# What is the difference between natural numbers and positive integers?

I was reading sets and came to some reserved letters for a few sets. Two of them really confused me. They were -

$\mathbb N$ : For the set of natural numbers.

$\mathbb Z^+$ : For the set if all positive integers.

In my sense, both the sets contain $\{1,2,3,\dots\}$ Then, why are they considered different?

I searched a little on this topic and got this, but it doesn't tell anything about significance of two different sets.

• Often N is defined as $\{0,1,2,3,..\}$; then the sets are different. Look up the definition of it in the book/article you are reading. – Karl Apr 5 '15 at 7:36
• – Marcus Andrews Apr 5 '15 at 7:38
• @Karl Natural numbers don't contain 0. – Hritik Apr 5 '15 at 7:39
• @Hritik Please read en.wikipedia.org/wiki/Natural_number ,before you make such apodictic statements. – Karl Apr 5 '15 at 7:42
• @Karl WTH ! Wiki page says: "There is no universal agreement about whether to include zero in the set of natural numbers. Some authors begin the natural numbers with 0 , corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corresponding to the positive integers 1, 2 , 3, ..." I'm seriously disappointed with my elementary school. – Hritik Apr 5 '15 at 7:48

You should be aware that some authors define $\mathbb{N}$ to include zero. This isn't of much consequence in itself since the properties of the set are preserved: there is a bijection between $\mathbb{N}$ with zero and $\mathbb{N}$ without zero, both are well-ordered, and so forth—effectively, we've done nothing but "relabel" the elements.

Only when we start adding structure to these elements does the distinction become important. For instance, if we define an addition $+: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, we might make $0$ an additive identity. Therefore, when one writes "$\mathbb{N}$" in such a scenario (most scenarios), then it should be made clear which definition is intended.

Now, if we take both to mean the set $\{1, 2, 3, \cdots\}$, then whether one writes $\mathbb{N}$ or $\mathbb{Z}^+$ is immaterial. However, using $\mathbb{Z}^+$ removes ambiguity since $\mathbb{Z}^+$ definitively does not include zero, and we would not have to go out of our way defining $\mathbb{N}$.

• thats the answer i was waiting @Kaj Hansen – user210387 Apr 5 '15 at 7:42
• What if you are questioned: Write the elements of set N? Will you include zero? Why is this ambiguity? – Hritik Apr 5 '15 at 7:46
• @Hritik the ambiguity is due to history, some places had $0$ included and others didn't. If I was asked to write the natural numbers, I'd ask them for which definition I should use. – Alice Ryhl Apr 5 '15 at 7:47
• Alternatively, if you were asked to define the natural numbers, you could do so axiomatically. With this method, you begin by declaring an element is a natural number, and it really doesn't matter what this element is called, be it $0$ or $1$ or $\Delta$ - this is the idea I'm trying to convey. Any set that satisfies the axioms we call the "natural numbers". See here for more information: en.wikipedia.org/wiki/Natural_number#Peano_axioms – Kaj Hansen Apr 5 '15 at 7:51
• @Hritik yes, different people mean different things when they say "natural number", also if you want to write fancy maths, look here. – Alice Ryhl Apr 5 '15 at 8:00

The positive integers are $\mathbb Z^+=\{1,2,3,\dots\}$, and it's always like that.

The natural numbers have different definitions depending on the book, sometimes the natural numbers is just the postivite integers $\mathbb N=\mathbb Z^+$, but other times the natural numbers are actually the non-negative numbers $\mathbb N=\{0,1,2,\dots\}$.

Some people also write $\mathbb N_0=\{0,1,2,\dots\},\mathbb Z^+=\{1,2,3,\dots\}$ and completely avoid $\mathbb N$ due to this ambiguity.

If you want to be completely unambigiuous, you should use the words positive integers and nonnegative integers for these sets.

• What ? Natural numbers contain 0 ? I'm not talking about some books, laws of Maths must be universal. – Hritik Apr 5 '15 at 7:42
• @Hritik some people use different definitions for $\mathbb N$, what you call things doesn't change the laws of maths. Math is invariant under notation. – Alice Ryhl Apr 5 '15 at 7:44
• @Hritik, ...but definitions contain a certain degree of arbitrariness! – Karl Apr 5 '15 at 7:46
• This just shows how bad my education system is. Alas, they never told us about the ambiguity in natural numbers. We were taught that natural number does not contain 0 whereas whole numbers contain 0. Happy that I learned something new today. – Ardent Sep 11 '19 at 8:06
• @Kaushik well the whole numbers typically also include the negative ones, so that's yet another one: $\mathbb Z = \{ \dots, - 2,-1,0,1, 2,\dots\}$. – Alice Ryhl Sep 12 '19 at 9:09

Regarding the question of whether or not the natural numbers should include zero, there are two arguments in favor of doing so that I find compelling:

1) By including zero, the natural numbers can then be used to indicate cardinalities for all finite sets. If zero is not included, then the cardinality of the empty set is missing.

2) As John Conway pointed out, we already have a perfectly good way to describe the set $\{1, 2, 3, \ldots \}$, namely the positive integers. (JC was arguing why not to exclude zero from the natural numbers.)

• We also have a perfectly good way to describe the set $\{0, 1, 2, 3, \ldots\}$, namely the non-negative integers. Therefore I do not accept argument 2. – celtschk Apr 6 '15 at 22:11
• Monoids are often more useful than semigroups without additive identities. Counting numbers (natural numbers without "zero") are a semigroup that can be used to define 1x=x and (N+1)x = Nx+x for any semigroup element x. Natural numbers including zero can be used to extend that definition so that applying 0x to any monoid element x will yield its additive identity. Being able to apply Nx to semigroups (which won't work if N can be zero) can be helpful, but so too can be the ability to use 0x to get the additive identity for x's monoid. – supercat Jun 9 '15 at 16:31

I think both are same. Because zero is not included in a set of natural numbers.

• That depends on your definition of natural numbers. – Glorfindel Jan 28 '17 at 10:08