4 ms·
It is somewhat controversial: according to Wikipedia There is no universal agreement about whether to include zero in the set of natural numbers. Today
by ot 12y ago
It is somewhat controversial: according to Wikipedia
There is no universal agreement about whether to include zero
in the set of natural numbers. Today some textbooks,
especially college textbooks, define the natural numbers to
be the positive integers {1, 2, 3, ...}, while others,
especially primary and secondary textbooks, define the term
as the non-negative integers {0, 1, 2, 3, ...}.
Among mathematicians, the most common (by far) definition is the one given by Peano axioms, which include 0.
Including 0 is very convenient for a number (pun intended) of reasons, for example it makes the naturals a monoid, and it is easier to define the integers.
- ska 12y agoPedantic: they are still a commutative monoid without 0, under multiplication with identity 1. You need 0 as the identity for addition, which is fundamental in these constructions.
- OscarCunningham 12y agoI'm not sure if it is more common to say that 0 is a natural number. There's a math.stackexchange thread about this which seems to have equally many people saying that each is the more common one. http://math.stackexchange.com/questions/283/is-0-a-natural-number http://math.stackexchange.com/questions/283/is-0-a-natural-n... (I count 4 people saying {1,2,3,...} is more common vs 3 saying {0,1,2,...} is more common. I also notice that N_0 is used by both sides to refer "unambiguously" to the convention of the other side.) In my experience {1,2,3,...} is used more in number theory, and {0,1,2,...} is used more in logic.