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Mastering Algebra: Essential Properties & Axioms

Explore properties of algebra, including Commutative, Associative, Identity, Inverse, Distributive & Equality. Learn key axioms and how they underpin rational numbers to enhance your math skills. Complete practice problems on page 105 (1-18) to reinforce learning.

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Mastering Algebra: Essential Properties & Axioms

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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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