# Zero is a natural number **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [February 18, 2004, 7:27am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762 "2004-02-18T07:27:01Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [February 18, 2004, 7:27am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/1 "2004-02-18T07:27:01Z")
In [Is zero a number?](http://www.straightdope.com/mailbag/mzero.html) it is stated that the natural numbers start with the unit. This certainly comes as a shock, given that the second-order Peano axioms (the definition of the structure of **N** ) start with “There exists an element 0”. ---
**Author:** ![C\_K\_Dexter\_Haven](https://avatars.discourse-cdn.com/v4/letter/c/b2d939/32.png) [@C\_K\_Dexter\_Haven](https://boards.straightdope.com/u/C_K_Dexter_Haven)\ **Post date:** [February 18, 2004, 1:55pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/2 "2004-02-18T13:55:41Z")
First, **Mathochist** , welcome to the Straight Dope Message Boards, and thanks for providing a link to the column! Interestingly, it looks like this is a definitional question with different authorities using different definitions of the natural numbers. When I wrote the Staff Report, I used John Olmsted’s classic _The Real Number System_ (1962) which defines **N** the Natural Numbers as the set of positive integers {1, 2, …}. And I’ve also now checked H.L.Royden’s _Real Analysis_ (1968) which also identifies the natural numbers as the positive integers. Admittedly, my texts are a little old, back from the days when I was doing my math Ph.D. in the early 1970s. Background: Peano (in 1889 and after) was trying to define a number set, let’s call it the “Natural Numbers.” His axioms are: (1) That there is a starting point, some “initial number” that is in the set (we’ll come back in a minute) (2) That there is an order – that is, given a number n in the set, there is a “successor” number n’ that comes next (3) That successor numbers are distinct – that is, if n and m are distinct (unequal), then n’ and m’ are distinct (4) That the initial starting point is not a successor to any number in the set (5) The Induction Axiom: If there is some property such that (a) the initial number has that property, AND (b) if any number n has that property, then its successor number n’ also has that property; THEN the entire number set has that property. OK, now where it gets interesting is that a quick websearch for Peano’s Axioms shows two different versions of the first axiom. Some list 0 as the initial number for the set, and others list 1. The remaining axioms are identical in all lists, just slightly reworded. So, I guess it depends on how you want to define things. If you use 0 as the initial number, then you get a different set of natural numbers than if you use 1 as the initial point. In the Staff Report, I made it fairly clear that I was defining “natural numbers” to be equivalent to “counting numbers”, and I don’t think there’s any ambiguity about “counting numbers.” I’m kind of time pressed at the moment, so I’m not able to resolve this. There are several possibilities: A. The definition of Natural Numbers used by Peano in 1889 started with 0, and was changed sometime after that but before the 1960 to start with 1 B. The definition of Natural Numbers used in my texts has changed recently (since the 1960s) C. There is no universally agreed definition of the term “Natural Numbers.” ---
**Author:** ![Pop1](https://avatars.discourse-cdn.com/v4/letter/p/5fc32e/32.png) [@Pop1](https://boards.straightdope.com/u/Pop1)\ **Post date:** [February 18, 2004, 4:40pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/3 "2004-02-18T16:40:43Z")
> [@C K Dexter Haven](#): > > (5) The Induction Axiom: If there is some property such that (a) the initial number has that property, AND (b) if any number n has that property, then its successor number n’ also has that property; THEN the entire number set has that property. It seems to me that starting with zero would give you a problem here. Take n[sup]2[/sup]=n 0_0 = 0 1_1 = 1 Axiom 5 would then have it that all integers would follow this rule. But 2\*2 =/= 2 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 18, 2004, 9:35pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/4 "2004-02-18T21:35:10Z")
> [@Pop1](#): > > It seems to me that starting with zero would give you a problem here. > > Take n[sup]2[/sup]=n > 0_0 = 0 > 1_1 = 1 > > Axiom 5 would then have it that all integers would follow this rule. > But 2\*2 =/= 2 No it wouldn’t. Induction requires that n[sup]2[/sup] = n implies that (n + 1)[sup]2[/sup] = n + 1 for all n. There’s no implication from 1 to 2, so there’s no problem. Anyway, there are constructions of **N** that contain 0, and constructions that don’t. The standard construction has 0 = {}, n’ = n U {n} (the successor of n is denoted n’), and addition and multiplication as defined below: x + 0 = x x + y’ = (x + y)’ x \* 0 = 0 x \* y’ = x + (x \* y) But, there’s no reason why you couldn’t take 1 = {}, n’ = n U {n}, and do some alternate definitions: x + 1 = x’ x + y’ = (x + y)’ x \* 1 = x x \* y’ = x + (x \* y) This has all the properties of the naturals without 0, but most people use the first construction–that’s why I called it the standard. Anyway, as soon as you construct **Z** from **N** , you do get 0 no matter which version of **N** you start with. ---
