Summary for what I have *read* last night.. take note “read” only, I haven’t internalize the theorems and axioms *yet*.. =)

I just type them down, so correct me if I have mistaken anything :)

**Closure**– the sum of any two natural numbers is a natural number.

**Associative axioms**– a chain of addition and multiplication of numbers are grouped, the result is *the same* no matter how the numbers are grouped.

**Commutative axioms**– the order of the numbers involved in addition or multiplication is immaterial.

**Distributive axioms**– multiplication is distributed over addition.

**Identity axiom**: a x 1 = a, 1 is a unique natural number.

**Additive identity**: 0 is the additive identity.

**Additive inverse**: (-a) is the additive inverse of a.

**Theorem 1**: additive inverse of the additive inverse of a number itself.

**Theorem 2**: cancellation law for addition.

**Theorem 3**: the product of any number and 0 is 0.

**Theorem 4**: the product of a number and the additive inverse of another number equals the additive inverse of their product.

**Theorem 7**: the product of the additive inverse of two numbers equals to their product.

**Theorem 8**: -(a+b) = (-a) + (-b)

**Theorem 9**: b – a is the unique solution to a + x = b

**Theorem 10**: a – b = c if and only if a = b + c, a = b + c if and only if a – b = c

**Multiplicative inverse**: 1/a is the multiplicative inverse of a

**Theorem 11**: 1 over reciprocal of a equal a.

**Theorem 12**: cancellation law for multiplication. ac = bc —-> a = b

**Theorem 13**: ab = 0, if the product of 2 numbers is zero, then at least one of them is zero. a = 0 or b = 0

**Theorem 14**: b/a is the unique solution to ax = b

**Theorem 15**: a/b = c if and only if a = bc

**Theorem 16**: a/1 = a

**Theorem 17**: c/c = 1

**Theorem 18**: products of quotients. (b/a) (d/c) = bd/ac

**Theorem 19**: equality of quotients. b/a = d/c —-> ad = bc

**Theorem 20**: quotient of quotients. d/c ÷ b/a = d/c x a/b <— reciprocal

**Theorem 21**: sums of quotient. b/a + c/d = bd + ac÷ad

**Theorem 22**: d = ac + b, d/a = c + b/a

ughhh I can’t continue no more, I’m reading the last Theorem (27).

anything you want to ask?

Ask me ^^

Reyes, Fe N. (2002) *College Algebra, Vol. 1* Philippines: UPOU OASIS

(repost from my tublr)