Arithmetics

These are the only rules that we need in order to develop arithmetic/algebra.

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

Inverses

Division as inverse operation If ab = c with a 0 and b 0 then and .

Multiplication by Zero Law a 0 = 0

Zero Factor Law If ab = 0, then a = 0 or b = 0.

Combining the two operations we have the

Distributive Law a ( b + c ) = ab + ac and its cousin

Distributing over Negatives a ( b - c ) = ab - ac

Remember, the Zero Factor Law is very important since it lets us find roots of polynomials (or more general functions) by factoring.