Natural Numbers

The natural numbers, commonly denoted as ℕ, form the backbone of the entire number system and much of abstract algebra.

Definition

The natural numbers start at 1 and continue indefinitely in the positive direction:

ℕ = {1, 2, 3, 4, 5, ...}

The definition of natural numbers is closely linked to the Peano axioms, a set of axioms for the natural numbers proposed by Giuseppe Peano.

Peano Axioms

  1. There is a natural number 1.
  2. Every natural number has a unique successor, which is also a natural number.
  3. There is no natural number whose successor is 1.
  4. Distinct natural numbers have distinct successors.
  5. If a property belongs to 1, and if the implication (K(n) implies K(n+1)) holds for every natural number n, then the property belongs to every natural number.

Properties of Natural Numbers

Natural numbers have several important properties:

  1. Closure Property: If you add or multiply two natural numbers, the result is always a natural number.

  2. Associative Property: For all natural numbers a, b, and c, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c).

  3. Commutative Property: For all natural numbers a and b, a + b = b + a and a * b = b * a.

  4. Distributive Property: For all natural numbers a, b, and c, a * (b + c) = a * b + a * c.

  5. Identity Property: The number 1 is the multiplicative identity in the set of natural numbers. There is no additive identity in the set of natural numbers.

  6. No Negatives: There is no natural number that, when added to another natural number, results in 0.

Well-Ordering Principle

The well-ordering principle is a fundamental property of the natural numbers. It states that any non-empty set of natural numbers has a least element. This principle is the foundation of many proofs in number theory.

Note

Zero is not traditionally considered a natural number, but some mathematicians include it, depending on the context. For the purpose of this explanation, we've defined natural numbers as starting from 1.