Real Numbers:Important Definition

Real Numbers:Important Definition

Natural and Whole Number


What is Natural Numbers?
Set of counting numbers is called the Natural Numbers
N = {1,2,3,4,5,...}

What is whole number?
Set of Natural numbers plus Zero is called the Whole Numbers
W= {0,1,2,3,4,5,....}
Note: 
So all natural Number are whole number but all whole numbers are not natural numbers

Integers

What are Integers Numbers
Integers is the set of all the whole number plus the negative of Natural Numbers
Z={..,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...}
Note
1) So ,integers contains all the whole number plus negative of all the natural numbers
2)the natural numbers without zero are commonly referred to as positive integers
3)The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer 
4)natural numbers with zero are referred to as non-negative integers
5) The natural numbers form a subset of the integers.


Rational and Irrational Numbers

Rational Number
: A number is called rational if it can be expressed in the form p/q where p and q are integers ( q> 0).
Example : 1/2, 4/3 ,5/7 ,1 etc.
Important Points to Note
.every integers, natural and whole number is a rational number as they can be expressed in terms of p/q
.There are infinite rational number between two rational number
.They either have termination decimal expression or repeating non terminating decimal expression
.The sum, difference and the product of two rational numbers is always a rational number. The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure property under addition, subtraction, multiplication and division.


Irrational Number
: A number is called rational if it cannot be expressed in the form p/q where p and q are integers ( q> 0).
Example : √3,√2,√5,p etc.
Important Points to Note
Pythagoras Theorem: .In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using this theorem we can represent the irrational numbers on the number line.

.They have non-terminating and non-repeating decimal expression.

.The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division

Real Numbers:

.All rational and all irrational number makes the collection of real numbers. It is denoted by the letter R.

.We can represent real numbers on the number line. The square root of any positive real number exists and that also can be represented on number line
The sum or difference of a rational number and an irrational number is an irrational number.

.The product or division of a rational number with an irrational number is an irrational number.

.This process of visualization of representing a decimal expansion on the number line is known as the process of successive magnification