NUMBERS
TYPES OF NUMBERS
NATURAL NUMBERS :- Numbers used for counting are called natural numbers or counting numbers. The collection of natural numbers represented by 'N'.
N={1,2,3,4,5,6, ... }
The least natural number is 1.
The largest natural number does not exist.
The difference between any two consecutive natural numbers is one.
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WHOLE NUMBERS :- Natural numbers along with zero are called whole numbers. The collection of whole numbers represented by 'W'.
W={0,1,2,3,4,5,6, ...}
The least whole number is 0.
The largest whole number does not exist.
The whole number but not a natural number is 0.
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INTEGERS :- Whole numbers along with negative signed natural numbers are called integers. The collection of integers represented by 'I".
I={....,-3,-2,-1,0,1,2,3,4,...}
The least integer does not exist
The largest integer does not exist.
Zero is neither positive nor negative.
Integers but not natural numbers are 0,-1,-2,-3,-4,-5,.....
Integers but not whole numbers are -1,-2,-3,-4,-5,....
Collection of integers also represented by 'Z'. It is taken from the word 'Zahlen' a German word for number. This is because mathematicians made contributions to number theory, the branch of mathematics concerning theorems about the integers ( in German called Zahlenthoerie)
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RATIONAL NUMBERS :-The word 'rational' is derived from the word 'ratio'. The numbers in the form of p/q where p,q are integers mutually prime and q not equal to zero. The p/q form is fraction form or quotient form. Rational numbers can also be called fractions or Quotient numbers. Hence the collection of quotient numbers is denoted by 'Q'.
Q={p/q /p,q are integers mutually prime and q not equal to zero}
Every integer is also a rational number because 3 = 3/1 , -5= -5/1, 0=0/1 etc. which are in p/q form.
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The rational numbers after performing division the decimal value attained. The decimal value contains two parts separated by a dot. The part on the left side of the dot is whole part and on the right side is the fractional part. The whole part is called integral part of the number and fractional part is called decimal part of the number. This decimal value may terminating or non terminating with repetition.
For example :- 1) 3/4 = 0.75 , 1/35 = .02857 etc. are terminating
2) 1/3 = .3333.... , 13/7 = 1.857142857... etc. are not terminating and repeating.
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IRRATIONAL NUMBERS :- The numbers which cannot be written in the form of p/q where p,q are integers and q not equal to zero are called irrational numbers. These numbers are also called as surds. The collection of irrational numbers is represented by 'S'.
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REAL NUMBERS :- The collection of rational and irrational numbers are called Real numbers. The collection of real numbers is represented by 'R'.
R = Q U S
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EVEN NUMBERS :- Numbers which are exactly divisible by 2 are called even numbers.
Example:- 2,4,6,8,10,12,14,16, ...
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ODD NUMBERS :- Numbers which are not exactly divisible by 2 are called odd numbers.
Example:- 1,3,5,7,9,11,13,15, ...
Note:- Zero is neither even nor odd.
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FACTORS:- Exact divisors of a number are called its factors.
Example:- Factors of 24 are → 1,2,3,4,6,8,12,24.
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PRIME FACTORS :- A factor which is a prime number is called a Prime Factor.
Example:- Factors of 24 are → 1,2,3,4,6,8,12,24.
Of these 2 and 3 are prime numbers. These two numbers are called prime factors of 24.
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MULTIPLES:- Numbers obtained on multiplying a given number by whole numbers are called its multiples.
Example:- Multiples of 3 are → 3,6,9,12,15,18, ...
Note:- Zero is multiple for all numbers.
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PRIME NUMBERS :- A numbers having exactly two factors, itself and 1 are called prime numbers. They are not divisible by any natural number other than itself and 1.
Example:- 2,3,5,7,11,13,17,19, ...
Note:- One is not a prime number as it has only itself as factor.
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COMPOSITE NUMBERS :- The numbers having more than two factors ( other than 1 and itself ) are called composite numbers. (or) The other natural numbers which are not prime numbers are called composite numbers.
Example:- 4,6,8,9,10,12, .....
Note:- One is neither a prime number, nor a composite number.
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MUTUAL PRIME NUMBERS :- A pair of two natural numbers having no common factors, other than 1, are called mutual prime numbers (or) co-primes.
Example:- (3,5), (5,6), (11,17), ....
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TWIN PRIME NUMBERS :- A pair of Prime numbers differing by two are called twin prime numbers.
Example:- (3,5), (5,7), (11,13), ....
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