Permalink Submitted by Anonymous on February 27, 2014

The closure of the natural numbers under addition means that the sum of any two natural numbers is a natural numbers. We can apply this principle again and again (finitely many times) to see that the sum of any finite number of natural numbers is a natural number. This however says nothing of the sum of an infinite number of naturals, which we have in the case of an infinite series; in fact we see that this need not exist at all.

Similarly, whilst rational numbers are closed under addition also, the sequence S given in the article for pi/6 gives an example of a series of rational numbers converging to an irrational number (the rational numbers are also not closed in another sense, as a sequence of rational numbers can converge to an irrational number).

## The closure of the natural

The closure of the natural numbers under addition means that the sum of any

twonatural numbers is a natural numbers. We can apply this principle again and again (finitely many times) to see that the sum of any finite number of natural numbers is a natural number. This however says nothing of the sum of an infinite number of naturals, which we have in the case of an infinite series; in fact we see that this need not exist at all.Similarly, whilst rational numbers are closed under addition also, the sequence S given in the article for pi/6 gives an example of a series of rational numbers converging to an irrational number (the rational numbers are also not closed in another sense, as a sequence of rational numbers can converge to an irrational number).