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Properties of Algebra

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!!!! Axiom is just a property that has not been proven but we accept and use to do algebra and prove things.

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Properties of Algebra

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

  2. Axioms for Rational Numbers • 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

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

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

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

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

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

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

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

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

  11. Distributive Property • To distribute means to give out • 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

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

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

  14. Homework • Page 105 (1-18) all

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