Key Properties and Axioms in Real Number Algebra

Reflexive Property

a = a

Symmetric Property

If a = b, then b = a.

Transitive Property

If a = b and b = c, then a = с.

Substitution Axiom

If a = b, then in any true sentence involving a we may substitute b for a and obtain another true sentence.

Closure Axiom of Addition

For all real numbers a and b, a + b is a unique real number.

Associative Axiom of Addition

For all real numbers a, b, and c, (a + b) + c = a + (b + c).

Additive Identity

There exists a unique real number 0 (zero) such that
a + 0 = 0 + a = a for every real number a.

Additive Inverses

For each real number a, there exists a real number

Commutative Axiom of Addition

For all real numbers a and b, a + b = b + a.

Closure Axiom of Multiplication

For all real numbers a and b, ab is a unique real number.

Associative Property of Multiplication

For all real numbers a, b, and c, (ab)c = a(bc).

Multiplicative Identity

There exists a unique nonzero real number 1 (one) such that 1· a = a· 1 = a.

Multiplicative Inverses

For each nonzero real number a there exists a 1 real number (the reciprocal of a) such that a(1/a) = (1/a)a =1

Commutative Property of Multiplication

For all real numbers a and b, ab = ba.

The Distributive Axiom of Multiplication over Addition

For all real numbers a, b, and c, a(b + c) = ab + ac.

Cancellation Law for Addition

a = b if and only if a + c = b + c.

Cancellation Law for Multiplication

a = b if and only if ac = bc (c ≠ 0)

Theorem 1 Part 3

If a = b, -a = -b.

Theorem 1 Part 4

-(-a) = a

Theorem 1 Part 5

a * 0 = 0

Theorem 1 Part 6

-0 = 0

Theorem 1 Part 7

-a = -1(a)

Theorem 1 Part 8

- (ab) = a(−b) = (-a)b

Theorem 1 Part 9

-(a + b) = −a + (-b)

Theorem 1 Part 10

If a ≠ 0, 1/-a = -1/a = -(1/a)

Theorem 2

For all real numbers a and b: ab = 0 if and only if a = 0 or b = 0.