MA112 – R.1 The Real Number Line
Today we will cover:
Announcements







the real number line
(HW 1-3)
properties of the real #s (HW4-5)
interval notation (worksheet, HW 8-10)
absolute value
(HW 6-7)
turn in the student query by the end of class
Homework R.1 assigned on MyMathlab
Quiz Friday on a problem like HW#9.
The Real Number Line
Constructing the Real Line 
- - - - - - - - - - - - - - - - - - - - - - - - -
How many numbers do we need to ‘fill up’ the real line?
ℕ = the counting numbers = the natural numbers = {1, 2, 3, …}
 “closed” under addition because if you add any two natural numbers, you still get a
natural number
 also closed under multiplication (which is repeated addition)
 not “closed” under subtraction because subtracting two natural numbers doesn’t
necessarily give you a positive number: 1-100 = -99
 = the whole numbers = the natural numbers and 0
 adds 0 to the real number line
 = integers = {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4 …}
 The integers extend the real line to the left of zero, by adding the negative whole
numbers.
 now closed under subtraction!
 not closed under division: 1/3 is not an integer
 = rational numbers = set of all numbers that can be expressed as a fraction
closed under addition, subtraction, multiplication AND division
Recognizing rational numbers: A number is rational if and only if its decimal
representation terminates or repeats.
18
13
1.8 =
, 1.444… =
10
9
Is the line filled up now?
Amazing fact: Most of the line is still empty! Most points on the real line are not rational
(cannot be exactly expressed as a fraction of integers)
Example:  cannot be expressed as a fraction.
What are these other numbers??? We call them the irrationals. Example: , e,
 = {all numbers of the line} = {rational numbers, irrational numbers}
Study Aid: ℕ        
(note that ‘’ means ‘subset of’)
Properties of the Real Numbers for Addition and Multiplication
Property
Addition
Multiplication
commutativity
a+b=b+a
ab=ba
associativity
a+(b+c)=(a+b)+c
a(bc)=(ab)c
identity
a+0=a
a×1=a
inverse
a-a=0
a÷a=1
distributive
a(b+c)=ab+ac
Describing Parts of the Real Line
We can describe segments of the real line using inequalities and intervals.
Inequalities:
x<3 means that x is less than 3
x>3 means that x is greater than 3
x=3 means that x is 3
Figure out some way to remember
the difference between ‘<’ and ‘>’!!
3 ways of describing a parts of the line: (See class worksheet.)
Intervals
[-1,5]
[-1,5)
(-1,5]
(-1,5)
Inequalities
-1  x ≤ 5
-1 ≤ x < 5
-1 < x ≤ 5
-1 < x < 5
Graph
[------------------]
[------------------)
(------------------]
(------------------)
Absolute Value
Every number has a sign (positive or negative) and a magnitude (distance from 0).
The absolute value function takes just the magnitude of a number.
example: |4| = 4 and |-4| = 4
 x if x  0
Absolute value in precise mathematical language: x  
.
 x f x  0
2.