# Name of the set of negative integers

I know that N (natural numbers) is the set of non-negative integers, 0, 1, 2, 3, 4...infinity, and that Z is the set of all integers, both positive and negative. But what is the name or representation of the set of negative integers?

## Answers and Replies

Hurkyl
Staff Emeritus
Gold Member
I don't recall seeing it given a specific name. I sometimes see decorations like
$$\def\ZZ{\mathbb{Z}} \ZZ^+ \, \ZZ^> \, \ZZ^{\geq} \, \ZZ^- \, \ZZ^{\leq}$$​
to specify various subsets of the integers (the positive elements, the elements greater than 0, the elements greater-than-or-equal-to zero, et cetera).

Ugh, it would seem logical that it would have a name independent of denoting a particular part of Z.

CRGreathouse
Homework Helper
Ugh, it would seem logical that it would have a name independent of denoting a particular part of Z.
Yeah, it really doesn't have one. Frankly it's more common to pull an element n from N and write -n, rather than pull an element m from $$\mathbb{Z}^{<}$$ and write m.

HallsofIvy
Homework Helper
Ugh, it would seem logical that it would have a name independent of denoting a particular part of Z.
Why would that seem logical? What do you perceive as the reason for "naming" sets of numbers?

Why would that seem logical? What do you perceive as the reason for "naming" sets of numbers?
Because the negative integers are not the non-negative integers?

Actually N often denotes the positive integers, which again begs the question of why does labeling any set of numbers matter at all.

Actually N often denotes the positive integers, which again begs the question of why does labeling any set of numbers matter at all.
There is no strong convention, but Euler considers 0 to be part of the set of natural numbers; I'll go with him on it ;)