1. Numbers Sets
Natural Numbers – Counting numbers. Does not include 0 (example: 1,2,3,4…)
Whole Numbers – All Natural numbers and the number zero (example: 0,1,2,3,4…)
Why is zero important?
Additive Identity – Any number added to zero keeps it’s identity
Place Value – Holds a place when there isn’t a quantity for that value

2. Integers – All whole numbers AND THEIR opposites (example: … -3, -2, -1, 0, 1, 2, 3…)
Rational Numbers – Any number that can be written in the form a/b where both a and b are integers and b is not equal to zero.
In simple terms – all numbers that you have ever dealt with EXCEPT NON-terminating, NON-repeating decimals (example: π )

4. Open and Closed Sets
A set is a grouping of numbers (example: whole numbers, integers, rational numbers, etc)
A set is considered to be closed under an operation if the result of the operation on two numbers in the set is another member of the set
Integer + Integer = Integer ALWAYS TRUE
Natural num + Nat. Num = Nat Num ALWAYS TRUE
Therefore these sets are closed operations

5. Properties
Additive Identity – A number such that when you add it to a second number, the sum is equal to the second number (example : = 4)
Multiplicative Identity – A number such that when you multiply it by a second number, the product is equal to the second number (example: 4 x 1 = 4)
Additive Inverse – Two numbers are additive inverses if their sum is the additive identity ( or 0)
(example: 3 + (-3) = 0 )
Multiplicative Inverse – Two numbers are multiplicative inverses if their product is the multiplicative identity (or 1) (example: 5 x 1/5 = 1 )

6. Commutative Property of Addition: Changing the order of two or more addends in an addition problem does not change the sum
( a + b = b + a)
Commutative Property of Multiplication: Changing the order of two or more factors in a multiplication problem does not change the product
( a x b = b x a )
Associative Property of Addition: Changing the grouping of the addends in an addition problem does not change the sum
( a + b ) + c = a + ( b + c )
Associative Property of Multiplication: Changing the grouping of the factors in a multiplication problem does not change the product
( a x b ) x c = a x ( b x c )

7. Powers of Rational Numbers
Power – a number written using a base and an exponent
4 5 = 4 x 4 x 4 x 4 x 4
base exponent
When calculating the power of a rational number
( ¾ )4 = 3 4 / 4 4

8. Operations with Powers
Multiplication:
(1/4)2 x (1/4)3 = (1/4)2+3 Add the exponents
(5)4 x (5)2 = (5)4 + 2
Division:
(1/4)4 ÷ (1/4)2 = (1/4)4 – Subtract the exponents
(5)4 ÷ (5)2 = (5)4 - 2