# What is the smallest natural number 1

The natural numbers = {0, 1, 2, 3, 4, ....} are the numbers of counting.

Properties of natural numbers:

• 0 is the smallest natural number.
• Every natural number has a unique one successor.
• There is no other natural number between a natural number n and its successor n + 1.
• There is no greatest natural number.
• The natural numbers are ordered.
• The representation on the number line results in isolated points.

Basic arithmetic rules (axioms) for natural numbers:

Seclusion
• If you add two natural numbers, you get a natural number again. The natural numbers are relative to the Addition completed.
• If you multiply two natural numbers, you get a natural number again. The natural numbers are relative to the Multiplication complete.
• With regard to subtraction and division, the natural numbers are not closed.

Neutral element

• 0 is the neutral element in terms of addition because n + 0 = n
• 1 is the neutral element in terms of multiplication because n * 1 = n

Commutative law

 n + m = m + n It is true Commutative law regarding the addition n * m = m * n It is true Commutative law regarding the multiplication

Associative law

 (n + m) + p = n + (m + p) It is true Associative law regarding the addition (n * m) * p = n * (m * p) It is true Associative law regarding the multiplication

Distributive law

 (n + m) * p = n * p + m * p It is true Distributive law.

: i.e .: n, m, p are any natural numbers.

Special natural numbers

• Even numbers are natural numbers that are divisible by 2.
• Uneven numbers are natural numbers that are not divisible by 2.
• A Prime number is a natural number that is greater than 1 and can only be divided by itself or 1.
Every natural number greater than 1 is either a prime number or can be represented as a product of prime factors.