Classification of Numbers

When we start learning maths we start it with counting the numbers. Numbers are quite simple to see but plays very important role in maths and statistics.

In short, we can say that without number nothing is there in mathematics.

In our daily life we use numbers for counting which are starting from 1,2,3,… and so on. These numbers are called as the counting numbers.

Mainly numbers are classified as real numbers and imaginary numbers.

Real numbers are those which are having only real existence. While imaginary numbers are those which has both real and imaginary parts.

In this article we see all about real numbers and their classification.

Mainly real numbers are classified as

  1. Natural numbers
  2. Whole numbers
  3. Integers
  4. Rational numbers
  5. Prime numbers
  6. Composite numbers
  7. Even numbers
  8. Odd numbers

1.) Natural numbers:

  • Natural numbers are the counting numbers which starts from 1, 2, 3, 4, 5,…up to infinity.
  • Set of natural numbers contains all positive numbers starting from 1.
  • And set of natural numbers is represented by N.
  • We have to note that set of natural numbers doesn’t includes zero.

2.) Whole numbers:

  • As the name is whole numbers it includes all the natural numbers including zero also.
  • Hence, set of whole numbers is starting from 0, 2, 3, 4, 5,…up to infinity.
  • Set of natural numbers is the proper subset of whole numbers.
  • Set of whole numbers is represented by W.

3.) Integers:

  • Set of integers is the largest subset of the set of real numbers.
  • Set of integers is represented by I.
  • It includes all positive and negative real numbers including zero also.
  • That means, set of integers contains positive integers (non negative) and negative integers (non positive) including zero.
  • It includes …-3, -2, -1, 0, 1, 2, 3,…
  • Here, the set of whole numbers and set of natural numbers both are the proper subset of set of integers.

4.) Rational numbers:

  • Those numbers which are not written in whole number form are the rational numbers that means they are written in the form of fractions or ratios.
  • The form of rational numbers is as p/q where p and q both are the integers and q should not be zero.

For example:

  • 3/5, 1/2 are the rational numbers because if we solve these fractions then we get the answer in decimal form not in the whole number form.

Such as 1/2= 0.5 and 3/5= 0.6

Hence, these are the rational numbers.

  • Rational numbers includes both positive and negative rational numbers.

 

For example:

  • 7 is rational or irrational number?

We know that, every number written in the form of p/q where p and q are both integers and q should not be zero are the rational numbers.

So, 7 can be written as 7/1 where 7 and 1 are the integers and denominator is not zero here.

So, we have written 7 in the form of ratio and hence it is also the rational number.

  • Similarly, 8 can be written as 8/1 which is also the rational number.
  • Thus, we can say that all the integers can be written in the form of ratio as we written above hence all the integers are the rational numbers.
  • Hence, we can say that the set of integers is the proper subset of set of rational numbers.

Note:

π is the rational or irrational number?

Ans: we know that,

π~22/7

As the value of π is not exactly equal to 22/7 but it is nearly equal to 22/7.

Hence, π is the irrational number or π is not the rational number.

Note:

Those numbers which are not rational numbers are called as the irrational numbers.

For example:

  • √4= 2 and 2 can be written as 2/1 hence, we can say that √4 is the rational number.
  • But, √3= 1.732 and which cannot be written in the form of ratio.

Hence, √3 is the irrational number.

  • Similarly, √2= 1.414 which is also not written in the form of ratio so √2 is also the irrational number.
  • In this way, we can say that, all the square roots of the non perfect squares are the irrational numbers.

Natural numbers are again classified into prime numbers and composite numbers.

1.) Prime numbers:

  • Prime numbers are those natural numbers which are only divisible by 1 and the number itself.
  • Prime numbers starts from 2, 3, 5, 7, 11,…and so on.
  • Also we can say that, prime numbers are those which are not expressed in the form of product of numbers other than one and the number itself.

For example:

  • 2 can be written as only in the form 2= 2*1
  • Also, 3 can be written as 3= 3*1
  • Similarly, 5 can be written as only 5= 5*1
  • And 7 also written in the form of 7=7*1 only.
  • Such natural numbers which are divisible by one and the number itself are the prime numbers.
  • Here, 2 is the lowest prime number and it is also the first and only one even prime number here.
  • All the other prime numbers other than 2 are the odd prime numbers only.

Note:

1 and 0 are neither the prime numbers nor the composite numbers.

2.) Composite numbers:

  • Composite numbers are those natural numbers which are divisible by 1, the number itself and also by the other numbers.
  • That means they have factors other than 1 and the number itself.
  • And those natural numbers which are not prime numbers are the composite numbers.

For example:

  • 4, 6, 8, 9, 10,…and so on are the composite numbers.
  • Because these numbers are written in the form of product of other numbers also.
  • Such as, 4= 2*2 and 4= 4*1
  • 6= 6*1 and 6= 3*2
  • Also, 8= 8*1 and 8= 4*2
  • Similarly,

9= 9*1 and 9= 3*3

  • In this way, we can express the composite numbers in the form of product of other numbers.

Note:

0 and 1 are neither prime nor composite numbers.

Integers are again classified into odd numbers and even numbers.

1.) Even numbers:

  • Even numbers are the both positive and negative integers.
  • Even numbers are those integers which are divisible by 2.

For example:

…, -6, -4, -2, 0, 2, 4, 6,…and so on all are the even numbers as these numbers are divisible by 2.

2.) Odd numbers:

  • Odd numbers are those positive and negative integers which are not divisible by 2.
  • Also, we can say that all the integers which are not even numbers are the odd numbers.

For example:

…-5, -3, -1, +1, +3, +5…and so on all are the odd numbers as they are not divisible by 2.


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