Question

Give the properties of real numbers?

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Answer 1

A Real number is a value of a number, representing a quantity of something. It can be represented by a point on an infinitely long and continuous line (real number line like X axis). Real numbers include rational numbers and irrational numbers.
Rational numbers include all integers, and fractions of integers.  Irrational numbers include square roots and nth roots of integers like:

A real number can be represented in decimal system also as:  1.2, or   132.92822223333322222.......  infinitely long. It can be expressed as a ratio or a proportion or a percentage like 3/4, √5+1, 1:3, 25%.

They are uncountable. There are infinitely many real numbers. In between two given real numbers, you can find infinitely many real numbers.  There are more real numbers than number of integers.

We have mathematical operators like +,-,/ and * to perform algebraic operations on real numbers.

Properties:

1.    Commutative property
A + B = B + A               But, the operations -, / do not obey this property.
A * B = B * A
Order of the operands does not matter.

2.    Associative property
A + ( B + C )  =  (A +B ) + C     the order of evaluation of which + is done first does not matter.
A * (B * C) = (A *B) * C               A / (B / C) = (A /B) / C

3.    Distributive property
A * ( B + C ) =  A * B + A * C                  A * ( B - C ) =  A * B - A * C 
But  / is not distributive over + and -.

4.    Identity property
For operators +, -, / and *, there exist identity numbers. Like a number added to identity is equal to the number itself.
6 + 0 = 6         6 * 1 = 6         9 / 1 = 9    11 – 0 = 11

5.    Density property
We can always find a real number lying between given two real numbers.  Given 222.221 and 222.222, we can find  222.2215 that lies between them.
There are infinite number of real numbers between given two numbers.