What is the reason why the natural numbers (1,2,3,4,5...) are not a group under addition ?

Question

What is the reason why the natural numbers (1,2,3,4,5...) are not a group under addition ?

Answer 1

Shouldn't this be in the math section?

First: I supposed there are two ways to define natural numbers, one which includes 0, the other which you show that does not include 0. (I myself prefer to include 0) In any case, a group requires an identity element, and in the case for addition, 0 is the identity element. That is, a + 0 = a for all a in the set of natural numbers. So first reason (with your definition of natural numbers) why it is not a group under addition is because it lacks the identity element.

Suppose we do add 0 into the set of natural numbers. Is it a group under addition? No. Another property which a group must include is that the inverse of each element must be in the group. What does this mean exactly? Let b be the inverse of a. Then there be some such b, such that b + a = 0. So if a = 2, then b = -2, but -2 is not in the set of natural numbers.

All that being said, the set of integers under addition is a group.