The arithmetic operations with real numbers are governed by the following axioms:

(1) Closure Axiom of Addition / Multiplication

For real numbers a and b,

a + b is a unique real number
ab is a unique real number

(2) Commutative Axiom of Addition / Multiplication

For real numbers a and b,

a + b = b + a
ab = ba

(3) Associative Axiom of Addition / Multiplication

For real numbers a, b and c,

( a + b ) + c = a + ( b + c )
(ab)c = a(bc)

(4) Identity Axiom of Addition

For any real number a,

a + 0 = 0 + a = a

(5) Identity Axiom of Multiplication

For any real number a,

a(1) = 1(a) = a

(6) Additive Inverse Axiom

For any real number a, there exists a unique real number -a such that

a + (-a) = -a + a = 0

The number -a is known as the additive inverse of a.

(7) Multiplicative Inverse Axiom

For any nonzero real number a, there exists a unique real number
( 1 / a ) such that

a ( 1 / a ) = ( 1 / a ) a = 1

The number ( 1 / a ) is known as the multiplicative inverse or reciprocal of a, where a ¹ 0.

(8) Distributive Axiom

For any real numbers a, b, and c,

a ( b + c ) = ab + ac
a ( b - c ) = ab - ac
( a + b) c = ac + bc
( a - b) c = ac - bc