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This does not really make sense to me for several reasons:

  • The integers are usually constructed using a given construction of the natural numbers
  • Historically natural numbers were conceived of before the integers
  • The notation $\mathbb{Z}^+$, is more cumbersome to use than $\mathbb{N}$.

However, I see some good reasons to use this notation, for instance if the basic object of study is the integers, then (for some reason) one might want so signify that the natural numbers is a subset of the integers and hence use $\mathbb{Z}^+$.

What is the history here (if any)? Why do some mathematicians insist on using the notation?

Henrik
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    For a great many of us $\Bbb Z^+$ and $\Bbb N$ are different sets: the latter contains $0$, and the former does not. Moreover, $\Bbb Z^+$ is unambiguous. – Brian M. Scott Dec 11 '14 at 07:15
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    @BrianM.Scott For me, $\mathbb N$ does not include $0$. $\mathbb N_0$ does. – 5xum Dec 11 '14 at 07:19
  • Yes, but for some of us $\mathbb{N}$ does not contain $0$, and even if we insist that it does this does not really answer the question, since then when might as well use $\mathbb{N}^+$ (which seems more natural to me). – Henrik Dec 11 '14 at 07:19
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    @user22705: Why on earth would $\Bbb N^+$ be more natural than $\Bbb Z^+$? (Natürlich wird die Menge der positiven ganzen $\mathbf{Z}$ahlen mit dem Symbol $\Bbb Z^+$ abgekürzt! :-)) – Brian M. Scott Dec 11 '14 at 07:30
  • This Mathworld link explains it all quite well: http://mathworld.wolfram.com/NaturalNumber.html Given the ambiguity with $\mathbb{N}$, I think the correct response would be for modern authors to completely eschew that notation and instead use either $\mathbb{Z}^+$ or $\mathbb{Z}^*$, depending on what is intended. If there is still any chance for confusion - e.g. I haven't seen the latter being used that frequently - then one's meaning should be clearly spelt out when first using the notation. – Deepak Dec 11 '14 at 07:37
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    @BrianM.Scott although Zahl is a good thought about $\mathbb{Z}^+$, I think $\mathbb{N}$ is about "Natural", not about "Number". – Ruslan Dec 11 '14 at 14:28

3 Answers3

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I really think it is just a notation thing. For sure humans considered the natural numbers before we considered the negatives and even zero.

Why I think it is a handy notation. There is no universal agreement on what we call the natural numbers, some people consider zero to be a natural number, while I have never had a book that did not start the enumeration of the naturals with the number one. So the notation $\mathbb{Z}^+$ rids us of this confusion.

H_B
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  • I have never had a book that did not start the enumeration of the naturals with the number one. - Exactly the opposite here. Notations like $\mathbb{C,R,Q,Z,N}$ usually denote groups and rings. But you can't have either without the existence of a neutral element. Like Switzerland. Or zero. – Lucian Dec 11 '14 at 07:46
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    When is $\mathbb{N}$ a group or a ring? – H_B Dec 11 '14 at 10:12
  • And I would say that notations you mentioned are far more used outside of ring and group theory than the other way around. I say this because they are used in almost every "major" branch of mathematics. – H_B Dec 11 '14 at 10:14
  • It's not a question of branches; it's a question of variation in mathematical notation depending on each country's sphere of influence. I was merely pointing out that the conventions you mention are not universal. And yes, I meant semigroup and semiring. Sorry. – Lucian Dec 11 '14 at 17:24
  • Did I not say that in my post? – H_B Dec 11 '14 at 22:14
  • You seem to be saying in your post that $\mathbb Z^+$ is universally accepted as meaning $\mathbb N^*$, and therefore unambiguous, which simply isn't the case. – Lucian Dec 11 '14 at 22:21
  • In the context one should know if we are talking about sets or not. This is obvious. – H_B Dec 11 '14 at 22:33
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There has always been some ambiguity as to whether $\Bbb N$, the natural numbers, includes zero or not due to inconsistent usage by past authors.

Where as using $\Bbb Z^+$, the positive integers, or $\Bbb Z^\ast$, the nonnegative integers, is unambiguous about the matter.

Graham Kemp
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$\mathbb{Z}^+$ means "positive integers."

$\mathbb{N}$ means "natural numbers," which includes zero, no, wait, it doesn't, yeah, it does... Look at this page: http://oeis.org/wiki/Latin_alphabet On the rightmost column of the table, it says:

$\mathbb{N}$ The set of natural numbers, including 0[31], though sometimes also used to mean the same thing but excluding 0.[15]

The "[31]" is a citation for Princeton University Press book, the "[15]" is for a Springer-Verlag book. Both books were published less than a decade ago. Both books have tables of notation, because readers are almost certain to be confused as to whether the author means for $\mathbb{N}$ to include 0 or not.

I can't remember when was the first time I saw $\mathbb{Z}^+$. But if it wasn't in a table of notation, I'm sure I instinctively understood that it means "positive integers."

Robert Soupe
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