Axioms/Postulates

Axioms always applies to complex and real numbers

Closure Under Addition

2 numbers in a set closed under addition added together should be a number from the same set

Closure Under Multiplication

2 numbers in a set closed under multiplication multiplied together should be a number from the same set

Commutativity of Addition

Numbers added together can change/switch positions and equal the same thing

a+b=b+a

Commutativity of Multiplication

Numbers multiplied together can change/switch positions and equal the same thing

ab=ba

Associativity of Addition

Numbers in an expression or equation can be grouped together or switch with grouped numbers to equal the same thing

(a+b)+c=a+(b+c)

Associativity of Multiplication

Numbers in an expression or equation can be grouped together or switch with grouped numbers to equal the same thing

a(bc)=(ab)c

Distributivity of Multiplication Over Addition

Numbers multiplied to numbers added together can be ‘distributed’ (multiplied into) the added numbers

a(b+c)=ab+ac

Additive Identity

‘0’ means additive identity

Multiplicative Identity

‘1’ means the multiplicative identity

Additive Inverses

When a number is added to the inverse of that number, it equals 0.

b is an inverse of a if and only if a+b=0

Multiplicative Inverses

When a number is multiplied to the inverse of that number, it equals 1.

b is an inverse of a if and only if ab=1

Reflexive Property of Equality

x=x

Symmetric Property of Equality

If x=y, then y=x

Transitive Property of Equality

If x=y and y=z, then x=z

Addition Property of Equality

If a=b, then a+c=b+c

Multiplication Property of Equality

If a=b then ac=bc

Zero Product Property

a*0=0

Multiplication Property of Negative 1

-a=(-1)a

Definition of Subtraction

a-b=a+(-b)

Definition of Division

a/b=a(1/b)