1. REAL
NUMBERS

2. If you were asked to count, the numbers you’d say are called counting numbers. These numbers can be expressed using set notation.
These are also called the natural numbers.
{1, 2, 3, 4, }
If we include 0 we have the set of whole numbers.
{0, 1, 2, 3, 4, }
{ …, -3, -2, -1, 0,1, 2, 3, }
Include the opposites of the whole numbers and you have the set of integers.

3. Whole numbers are a subset of integers and counting numbers are a subset of whole numbers.
rational numbers
integers
whole numbers
counting numbers
This means to divide one integer by another or “make a fraction”
If we express a new set of numbers as the quotient of two integers, we have the set of rational numbers

4. There are numbers that cannot be expressed as the quotient of two integers. These are called irrational numbers.
REAL NUMBERS
rational numbers
integers
irrational
numbers
whole numbers
counting numbers
The rational numbers combined with the irrational numbers make up the set of real numbers.

5. Translating English to Math
sum of two numbers
difference between two numbers
The product of two numbers
the quotient of two numbers is
a + b
a - b ab b a =

6. P E M D A S ORDER OF OPERATIONS arenthesis ultiplication xponents
When there is more than one symbol of operation in an expression, it is agreed to complete the operations in a certain order. A mnemonic to help you remember this order is below.
Complete multiplication and division from left to right
Complete addition and subtraction from left to right
Apply exponents
Do any simplifying possible inside of parenthesis starting with innermost parenthesis and working out
arenthesis
xponents
ultiplication
ivision
ddition
ubtraction
P E M D A S

7. PEMDAS PEMDAS PEMDAS PEMDAS PEMDAS exponents – apply the exponent now
complete addition and subtraction, left to right
complete multiplication and division, left to right
parenthesis – combine these first
PEMDAS
PEMDAS
PEMDAS
PEMDAS
PEMDAS

8. COMMUTATIVE PROPERTY
The operations of both addition and multiplication are commutative
When adding, you can “commute” or trade the terms places
When multiplying, you can “commute” or trade the factors places

9. ASSOCIATIVE PROPERTY
The operations of both addition and multiplication are associative
When adding, you can “associate” and add any terms first and then add the other term.
When multiplying, you can “associate” and multiply any factors first and then multiply the other factor.

10. DISTRIBUTIVE PROPERTY
The operation of multiplication distributes over addition
The distributive property also holds for a factor that is multiplied on the left.

11. CAUTION: Remember that the value for a and/or b could also be positive or negative.
Rules of Signs
A positive times a negative is
A negative times a positive is
NEGATIVE
NEGATIVE
The negative of a negative
POSITIVE
A negative divided by a positive is
NEGATIVE
A positive divided by a negative or
A negative divided by a negative is
POSITIVE