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)