These are the only rules that we need in order to develop
If you cannot figure out how to justify an algebraic statement
directly from these rules, then it is probably false!
Addition is (has):
Commutative a + b = b + a
Associative ( a + b ) + c = a + ( b + c )
Unital a + 0 = a
Inverses a + ( - a ) = 0
Subtraction as inverse operation If a + b = c , then a = c - b
and b = c - a .
Multiplication is (has):
Commutative ab = ba
Associative ( ab ) c = a ( bc )
Unital a 1 = a
Division as inverse operation If ab = c with
and b 0 then and .
Multiplication by Zero Law a 0 = 0
Zero Factor Law If ab = 0, then a = 0 or b =
Combining the two operations we have the
Distributive Law a ( b + c ) = ab + ac and
Distributing over Negatives a ( b - c ) = ab
Remember, the Zero Factor Law is very important since it lets
us find roots of polynomials (or more general functions) by