The standard properties of equality involving real numbers are: Reflexive property: For each real number a, a = a Symmetric property: For each real number a, for each real n
…umber b, if a = b, then b = a Transitive property: For each real number a, for each real number b, for each real number c, if a = b and b = c, then a = c The operation of addition and multiplication are of particular importance. Also, the properties concerning these operations are important. They are: Closure property of addition: For every real number a, for every real number b, a + b is a real number. Closure property of multiplication: For every real number a, for every real number b, ab is a real number. Commutative property of addition: For every real number a, for every real number b, a + b = b + a Commutative property of multiplication: For every real number a, for every real number b, ab = ba Associative property of addition: For every real number a, for every real number b, for every real number c, (a + b) + c = a + (b + c) Associative property of multiplication: For every real number a, for every real number b, for every real number c, (ab)c = a(bc) Identity property of addition: For every real number a, a + 0 = 0 + a = a Identity property of multiplication: For every real number a, a x 1 = 1 x a = a Inverse property of addition: For every real number a, there is a real number -a such that a + -a = -a + a = 0 Inverse property of multiplication: For every real number a, a ≠ 0, there is a real number a^-1 such that a x a^-1 = a^-1 x a = 1 Distributive property: For every real number a, for every real number b, for every real number c, a(b + c) = ab + bc The operation of subtraction and division are also important, but they are less important than addition and multiplication. Definitions for the operation of subtraction and division: For every real number a, for every real number b, for every real number c, a - b = c if and only if b + c = a For every real number a, for every real number b, for every real number c, a ÷ b = c if and only if c is the unique real number such that bc = a The definition of subtraction eliminates division by 0. For example, 2 ÷ 0 is undefined, also 0 ÷ 0 is undefined, but 0 ÷ 2 = 0 It is possible to perform subtraction first converting a subtraction statement to an addition statement: For every real number a, for every real number b, a - b = a + (-b) In similar way, every division statement can be converted to a multiplication statement: a ÷ b = a x b^-1. (MORE) 
