Properties of Algebra (aka all the rules that holds the math together!)

Axioms for Rational Numbers All of our axioms for rational numbers are for ONLY addition and multiplication!!!! All of our axioms for rational numbers are for ONLY addition and multiplication!!!! Axiom is just a property that has not been proven but we accept and use to do algebra and prove things Axiom is just a property that has not been proven but we accept and use to do algebra and prove things

Commutative Property Root word is: commute Root word is: commute To commute means to move To commute means to move The numbers move places The numbers move places

Commutative Property Addition: Addition: a + b = b + a a + b = b + a Example: Example: = = 3 +2 Multiplication Multiplication ab= ba ab= ba Example: Example: 2(3) = 3(2) 2(3) = 3(2)

Associative Property Root word: Associate Root word: Associate To associate means to group together To associate means to group together In math, our grouping symbols are the ( ) In math, our grouping symbols are the ( ) Keep the order of the numbers the same!!! Just change the ( ) Keep the order of the numbers the same!!! Just change the ( )

Associative Property Addition Addition a+(b+c)=(a+b)+c a+(b+c)=(a+b)+c Example: Example: 2+(3+5)=(2+3)+5 2+(3+5)=(2+3)+5 Multiplication Multiplication a(bc) = (ab)c a(bc) = (ab)c Example: Example: 2(3·5) = (2·3)5 2(3·5) = (2·3)5

Identity Properties Your identity is who you are Your identity is who you are The same goes for numbers and variables The same goes for numbers and variables 3 is who 3 is and x is who x is 3 is who 3 is and x is who x is The idea with the identity property is you want to get itself back The idea with the identity property is you want to get itself back

Identity Property Addition Addition a + 0 = a a + 0 = a Example: Example: = = 3 Multiplication Multiplication a (1) = a a (1) = a Example: Example: 3 (1) = 3 3 (1) = 3

Inverse Properties The inverse in math means the “opposite” The inverse in math means the “opposite” When we add the opposite of a positive is a negative and vice versa When we add the opposite of a positive is a negative and vice versa When we mult the opposite is the reciprocal When we mult the opposite is the reciprocal In an inverse we want our addition to = 0 and our mult to = 1 In an inverse we want our addition to = 0 and our mult to = 1

Inverse Property Addition Addition a + (-a) = 0 a + (-a) = 0 Example: Example: 3 + (-3) = (-3) = 0 Multiplication Multiplication a(1/a) =1 a(1/a) =1 Example: Example: 3 (1/3) = 1 3 (1/3) = 1

Distributive Property To distribute means to give out To distribute means to give out You are giving the # on the outside of the ( )’s to every # inside the ( ) You are giving the # on the outside of the ( )’s to every # inside the ( ) The distributive property is the only one that includes addition and mult at the same time The distributive property is the only one that includes addition and mult at the same time

Distributive Property of Multiplication over Addition a (b + c + d) = ab + ac + ad a (b + c + d) = ab + ac + ad Example: Example: 4 ( 3x + 2y – 5) 4 ( 3x + 2y – 5) = 4 (3x) + 4(2y) + 4 (-5) = 12x + 8y + -20

Properties of Equality Reflexive Property: a =a Reflexive Property: a =a Example: 4 =4 Symmetric Property : If a=b, then b=a Symmetric Property : If a=b, then b=a Example: If x= 3, then 3=x Transitive Property: If a=b Transitive Property: If a=b and b=c and b=c then a=c then a=c Example: If x=3 and 3=y then x=y

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