Order properties of real numbers

Answer 1

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 number 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.