1. Chapter 1: Tools of Algebra 1-1: Properties of Real Numbers
Essential Question: What are the subsets of the real numbers? Give an example of each.

2. 1-1: Properties of Real Numbers
Natural Numbers
Whole Numbers
Integers
1, 2, 3, 4, …
0, 1, 2, 3, 4, …
…-3, -2, -1, 0, 1, 2, 3, 4, …

3. 1-1: Properties of Real Numbers
All real numbers are either rational or irrational
Rational Numbers
Examples:
Can be written as a fraction using integers (Denominator can’t be 0)
Can also be written as either a terminating or repeating decimal
Irrational Numbers
Can’t be written as a fraction using only integers
Decimal form neither terminates or repeats

4. 1-1: Properties of Real Numbers
Example: Which set of numbers best describes the values for each variable?
The cost C in dollars of admission for n people C: n:
The maximum speed s in meters per second on a roller coaster of height h in meters (use the formula: ) S: h:
The park’s profit (or loss) P in dollars for each week w of the year P: w:
Cost is a terminating decimal (like $24.95), so it’s a rational number.
Since we can talk about 0 people, and never fractions of people, then n is going to be a whole number.
Speed is calculated using a square root, so speed will be an irrational number, unless the square root of h is a rational number
Height is measured in rational numbers.
Profit, as with anything involving money, is a rational number.
The week will be a natural number (1 – 52)

5. 1-1: Properties of Real Numbers
Real numbers can be graphed as points on a number line
Example: Graph the numbers
Graph the numbers
Use a calculator to find

6. 1-1: Properties of Real Numbers
Ordering Real Numbers
If a and b are real numbers, then either a = b, a < b, or a > b
There are a number of ways to prove that a < b
Compare a and b on a number line
Determine a positive number that can be added to a to get b
b – a is a positive number
Example: Compare
Because -0.1 – (-0.5) is positive, -0.1 must be greater, so

7. 1-1: Properties of Real Numbers
Finding Inverses
Opposite (additive inverse)
Flip the sign of the number
“Additive inverse” because a + -a = 0
Reciprocal (multiplicative inverse)
Convert the number to an improper fraction and flip the fraction
“Multiplicative inverse” because
Example
Find the opposite and reciprocal of -3.2
Opposite:
Reciprocal:
-(-3.2) = 3.2

8. 1-1: Properties of Real Numbers
Let a, b, and c be real numbers
Property
Addition
Multiplication
Meaning
Closure
a + b is a real number
ab is a real number
Real numbers produce real numbers
Commutative
a + b = b + a
ab = ba
Order doesn’t matter
Associative
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Grouping doesn’t matter
Identity
a + 0 = a, 0 + a = a
a • 1 = a, 1 • a = a
+0 or x1 produces original number
Inverse
a + (-a) = 0
Reciprocals cancel out
Distributive
a(b + c) = ab + ac
Multiply to everything on inside

9. 1-1: Properties of Real Numbers
Example
Which property is illustrated?
6 + (-6) = 0
(-4 • 1) – 2 = -4 – 2
Inverse property of addition
Identity property of multiplication

10. 1-1: Properties of Real Numbers
Finding Absolute Values
The absolute value of a number is it’s distance from 0 on a number line
Distance is always positive
Find |-4|, |0|, |5 • (-2)|, and -|5 • (-2)|
|-4| =
|0| =
|5 • (-2)| =
-|5 • (-2)| =
With absolute value signs, treat them like parenthesis and simplify everything inside the absolute value signs first. 4
|-10| = 10
-|-10| = -10

11. 1-1: Properties of Real Numbers
Assignment
Page 8-9
2 – 54, even problems