Numbers

Numbers Definition

Numbers are the most important constituents of Mathematics. Numbers are the symbols used to count, measure and label the objects. No concept of Mathematics would have existed without the concept of numbers. It is very difficult to imagine the existence of the modern world of technology without the concept of numbers. So, it is very important for all of us to understand and analyse the numbers definition, types, properties and uses. 


Types of numbers:

The different types of numbers used in Mathematics are as follows. 

  • Natural Numbers: All counting numbers are called natural numbers. It is represented by the set ‘N’. N = {1, 2, 3, 4, 5, ……}

  • Whole Numbers: All the natural numbers along with zero are called whole numbers. Whole numbers are represented by the set W. W = {0, 1, 2, 3, 4, 5, 6, ……}

  • Odd and Even Numbers: The numbers that are completely divisible by two are called even numbers and the numbers that cannot be completely divisible by two are called odd numbers. Ex: 7 is an example of an odd number and 8 is an example of an even number. 

  • Prime and Composite Numbers: The numbers that are divisible only by one and itself are called prime numbers and the numbers that are divisible by any other number apart from one and itself are called composite numbers. 

  • Integers: All the positive and negative whole numbers along with zero are called integers. It is represented by the set I.

  • Fractions and Decimals:  The numbers that can be expressed as a part of a whole or with the help of a decimal point are called fractions and decimals respectively. 

  • Rational and Irrational Numbers: All numbers that can be expressed in the form of a fraction whose denominator is not equal to zero are called rational numbers. Irrational numbers are those numbers that cannot be expressed in the form of a fraction. 

  • Real and Imaginary Numbers: All numbers that exist in reality are called real numbers and the numbers that are assumed to exist, just to explain some Mathematical concepts are called imaginary numbers. 


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Numbers in Words:

It is very easy to represent the numbers in words with the knowledge of place values and face values of numbers. For example, the number 574 can be written in words as: Five Hundred and Seventy four. The place value of a digit in the number to its extreme right has the place value equal to one and the next digit has the place value 10, 100, 1000 and so on. The numbers in words can also be written in numerical forms using the knowledge of these place values. 


Properties of Numbers Definition:

The basic properties of numbers in Mathematics are:

  1. Closure Property:

This property explains that the result of any Mathematical operations on the numbers is a number itself. i.e. The answer obtained by when 2 or more numbers are subjected to addition, subtraction, multiplication or division is in the form of a number only. 


  1. Commutative Property:

According to commutative law, the result obtained by performing Mathematical  operations on numbers does not change by changing the order of operands. Addition and multiplication of numbers satisfies commutative law whereas division and subtraction do not. If H and J are two numbers, then according to commutative law,


H + J = J + H

H x J = J x H


  1. Associative Property:

Associative property is one of the properties of numbers that is related to the grouping of numbers while performing the fundamental operations. According to this law, the order of grouping of numbers does not alter the result obtained. Associative law holds good for addition and multiplication but not for subtraction and division. If H, J and K are three numbers, then associative law states that


H + (J + K) = (H + J) + K

H x (J x K) = (H x J) x K


  1. Distributive Property:

Distributive property of numbers  explains the distribution of addition over multiplication or multiplication over addition. If H, J and K are any three numbers, then distributive property states that:


H (J + K) = (H x J) + (H x K)


  1. Identity Property:

Identity of a number is that number which gives the first number itself as the answer even after the Mathematical operation is performed. Additive identity is zero and multiplicative identity is one. i.e. The sum of zero and any number is the number itself. Similarly, the product of any number and one is the number itself.


H + 0 = H

H x 1 = H


  1. Inverse Property:

The inverse of a number is that number which gives the identity of the number when the Mathematical operation is performed. Additive inverse of a number is negative of that number and the multiplicative inverse of a number is the reciprocal of that number. 


H + (- H) = 0

H x 1/H = 1


Number Series:

There are numerous fascinating series and amazing patterns in the numbers of Mathematics. A list of number series examples is given below.

  • Series of natural numbers

  • Even and odd number series

  • Multiples of a number

  • Arithmetic number series pattern

  • Geometric progression

  • Harmonic progression

  • Square numbers

  • Cube numbers

In all the above mentioned examples, the succeeding terms of the number pattern can be determined by performing few Mathematical operations on the previous term.


Fun Quiz:

  1. Identify whether the following numbers are prime numbers or not. Give reasons for your answer. 

    1. 47

    2. 35

    3. 895

    4. 2791


  1. Fill in the blanks using the properties of numbers. Also state the property used.

    1. 38 + 18 = 18 + ___

    2. 5 + (8 + 7) = (__ + 8) + 7


  1. Write these numbers in words.

    1. 7658

    2. 342

FAQ (Frequently Asked Questions)

1. What are Different Types of Numbers Based on Their Divisibility by 2?

Numbers are the most basic elements of all scientific and Mathematical calculations. Numbers are classified into two types based on  their divisibility by 2. The two kinds of numbers are odd numbers and even  numbers. Even numbers are those numbers which are completely divisible by 2. i.e. the remainder is zero when the number is divided by 2. Odd numbers are the numbers which cannot be completely divided by 2. i.e the remainder is equal to one when the number is divided by 2. An even  number is generally represented as ‘2s’ and the general representation of an odd number is 2n + 1.

2. What is the Fundamental Theorem of Arithmetic?

Arithmetics is a branch of Mathematics that deals with numerical calculations. There are a wide range of number types and patterns in arithmetics. The fundamental theorem of arithmetic states that “Every composite number can be expressed as a product of its prime factors”. Prime numbers are those numbers which cannot be further simplified into simple factors. In other words, prime numbers are the numbers which can be completely divided only by one and itself.