What are some of the most used properties of real numbers.

You have asked too many questions at the same time. I have
edited your question as you are allowed to ask only one
question.

Some of the properties of real numbers
are:

Commutative properties: This applies to addition and
multiplication and states that for two real numbers a and b, a + b = b + a and a*b =
b*a

Associative properties: This states that for three real
numbers a, b and c, (a + b) + c = a + (b + c) and (a*b)*c =
(a*(b*c)

Distributive property: This states that
for three real numbers a, b and c, a*(b + c) = a*b +
a*c

Density property: This states that for any
two real numbers it is possible to find another real number that lies between the two.
Put another way, there are an infinite number of real numbers.

Identity properties: For any number a, a + 0 = a, a is
called the additive identity. Similarly for multiplication a*1 = a, 1 is the
multiplicative identity.