Number System : Definition with Examples

Number System 

Number system is a mathematical notation for representing numbers using different kind of symbols or digits and it allows us to operate arithmetic operations like addition, subtraction, multiplication.

Number System

Natural Numbers 

The natural numbers are the numbers used for counting and ordering, they are also known as counting numbers and they are represented by N.
e.g ➤ N= 1, 2, 3, 4, 5............ ∞ ( infinite )
Note ➠ zero should not be included 


Whole Numbers 

When we include zero in natural numbers then it becomes whole number, and we represent it by W.
Whole Number = Zero + Natural Numbers 
e.g ➤ W= 0, 1, 2, 3, 4, 5...........∞ ( infinite )
Note ➠ (i) All natural numbers are whole numbers, but not all whole numbers are natural numbers.
(ii) Zero is a whole number, but not a natural number.


Integer Numbers

When we represent whole numbers with positive and negative signs then we call it integer numbers or integers, and we represent it by I.
e.g ➤  I =   -∞ ......-3, -2, -1, 0, 1, 2, 3......∞

(i) Positive Integer Numbers

Natural numbers are called positive integer numbers. e.g ➤ 1, 2, 3, 4, ..............

(ii) Negative Integer Numbers 

If natural numbers are written with minus sign, then those numbers are called negative integer numbers. e.g ➤ -1, -2, -3, -4, ..............


Rational Numbers 

The numbers which can be written in the form of P/Q is called rational numbers.
e.g ➤ 1/2, 3/2, 7/8, √1.........
Where (i) in P/Q,  Q ≠ 0
(ii) P and Q are Integers


Irrational Numbers

The number which can not be written in the form of P/Q is called irrational numbers.
Where P and Q are integers and ( Q ≠ 0 )
Note ➠ π is a irrational number but the value of π (3.14) is a rational number. 


Real Numbers 

A group of rational numbers and irrational numbers is called real numbers, and we represent it by R.
e.g ➤ R= 0, -2, 2/3, -6/15, √2 etc


Imaginary Numbers 

Numbers that are not real but just imagined, those are called imaginary numbers.
e.g ➤ √-3, √-15
Note ➠ (i) Thus the square root of these negative numbers cannot be derived.
(ii) They are also written as follows in the language of mathematics (a+ib), (a-ib), (x+iy),(-ix+y), we add I with them and this is very important for higher mathematics.

Real numbers

Even Numbers 

Natural numbers which can be completely divided by 2 are called even numbers.
e.g ➤ 2, 4, 6, 8, 10, 12, ............
Commonly these are represent as 2m where m= Natural Number


Odd Numbers

Natural numbers which cannot be completely divided by 2 are called odd numbers.
e.g ➤ 1, 3, 5, 7, 9, ............
Commonly these are represent as 2m - 1 where m= Natural Number  


Prime Numbers

All numbers greater than 1, which only can be divided by itself or by 1 are called prime numbers.
e.g ➤ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ..............


Composite Numbers

All numbers greater than 1, which has at least one divisor other than 1 and itself, is called composite number.
e.g ➤ 4, 6, 9, 15, etc


Co - Prime Numbers

Two natural numbers whose HCF is equal to 1 is called co-prime numbers. 


Important Points

Identity Element of Addition

Zero is called identity element of addition because when we add zero to any number it do not change the result.

Identity Element of Multiplication 

1 is called identity element of multiplication because when we multiply 1 to any number it do not change the result.

Additive Inverse

Adding a number to the number which make the sum zero is called additive inverse.

Multiplicative Inverse 

Multiplying a number to the number which make the product 1, that number is called multiplicative inverse 
e.g ➤ Multiplicative inverse of x is 1/x


If you have any doubt or question let me know in the comments. 

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