Classification of Number System

Real numbers are those numbers which you can commonly identify and qualify.

For example: -10, -8.33, -1, -0, 1, 2, 5.77 etc.

Number line is a line on which all the positive and negative numbers can be marked in a sequence. It stretches from negative infinity to positive infinity.

All the numbers which can be represented on the number line are called real numbers.

Iimaginary numbers are those numbers which cannot be represented on number line. Imaginary numbers are those numbers about which we can just imagine but cannot physically perceive. 

For example: $\sqrt {-1} $, $\sqrt {-2} $, etc. Square roots of all negative numbers are imaginary.

$\sqrt {-1} $ is represented by the letter ‘i’.

Combination of real and imaginary numbers are called complex numbers.

For example, (2+3i) is a complex number.

All the real numbers can be divided into two parts. They are:

• Rational numbers
• Irrational numbers

All the numbers that can be expressed in the form of p/q, where p and q are integers and q not equals to zero, are called rational numbers.

For examples, -1, 2, $\frac {-2} {5} $, 0, 1, 1.7 etc.

We can illustrate the examples mentioned above, in p/q in the following manner:

-1 = $\frac {-1} {1} $ = $\frac {p} {q} $(q 0)

$\frac {2} {5} $ = $\frac {p} {q} $(q=0), 0=$\frac {0} {1} $

1.7 = $\frac {17} {10} $ = $\frac {p} {q} $

All the numbers that cannot be expressed in the form of $\frac {p} {q} $ are called irrational numbers.

For example, $\pi$, $\sqrt {2} $ etc.

Typically they are having non-terminating and non-recurring decimal part.

Rational numbers can be expressed as integers and fractions.

All the rational numbers which do not have any decimal or fractional part are called integers.

For example, -3,-2,-1, 0, 1, 2,3etc.

All the rational numbers which are in the form of $\frac{p}{q}$, where p and q are not equal to zero and p is not a multiple of q are called fractions. The number on the bottom of the fraction is called the denominator and the number on the top of the fraction is called the numerator.

For example, 1,2, $\frac{5}{2}$ , $\frac{4}{3}$, 1.7 etc.

In the fraction $\frac{5}{2}$, 5 is the numerator and 2 is the denominator.

proper, improper and mixed.

A proper fraction is a fraction whose numerator is smaller than its denominator.

For example, $\frac{2}{5}$ , $\frac{3}{11}$ are proper fractions.

An improper fraction is a fraction whose numerator is equal to or greater than its denominator.

For example, $\frac{5}{2}$ and $\frac{8}{3}$ are improper fractions.

A mixed fractions is an integer plus a fraction.

For example, 3$\frac{1}{5}$ or 7$\frac{1}{3}$. Mixed fraction are sometimes called mixed numbers.

Learn how to do fractions step by step, Here is an example which shows How to add fractions.

• Whole Numbers

• Natural Numbers

All the non-negative integers are whole numbers. The set of whole number contains 0,1,2,3,4 etc

Whole numbers except zero are called natural numbers. Natural numbers start from 1.

All the natural numbers which are multiple of 2 or in other words divisible by 2 are called even numbers such numbers are denoted by 2k where k is a natural numbers.

For example, 2,4,6,8,….

All the natural numbers which are not multiple of 2 or in other words which are not divisible by 2 called odd numbers.

Such numbers are denoted by 2k+-1 where k is a natural number.

On the basis of their origin:

All the natural numbers greater than 1 which are only divisible by 1 and the number it is called prime numbers. In other words we can say that each prime number is divisible by only two numbers.

For example, 2,3,5,7,11,13,17 etc.

Prime numbers are randomly distributed in the set of natural numbers.

All the natural numbers greater than 1 which are divisible by at least one more number, other than 1 and the number itself are called composite numbers.

In other words the natural numbers which are not prime are composite numbers.

Note: 1 is neither prime nor composite.

How to find whether a number is prime or not?

For small numbers, we could find by checking, if that number is divisible by any other number till that number itself.

But for the larger numbers like, say 631, there is an alternative method.

Step 1: find the approximate square root of the given number, i.e. 25.

Step 2: check if any prime number from 2 to 25 divides 631.

The prime numbers from 2 to 25 are 2, 3, 5,7,11,13,17,19 and 23. Since none of these numbers divides 631 exactly, 631 must be prime number.