Closure of the natural numbers under addition and multiplication

The set of natural numbers $ \mathbb{N} $ is closed under addition and multiplication.

In rigorous mathematical terms, this means that whenever addition or multiplication is applied to two elements of $ \mathbb{N} $, the result is again an element of $ \mathbb{N} $, that is, a natural number.

For this reason, addition and multiplication are referred to as internal operations, or equivalently as binary operations that are closed on the set $ \mathbb{N} $ of natural numbers.

Closure under addition

Addition is an internal operation on $ \mathbb{N} $ because, for every pair of natural numbers $ a $ and $ b $, their sum $ a + b $ belongs to $ \mathbb{N} $.

$$ \forall a, b \in \mathbb{N}, \quad a + b \in \mathbb{N} $$

This formal statement expresses the fact that adding two natural numbers always yields a natural number, so the operation never produces an element outside the set $ \mathbb{N} $.

Example. If we compute $ 3 + 5 = 8 $, the result $ 8 $ is a natural number. This confirms that $ \mathbb{N} $ is closed under addition.

Closure under multiplication

In the same way, multiplication is an internal operation on $ \mathbb{N} $ because, for every pair of natural numbers $ a $ and $ b $, their product $ a \cdot b $ also belongs to $ \mathbb{N} $.

$$ \forall a, b \in \mathbb{N}, \quad a \cdot b \in \mathbb{N} $$

Consequently, multiplying two natural numbers always produces a natural number.

Example. If we multiply $ 3 \cdot 5 = 15 $, the result $ 15 $ is again an element of the set of natural numbers. This confirms closure under multiplication.

It is important to emphasize, however, that although the set of natural numbers is closed under addition and multiplication, it fails to be closed under the corresponding inverse operations.

The set of natural numbers is not closed under subtraction or division.

Example. The difference $ 5 - 8 $ is not a natural number. In situations of this kind, in order to restore closure, it is necessary to extend the set $ \mathbb{N} $ to the set of integers $ \mathbb{Z} $, which also includes negative integers.

The closure of the natural numbers under addition and multiplication plays a significant role in several mathematical contexts:

Definition of algebraic structures: Closure with respect to one or more operations is a fundamental requirement in the definition of algebraic structures such as groups, rings, and fields. Although the natural numbers do not form a group, since additive inverses are not available, closure under addition and multiplication makes it possible to construct algebraic structures such as monoids, namely sets equipped with an associative operation and an identity element.
Use in mathematical proofs: The closure property is frequently invoked in proofs to guarantee that the result of an operation performed on elements of a set remains within the same set. For instance, proofs by mathematical induction rely on the fact that the sum of natural numbers is itself a natural number.

And so on.