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Natural or Counting Numbers
1, 2, 3, 4...

Whole Numbers
0, 1, 2, 3,4...

Integers
2, 1, 0, 1, 2, 3...

Rational Numbers
 Fractions
 Real numbers can be written as fractions:
 5 = 5/1
 Thus all integers are rational numbers
 Terminating (.5) and repeating numbers (.333) decimals are rational numbers as they can be written as fractions

Irrational Numbers
 They cannot be written as a fraction
 i.e. Pi or the square root of 3

Properties (axioms) of addition
 Closure
 Commutative
 Associative
 Identity Element
 Additive Inverse

Closure
Is when all answers fall into the original set:
(2 + 4 = 6) the set of even numbers is closed
(3 + 5 = 8) two odds equal even, thus it is open

Commutative
Means that the order does not make any difference:
a + b = b + a
Does not hold for subtraction

Associate
Means that the grouping does not make any difference:
(a + b) + c = a + (b + c)
Does not apply to subtraction

Identity Element
Any number added to zero equals the original number:
a + 0 = a

Additive Inverse
Is the opposite (negative) of the number
3 and 3

Properties (axioms) of multiplication
 Closure
 Commutative
 Associative
 Identity Element
 Multiplicative Inverse
 Distributive Property

Closure
Is when all answers fall into the original set:
2 x 4 = 8 (closed)
3 x 5 = 15 (closed)

Commutative
Means that the order does not matter:
a x b = b x a

Associative
Means that the grouping does not make any difference:
(a x b) x c = a x (b x c)
Does not hold for division

Identity Element
For multiplication is 1. Any number multiplied by 1 gives the original number.
a x 1 = a

Multiplicative Inverse
Is the reciprocal of the number. Any number multiplied by its reciprocal equals 1.
a x 1/a = 1; a and 1/a are multiplicative inverses or reciprocals (provide that a does not equal zero)

A Property of Two Operations: Distributive Property
Is the process of distributing the number on the outside of the parentheses to each term on the inside: a(b + c) = a(b) + a(c)
Note: You cannot use the distributive property with only one operation: a(bcd) does not equal a(b) x a(c) x a(d)

