Add new comment
Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.
A game you're almost certain to lose...
What are the challenges of communicating from the frontiers of mathematical research, and why should we be doing it?
Celebrate Pi Day with the stars of our podcast, Maths on the move!
Maths meets politics as early career mathematicians present their work at the Houses of Parliament.
Celebrate this year's International Women's Day with some of the articles and podcasts we have produced with women mathematicians over the last year!
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).