VARIOUS TYPES OF NUMBERS

NUMBER SYSTEM

Natural numbers: Counting numbers are called natural number.

Thus, 1, 2, 3, 4, ...... are all natural number.

Whole numbers: All counting numbers, together with 0, form the set of whole numbers. 

Thus, 0, 1, 2, 3, 4, ..... are all whole numbers.

Integers: All counting numbers, zero and negatives of counting numbers, form the set of integers.

Thus, ....., -3, -2, -1, 0, 1, 2, 3,...... are all integers.

Set of positive integers = { 1, 2, 3, 4, 5, .......... }

Set of negative integers = { -1, -2, -3, -4, ....... }

Set of non-negative integers = { 0, 1, 2, 3, 4, .......}

Set of non-positive integers = { 0, -1, -2, -2, -4, .......}

Even numbers: All multiples of two are called even numbers.

Since, 2, 4, 6, 8, 10, 12, 14, 16, 18, ...... is a multiple of 2.

Therefore each of 2, 4, 6, 8, 10, 16, 18, is an even number.

Every even number is divisible by 2 or 2 is a factor of every even number.

For example: 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, etc. therefore 2, 4, 6, 8, etc. are the multiples of 2.

Multiples: The number obtained by multiplying the given number is called the multiple of given number.

Odd numbers: Numbers which are not divisible by 2 are called odd numbers.

Since, none of 1, 3, 5, 7, 9, 11, 13, .... is divisible by 2, so each of 1, 3, 5, 7, 9, 11, 13, ...... is an odd number.

• Every number is either even or odd.
• A number cannot be both even as well as odd.
• All prime numbers except 2 are odd.

Prime number: A natural number that is divisible by not one (1) and itself is called a prime number.

For example: 2 is divisible only by 1 and itself, therefore 2 is prime number; 5 is divisible by only 1 and itself, therefore 5 is prime number.

To be more clear, note:

• If a natural number has only two factors, it is a prime number.
• 1 (one) is not a prime number as it has only one factor.
• 3 is a prime number as it has only two factors, which are 1 and 7.
• 10 is not a prime number as it has more than two factors i.e.1, 2, 5, 10.

All prime numbers less than 100 are 2, 3, 5, 7, 11, 13 ,17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Composite numbers: Every natural number that has more than two factors is called a composite number.

Since, factors of 10 are 1, 2, 5 and 10.

Therefore, 10 is a composite number.

Similarly, each of 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, etc. is a composite number.

Every even number greater than 2 is a composite number.

Perfect number: A number, the sum of whose factors (except the number itself), is equal to the number, is called a perfect number.

For example: The factors of 6 are 1, 2, 3 and 6. And, 1+ 2+ 3 = 6, so 6 is perfect number.

Co-prime numbers: Any two numbers that do not have a common prime factor are called co-prime numbers. 

For example: 2 and 3,  5 and 7 are the examples of co-prime numbers.

Twin prime: The two prime numbers whose difference is 2 are called twin prime numbers.

Important facts:

• The smallest prime number is 2.
• The only even prime number is 2.
• The first odd prime number is 3.
• 1 is unique number which is neither prime nor composite.
• The least composite number is 4.
• The least odd composite number is 9.
• All natural numbers are whole numbers.
• All whole numbers are not natural numbers as 0 is a whole number but not a natural number.