line.
a lies to the left of b or is equal to b.
b lies to the right of a on the number line.
b lies to the right of a or is equal
to a.
The two symbols < and > are called strict inequality signs, while the symbols ≤ and ≥ are non-strict inequality signs.

10. Example 3: Order on the Real Number Line
What can we say about the following relationship?
| | | | | | |
a. lies to the left of .
b , or is strictly less than .
c , or is less than or equal to .
d , or is strictly greater than .
e , or is greater than or equal to .

11. Example 4: Order on the Real Number Line
The statement “a is less than or equal to b + c” is written:
The negation of the statement a ≤ b is
Why is this true? a ≤ b means that “a is less than or equal to b.” The negation of a statement is to say that the statement is not true. So, if a is NOT less than or equal to b then a must be greater than b and a cannot be equal to b. Thus, a must be strictly greater than b.
If a ≤ b and a ≥ b then it must be the case that
Why is this true? Consider each case on a real number line.
a ≤ b | | | | | | |
a ≥ b | | | | | | |
Where can you place a and b in the second case so that the first case holds true? You should notice that this is impossible unless you make a = b.

12. Set-Builder Notation and Interval Notation
Set-builder notation is a general method of h describing the elements that belong to a given set.
The notation {x|x has property P} is used to describe a set of real numbers, all of which have the property P. This can be read “the set of all real numbers x having property P.”
Interval notation is a way of describing certain subsets of the real line.

13. Example 5: Set-builder Notation and Interval Notation
What sets of real numbers do the following properties describe? a. {x|x is an even integer} = {…,-4,-2,0,2,4…} b. {x|x is an integer such that -4 ≤ x <1} = {-4,-3,-2,-1,0} c. {x|x > 2 and x ≤ -5} = { } or
This set could also be described as {2n|n is an integer} since every even integer is a multiple of 2.
These symbols denote the empty set. This property describes the empty set because no real numbers satisfy it.

14. Set-builder Notation and Interval Notation
Sets that consist of all real numbers bounded by two endpoints are called intervals. Intervals can also extend indefinitely in either direction.
Intervals of the form (a,b) are called open intervals.
Intervals of the form [a,b] are called closed intervals.
The intervals (a,b] and [a,b) are called half-open or half-closed.
The symbols and indicate that the interval extends in the left and right directions, respectively.

15. Set-builder Notation and Interval Notation
(a,b)
[a,b]
(a,b]
( ,b)
[a, ]
Set-Builder Notation
{x|a < x < b}
{x|a ≤ x ≤ b}
{x|a < x ≤ b}
{x|x < b}
{x|x ≥ a}
Meaning
All real numbers strictly between a and b.
All real numbers between a and b, including both a and b.
All real numbers between a and b, including b but not a.
All real numbers less than b.
All real numbers greater than or equal to a.

16. Example 5: Set-builder Notation and Interval Notation
Describe each of the following properties using the chart below:
All real numbers strictly between -5 and 8.
All real numbers greater than or equal to 2.
All real numbers between -10 and 3, including 3 but not -10.
The entire set of real numbers.
Interval Notation
Set-builder
Notation
Interval is ______ at the left endpoint
Interval is ______ at the right endpoint a. b. c. d.

17. Absolute Value and Distance on the Real Number Line
The absolute value of a real number a, denoted as |a|, is defined by:
The absolute value of a number is also referred to as its magnitude; it is the non-negative number corresponding to its distance from the origin.
Given two real numbers, the distance between them is defined to be |a−b|. In particular, the distance between a and 0 is |a−0| or just |a|.

18. Absolute Value and Distance on the Real Number Line
Properties of Absolute Value
For all real numbers a and b: 1. 2. 3. 4. 5. 6. 7.
(This is called the triangle inequality because it is a reflection of the fact that one side of a triangle is never longer than the sum of the other two sides.)

19. Example 6: Absolute Value and Distance on the Real Number Line
Simplify the following expressions using your knowledge of absolute values: a. b. c. d. e. f.
Both and – are units from 0.
How does this compare to ?
is greater than 3, so this must be a positive number.
is less than 13, so this expression must be negative. So, its absolute value is
Note the properties of absolute value.

20. Working with Repeating Decimals
A rational number that appears with a repeating pattern of digits can be written as a ratio of integers by following the procedure outlined below: Suppose we wish to write as a ratio of integers. We know that Now, let Substitute
So, together we have