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Math 60 Properties of the Real Numbers Elementary Algebra Properties of Real Numbers Properties of Addition Properties of Multiplication Commutative Property For all real a, b: ab ba 3 4 43 7 (Order in which we add does not matter) Associative Property For all real a, b, c: a b c a b c Commutative Property For all real a, b: a b ba 3 4 4 3 12 (Order in which we multiply does not matter) Associative Property For all real a, b, c: a b c a b c 2 3 4 2 3 4 9 (Grouping numbers during an addition process is okay) Note: The positions of the numbers DO NOT change. Identity Element There exists a real number, 0, such that for every real a: a0 a, 0a a 2 3 4 2 3 4 24 (Grouping numbers during a multiplication process is okay) Note: The positions of the numbers DO NOT change. Identity Element There exists a real number, 1, such that for every real a: a 1 a , 1 a a (Technical, definition and use of “0”.) Additive Inverse (Opposite) For every real number a there exists a real number, denoted (–a), such that: a a 0 (Technical, definition and use of “1”.) Multiplicative Inverse (Reciprocal) For every real number a except 0 there exists a real number, denoted 1a , such that: 50 05 5 5 1 1 5 5 a 1a 1 3 3 0 (Definition of the term “opposite”.) 3 13 1 (Definition of the term “reciprocal”.) Distributive Property (of Multiplication over Addition/Subtraction) For all real a, b, c: a b c ab ac , a b c ab ac a b c ac bc , a b c ac bc 2 x 3 2 x 6 , y 4 3 3 y 12 (Do not confuse this with the “Associative Property”. “Distributive Property” has two different operations: “”& “+”) Note: We can distribute from the left or the right because of the “commutativity” of multiplication. The commutative and associative properties DO NOT hold for subtraction and division. That is, subtraction and division are NOT commutative and NOT associative. Hence: a b is not equal to b a a b is not equal to b a a b c is not equal to a b c a b c is not equal to a b c 2 3 1 is not equal to 3 2 1 1 2 12 is not equal to 2 1 2 2 3 4 3 is not equal to 2 3 4 5 2 3 4 83 is not equal to 2 3 4 16 These properties mean that order and grouping (association) don’t matter for addition and multiplication, but they certainly do matter for subtraction and division. Properties of Equality Reflexive Property: aa 5=5 Uses: Mainly used for formal proofs in Geometry, and higher mathematics. Symmetric Property: If a b , then b a If 3 = x, then x = 3 Uses: Used frequently in all areas of mathematics to: rewrite portions of our work in more traditional forms, AND/OR to adjust the final form of an answer to be consistent with accepted notation. Examples: Instead of solving 5 = 2x – 3, the symmetric property let’s us solve 2x – 3 = 5. Instead of writing –6 = x for a final answer, the symmetric property let’s us write our final answer as x = –6. Transitive Property: If a b and b c , then a c If x = 5 and 5 = y, then x = y. Uses: Used frequently in formal proofs in Geometry, and higher mathematics. Also used in logical reasoning. Sometimes, used as the basis for making substitutions when solving two equations and two unknowns. (Systems of equations with involving two variables.)