# Arithmetic/Types of Numbers/Natural Number

The natural numbers (i.e. counting numbers) are numbers which are used for counting and ordering.

They can be expressed mathematically as:

ℕ = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... }

Some mathematicians consider 0 to be a natural number. This convention is common in logic and computer science. ℕ⁰, ℕ₀ and ℤ₀+ unambiguously denote the set of non-negative integers, while ℕ*, ℕ⁺, ℕ₁ and ℤ+ unambiguously denote the set of positive integers.

The greatest natural number does not exist: for every possible natural number n, there exists n+1 which is also a natural number.

Natural numbers can be considered the basis of most if not all common number sets. For example, the integers (ℤ) are simply the natural numbers ℕ, 0, and the negatives of the natural numbers. Rational numbers (ℚ) can be defined as the quotients of integers, that is, fractions.

## Even Number

Even numbers are natural numbers that are divisible by 2. That is to say, natural number n is even if there exists another natural number m such that n = 2m

2ℕ = { 2, 4, 6, 8, 10, 12, 14, ... } = { 2⋅1, 2⋅2, 2⋅3, 2⋅4, ..., 2⋅n, ... }

## Odd Number

Odd numbers are natural numbers that are not divisible by 2. That is to say, natural number n is odd if there exists another natural number m such that n = 2m + 1

2ℕ + 1 = { 1, 3, 5, 7, 9, 11, 13, 15...} = { (2⋅1) + 1, (2⋅2) + 1, (2⋅3) + 1, (2⋅4) + 1, ..., (2⋅n) + 1, ... }

## Prime and Composite Numbers

### Prime Number

Prime numbers are natural numbers that are only divisible by 1 and by itself.

P = { 2, 3, 5, 7, 11, 13, 17, 19,...}

### Composite Number

Composite numbers are natural numbers that are the product of some prime numbers. For example:

4 = 2 ⋅ 2
12 = 2 ⋅ 2 ⋅ 3
15 = 3 ⋅ 5

Every natural number, except prime numbers and 1, is composite.

### One

1 is neither a prime nor composite number, as the number is only divisible by itself.