1. Be prepared to take notes when the bell rings.
P.1 Real Numbers
Be prepared to take notes when the bell rings.

2. Real Numbers
Set of numbers formed by joining the set of rational numbers and the set of irrational numbers
Subsets: (all members of the subset are also included in the set)
{1, 2, 3, 4, …} natural numbers
{0, 1, 2, 3, …} whole numbers
{…-3, -2, -1, 0, 1, 2, 3, …} integers

3. Rational and Irrational Numbers
A real number that can be written as the ratio 𝑝 𝑞 of two integers, where q ≠0
Example:
1 3 =0.3333
Repeats
1 8 = 0.125
Terminates
= =1. 126
A real number that cannot be written as the ratio of two integers
*infinite non-repeating decimals
Example:
2 = …
𝜋= …

4. Non-Integer Fractions
Real Numbers
Irrational Numbers
Rational Numbers
Non-Integer Fractions
Integers
Negative Integers
Whole Numbers
Natural Numbers
Zero

5. Real Number Line Negative Positive Origin Coordinate:
Positive
Negative
Coordinate:
Every point on the real number line corresponds to exactly one real number called its coordinate

6. Ordering Real Numbers Inequalities <𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 >𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛
<𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 >𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛
≤𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 ≥𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 a b
Example:
a < b
− 14 3 _______− 26
>

7. Describe the subset of real numbers represented by each inequality.

8. Interval: subsets of real numbers used to describe inequalities
Notation
𝑎,𝑏
𝑎, 𝑏
Interval Type
Closed
Open
Inequality
𝑎≤𝑥≤𝑏
𝑎<𝑥<𝑏
𝑎≤𝑥<𝑏
𝑎<𝑥≤𝑏
The endpoints of a closed interval ARE included in the interval. The endpoints of an open interval are NOT included in the interval.
*Unbounded intervals using infinity can be seen on page 4

9. Properties of Absolute Value
𝑎 ≥0
−𝑎 = 𝑎
𝑎𝑏 = 𝑎 𝑏
𝑎 𝑏 = 𝑎 𝑏 , 𝑏≠0

10. Absolute value is used to define the distance (magnitude) between two points on the real number line
Let a and b be real numbers. The distance between a and b is:
𝑑 𝑎,𝑏 = 𝑏−𝑎 = 𝑎−𝑏
The distance between -3 and 4 is:
−3−4 = −7 =7

11. Algebraic Expressions
Variables:
letter that represents an unknown quantity
Constant:
Real number term in an algebraic expression
Algebraic Expression:
Combination of variables and real numbers (constants) combined using the operations of addition, subtraction, multiplication and division
Examples of algebraic expressions:
5𝑥, 2𝑥−3, 4 𝑥 2 +2 , 7𝑥+𝑦
Terms:
Parts of an algebraic expression separated by addition
i.e. 𝑥 2 −5𝑥+8= 𝑥 2 + −5𝑥 +8
𝑥 2 , −5𝑥:𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑡𝑒𝑟𝑚𝑠
8:𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚
Coefficient:
Numerical factor of a variable term
Evaluate:
Substitute numerical values for each variable to solve an algebraic expression

12. Examples of Evaluation
Expression
−3𝑥+5
3 𝑥 2 +2𝑥−1
Value of Variable
𝑥=3
𝑥=−1
Substitute
−3(3)+5
3 (−1) 2 +2(−1)−1
Value of Expression
−9+5=−4
3−2−1=0
Used Substitution Principle: If a=b, then a can be replaced by b in any expression involving a.

13. Basic Rules of Algebra 4 Arithmetic operations: Addition, +
Subtraction, -
Division, / ÷
Multiplication, × ∙
Addition and Multiplication are the primary operations. Subtraction is the inverse of Addition and Division is the inverse of Multiplication.

14. Basic Rules of Algebra 𝑎−𝑏=𝑎+ −𝑏
Subtraction: add the opposite of b
Division: multiply by the reciprocal of b; if b≠0, then
𝑎−𝑏=𝑎+ −𝑏
−𝑏 is called the additive inverse (opposite of a real number)
𝑎 𝑏 =𝑎 1 𝑏 = 𝑎 𝑏
1 𝑏 is called the multiplicative inverse (reciprocal of a real number)
𝑎 𝑏 : 𝑎 𝑖𝑠 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟, 𝑏 𝑖𝑠 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟

15. Let a, b and c be real numbers, variables or algebraic expressions.
Commutative Property of Addition
𝑎+𝑏=𝑏+𝑎
Commutative Property of Multiplication
𝑎𝑏=𝑏𝑎
Associative Property of Addition
𝑎+𝑏 +𝑐=𝑎+ 𝑏+𝑐
Associative Property of Multiplication
𝑎𝑏 𝑐=𝑎 𝑏𝑐
Distributive Property
𝑎 𝑏+𝑐 =𝑎𝑏+𝑎𝑐
Additive Identity Property
𝑎+0=𝑎
Multiplicative Identity Property
𝑎∗1=𝑎
Additive Inverse Property
𝑎+ −𝑎 =0
Multiplicative Inverse Property
𝑎 1 𝑎 =1

16. Let a, b and c be real numbers, variables or algebraic expressions.
Properties of Negation and Equality
−1 𝑎=−𝑎
− −𝑎 =𝑎
−𝑎 𝑏=− 𝑎𝑏 =𝑎 −𝑏
−𝑎 −𝑏 =𝑎𝑏
−(𝑎+𝑏)=(−𝑎)+(−𝑏)
𝐼𝑓 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎+𝑐=𝑏+𝑐
𝐼𝑓 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐=𝑏𝑐
𝐼𝑓 𝑎+𝑐=𝑏+𝑐, 𝑡ℎ𝑒𝑛 𝑎=𝑏
𝐼𝑓 𝑎𝑐=𝑏𝑐, 𝑡ℎ𝑒𝑛 𝑎=𝑏 𝑖𝑓 𝑐≠0

17. Let a, b and c be real numbers, variables or algebraic expressions.
Properties of Zero
𝑎+0=𝑎, 𝑎−0=𝑎
𝑎∗0=0
0 𝑎 =0, 𝑖𝑓 𝑎≠0
𝑎 0 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝐼𝑓 𝑎𝑏=0, 𝑡ℎ𝑒𝑛 𝑎=0 𝑜𝑟 𝑏=𝑜

18. Homework Problems
Page 9 #’s 1-25 odd, 29, odd, 43-47, 51-55, 59, , ,