TYPES OF NUMBERS

TYPES OF NUMBERS
Numerals : Any figure or symbol that is used to indicate a number is a numeral.
Digits: Any arabic numeral (0-9) is a digit. Digits are used to form natural numbers and whole numbers.
When a natural number or a whole number is 9 or lower, it is also a digit. Natural or whole number higher than nine are formed by combining multiple digits
What are Numbers?
A number is a basic component of mathematics. Numbers are used for counting, measuring, keeping things in order, indexing, etc. We have different types of numbers based on their properties such as natural numbers, whole numbers, rational and irrational numbers, integers, real numbers, complex numbers, even and odd numbers, etc. We can apply the basic fundamental arithmetic operations of numbers and determine the resulting number. 
Different Types of Numbers :
Natural Numbers :
Natural numbers are also called “counting numbers” which contains the set of positive integers from 1 to infinity. The set of natural numbers is represented by the letter “N”. The natural number set is defined by:
N = {1, 2, 3, 4, 5, ……….
Whole Numbers
Whole numbers are also known as natural numbers with zero. The set consists of non-negative integers where it does not contain any decimal or fractional part. The whole number set is represented by the letter “W”. The whole number set is defined by:
W = {0,1, 2, 3, 4, 5, ……….}
Integers
Integers are defined as the set of all whole numbers with a negative set of natural numbers. The integer set is represented by the symbol “Z”. The set of integers is defined as:
Z =  {..............-3, -2, -1, 0, 1, 2, 3
Rational Numbers
Rational numbers are the numbers that can be written in the form p/q, when p and q are integers and 
 Examples: 7/1, 10/2, 1/1, 0/1, etc.
Irrational Numbers
The number that cannot be expressed in the form of p/q. It means a number that cannot be written as the ratio of one over another is known as irrational numbers. It is represented by the letter ”P”.
Examples: \(\sqrt{2}\) , p, Euler’s constant, etc 
Real Numbers :
Any number such as positive integers, negative integers, fractional numbers or decimal numbers without imaginary numbers are called the real numbers. It is represented by the letter “R”.
Examples: ¾, 0.333,\(\sqrt{2}\) , 0, -10, 20, etc.
Complex Numbers :
A number that is in the form of a+ib is called complex numbers, where “a and b” 
should be a real number and “i” is an imaginary number.
Examples: 4 + 4i, -2 + 3i, 1+ i\(\sqrt{2}\), etc
Along with the above numbers we come across different numbers like cardinal numbers, ordinal numbers, consecutive numbers, even and odd numbers, factors and multiples, prime numbers, composite numbers, co prime numbers, perfect numbers, prime factorization, LCM, HCF, fractions and decimals.
Imaginary numbers : It is the product of real number with the imaginary unit ‘i’.
Ex : i², 3i etc
where i = \(\sqrt{-1}\) so i² = -1
        i³ = -i, i⁴ = +1
Cardinal numbers & ordinal numbers : 
    A cardinal number is a number that denotes the count of any object. Any natural number such as 1, 2, 3, etc., is referred to as a cardinal number, whereas, an ordinal number is a number that denotes the position or place of an object. For example, 1st, 2nd, 3rd, 4th, 5th, etc. It indicates the order of things or objects, such as first, second, third, fourth, and so on|
Consecutive Numbers: 
    Consecutive numbers are numbers that follow each other in order from the smallest number to the largest number. They usually have a difference of 1 between every two numbers
Example: 23, 24, 25, 2
Even Numbers and Odd Numbers :
    Even numbers are those numbers that can be exactly divided by 2 by leaving the remainder as 0. 
Ex: 0, 2, 4, 6, 8, 10, ............ 
    Odd numbers are whole numbers that cannot be completely divided by 2.
    Odd numbers cannot be arranged in pairs. Interestingly, all the whole numbers except the multiples of 2 are odd numbers.
Ex : 1, 3, 5, 7.......
    When they are divided by 2 we get 1 as the remainder
Factors and Multiples
    Factors are the numbers that divide the given number completely without leaving any remainder, whereas the multiples are the numbers that are multiplied
by the other number to get specific numbers
Factors of a given number are numbers that can perfectly divide that given number.
Examples:
Factors of 6: 1, 2, 3, 6
Factors of 8: 1, 2, 4, 8
A multiple of a number is a number obtained by multiplying the given number by another whole number
Examples:
Multiples of 3: 3, 6, 9, 12, 15, ......
Multiples of 5: 5, 10, 15, 20, 25, .…
Prime Numbers and Composite Numbers
    A prime number is a number that has exactly two factors, 1 and the number itself. For example, 2, 5, 7, 11, and so on are prime numbers. 
    A composite number is a number that has more than two factors, which means it can be divided by the number 1 and itself, and at least one more integer.
Ex : 4, 6, 8, 9 ....... 
(factors of 6 other than 1 & 6 are 2, 3)
factors of 8 other than 1 & 8 are 2, 4
Co-prime Numbers : (Relatively prime numbers) 
    A pair of numbers whose HCF is 1 are co-primes.
If a pair of numbers has no common factor apart from 1, then they are called co-prime numbers.
Ex : (8, 15) = 1
    8 & 15 are co prime
Perfect Numbers :  Perfect numbers are the positive integers that are equal to the sum of its factors except for the number itself. In other words, perfect numbers are the positive integers that are the sum of their proper divisors.
Ex : Factors of 6 are 1, 2, 3 and 6 
    Sum of factors except 6 is 1 + 2 + 3 =
Prime factorization : Prime factorization allows us to write any number as a product of prime factors. It is a way of expressing a number as a product of its prime factors. To do prime factorization, we need to break a number down to its prime factors.
Prime factorization of 12

12 = 2 x 2 x 3 = 22 x 3
    Where 2 & 3 are prime factors.
LCM : The smallest common multiple of given numbers is Least Common Multiple (LCM) of two numbers.
Ex : LCM of 8 & 10
    Multiples of 8 are 8, 16, 24, 32, 40, 48......
    Multiples of 10 are 10, 20, 30, 40, 50.....
    LCM of 8 & 10 = 40
HCF : Largest Common Factor of given number is Highest Common Factor or Greatest common factor (GCF) or Greatest Common Divisor (GCD)
Ex : HCF of 15 & 21
Factors of 15 are 1, 3, 5, 15
Factors of 21 are 1, 3, 7, 21
Common factors are 1, 3
Highest Common Factor is 3.
Fractions and Decimals :
    Fraction is a part of a whole. They are represented by numbers that have two parts to them.There is a number at the top, which is called the numerator, and the number at the bottom is called the denominator.

Ex : \({8\over 7},{2\over 7},{110\over 119}\) 
A decimal number has a whole number part and a fractional number part. These are separated by a decimal piont.
Ex : 27 .7
Whole number part    Fractional part
Decimal piont
    Decimal numbers can also be represented as fractions.
Ex : 27 .7=\(277 \over 10\)