Chapter 8: Integers
Chapter 9: Reals and Rationals
Practice problems:
8.1A (3, 6, 11)
8.2A (1, 2, 3)
I.
Practice problems:
9pt1A: 1, 3, 7, 9
9pt2A: 2, 24
Chapter 8 / Chapter 9
Position of Integers, Rationals and Reals in the Context of Other Types of Numbers
The integers extend the real line to the left of zero, by adding the negative whole
numbers.
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 natural number: 1-100 = -99
the whole numbers = the natural numbers and 0
 0 is an important digit in our place value system
 like the counting numbers, is closed for addition and multiplication but not
subtraction and division
 = integers = {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4 …}
 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
Chapter 8: Negative Numbers (Integers)
I.
What is a negative number?
For any number N, its negative is also a number and is denoted –N.
N and –N are opposites of each other because when they are added together they
yield zero.
They are additive inverses: N+ -N = 0. We have two ways of representing this:
(1) measurement model:
a positive number is modeled as a number of units to the right
a negative number is modeled as a number of units to the left
4 + -4 = 0 means 4 units to the right, then 4 units to the left
Note: The negative means you go in the opposite direction, so the negative of a
negative is positive. – (-5) = 5
(2) set model (jar model)
in the jar model, a  and a  will cancel each other out
4+ -4 =0
II.
Models for adding integers.
An integer is the set of whole numbers with their negatives.
 = integers = {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4 …}
2+3
2+(-3)
(-2)+3
(-2)+(-3)
III.
measurement model
jar model
Integers provide a new model for subtraction.
Old method: take away approach
5-3 means have 5 and take away 3
Old method: missing addend approach
5-3 means 3+? = 5
New method: subtract by adding the opposite
5-3 means 5+(-3) = 2 (using, for example, jar method)
IV.
Multiplying and dividing negative numbers.
We can use our model of multiplication as repeated addition to understand
multiplication with negative numbers.
3*5 means 5+5+5 = 15
3*(-5) means -5 + -5 + -5 = -15
So a positive times a negative is a positive.
Example: 10*-51 = -510
A negative times a negative is a positive.
Why?
Because –a*-b is –(-ab), which means the opposite of negative a*b.
Example: -5*-3 = -(5*-3) = - (-15) = 15
Another way: understanding multiplication as a pattern.
-4*3 = 12
-4*2 = -8
-4*1 = -4
-4*0 = 0
-4*-1 = 4
-4*-2 = 8
04*3 = 12
(a negative times a positive is negative)
(any number times 0 is 0)
(a negative times a negative is positive)