Order properties of real numbers?
The standard properties of equality involving real numbers are:Reflexive property: For each real number a,a = aSymmetric property: For each real number a, for each real number b,if a = b, then b = aTransitive property: For each real number a, for each real number b, for each real number c,if a = b and b = c, then a = cThe 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 + aCommutative property of multiplication:For every real number a, for every real number b,ab = baAssociative 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 = aIdentity property of multiplication: For every real number a,a x 1 = 1 x a = aInverse property of addition: For every real number a, there is a real number -a such thata + -a = -a + a = 0Inverse property of multiplication: For every real number a, a ≠ 0, there is a real number a^-1 such thata x a^-1 = a^-1 x a = 1Distributive property: For every real number a, for every real number b, for every real number c,a(b + c) = ab + bcThe 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 = aFor 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 = aThe definition of subtraction eliminates division by 0.For example, 2 ÷ 0 is undefined, also 0 ÷ 0 is undefined, but 0 ÷ 2 = 0It is possible to perform subtraction first converting a subtraction statement to an additionstatement: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.
