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
Irrational numbers include square roots and nth roots of
A real number can be represented in decimal system also as: 132.92822223333322222....... infinitely long. It can be expressed as a
ratio or a proportion or a percentage.
They are uncountable. There are infinitely many real
numbers. In between two given real numbers, you can find infinitely many real
There are more real numbers
than number of integers.
We have mathematical operators like +,-,/ and * to perform algebraic
operations on real numbers.
1. Commutative property: Order of the operands does not matter.
A + B = B + A
the operations -, / do not obey this property.
A * B = B * A
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 +
* is distributive over + or - .
4. Identity property
For operators +, -, / and *, there exist identity numbers. A number
added to the 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.
Sum or product of any two real numbers is a real number. Same with
division or subtraction.
A * B = AB is a real number A/ B
is a real number
For a real number a, there is a unique real number b, such that a + b = 0 = identity.
b = - a and is called inverse of a for operation +.
a⁻¹ or 1/a
is the inverse of a for the operation /. a + -a = 0 a * 1/a = 1
For a real number a, multiplication with 0 gives 0. a * 0 = 0
Division by 0
Division of a real number by 0 is not defined.
For mathematical purposes, it is denoted by and
is called infinity.
/ infinity = 0