**Author:** ![C\_K\_Dexter\_Haven](https://avatars.discourse-cdn.com/v4/letter/c/b2d939/32.png) [@C\_K\_Dexter\_Haven](https://boards.straightdope.com/u/C_K_Dexter_Haven)\ **Post date:** [February 19, 2004, 1:16pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/5 "2004-02-19T13:16:07Z")
**ultra** , if the most common use of **N** is now {0,1,2,…} then I will amend the Staff Report, but I’d like a little verification. As I said, when I did my degree back in the late 60s and early 70s, **N** was the same as the “counting numbers.” Do you know when or why it changed? ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [February 19, 2004, 1:26pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/6 "2004-02-19T13:26:46Z")
My undergraduate days were '90 to '93 and we were told then that some people counted 0 as a natural number and some didn’t. ---
**Author:** ![theunknown](https://avatars.discourse-cdn.com/v4/letter/t/dec6dc/32.png) [@theunknown](https://boards.straightdope.com/u/theunknown)\ **Post date:** [February 19, 2004, 1:29pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/7 "2004-02-19T13:29:08Z")
When I went to high school, just a few years ago (about 2) we learned that **N** is {1, 2, 3…} and that zero appears in the whole number unit **W** {0, 1, 2…} as well as some others (real numbers, intergers) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 19, 2004, 4:19pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/8 "2004-02-19T16:19:03Z")
> [@C K Dexter Haven](#): > > **ultra** , if the most common use of **N** is now {0,1,2,…} then I will amend the Staff Report, but I’d like a little verification. As I said, when I did my degree back in the late 60s and early 70s, **N** was the same as the “counting numbers.” Do you know when or why it changed? No, I don’t. In my undergraduate days ('98 to '02), I think I only encountered one author who excluded 0 from the natural numbers. For what it’s worth, the first construction I gave is due to von Neumann, and it may well be that his influence finally won out. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [February 19, 2004, 5:28pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/9 "2004-02-19T17:28:31Z")
I posted this yesterday and it got eaten by starving hampsters. Dang laundry baskets anyway. I pulled out my copy of the classic Hardy and Wright “An Intro. to the Theory of Numbers” 5th edition, 1983 printing, orig. pub. 1938. It uses only the phrases such as “integers”, “positive integers” etc. No notation for them at all. By the time I was in college, all undergrad and graduate courses (CS and Math) used **N** for {0,1,2,…} and **Z** [sup]+[/sup] for {1,2,3…}. The use of “counting” and “whole” numbers in el-hi is inconsistent. They are bad terms anyway and should be thrown out. [Wikipedia](http://en.wikipedia.org/wiki/Natural_number) does note that some people leave 0 out. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [February 19, 2004, 7:24pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/10 "2004-02-19T19:24:37Z")
Most of the books I’ve seen lately (especially undergraduate algebra texts at various levels) define the natural (= counting) numbers as **N** = {1, 2, 3, …} and the whole numbers as {0, 1, 2, 3, …}; but I do have at least one book (a graduate [modern/abstract] algebra text) that considers the natural numbers to be {1, 2, 3, …}. I guess the bottom line is, if you’re going to write about “natural numbers,” just be sure to specify exactly what you mean. ---
**Author:** ![bonzer](https://avatars.discourse-cdn.com/v4/letter/b/45deac/32.png) [@bonzer](https://boards.straightdope.com/u/bonzer)\ **Post date:** [February 19, 2004, 11:59pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/11 "2004-02-19T23:59:27Z")
> [@C K Dexter Haven](#): > > I’m kind of time pressed at the moment, so I’m not able to resolve this. There are several possibilities: > A. The definition of Natural Numbers used by Peano in 1889 started with 0, and was changed sometime after that but before the 1960 to start with 1 > … Judging by the translation from the Latin (\*) in _From Frege to Godel_ (Harvard, 1967, p94) ed. by Jean van Heijenoort, Peano’s 1889 version defined N as “number (positive integers)” and defined 1 as what you call the “initial number”. The use of N may just be an imposition on the part of the translator, but I’d be surprised if they’d changed a 0 to that 1. (\*) I’d have guessed Italian. So, as of this evening, now my best guess as to the last significant scientific work written in Latin. ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [February 20, 2004, 12:29am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/12 "2004-02-20T00:29:36Z")
My maths lecturers (Cambridge, UK if anyone cares) have done it both ways. When looking for maths cites online I generally rely on three sites: [http://planetmath.org/encyclopedia/NaturalNumber.html](http://planetmath.org/encyclopedia/NaturalNumber.html) , [http://mathworld.wolfram.com/NaturalNumber.html](http://mathworld.wolfram.com/NaturalNumber.html) , and [http://en.wikipedia.org/wiki/Natural\_numbers](http://en.wikipedia.org/wiki/Natural_numbers) . The consensus seems to be: [ul] [li]Peano’s original formulation included 0.[/li][li]Now, both ways are used. Typically set theorists like {}=0, and number theorests like to exclude 0. Both ways work fine, but have to include ‘except…’ or ‘…and 0’ in different places.[/li][li]Some people use ‘counting number’, ‘whole number’, or something else, to try to remove ambiguity, but not always the same way! (I rely soly on mathworld, here, I haven’t seen these used since school, and can’t remember how they were defined.) I think **N** 0 means N including 0, but it might be the other way round, I can’t remember.[/li][li]I think ‘positive integer’ including 0 and ‘non-negative integer’ not including zero are universal, but I’ve seen people use them wrong.[/li][li]Usually you can tell from context. If it matters I personally try to make it explicit, sometimes using subscript U{0} or {0}, but that’s just me.[/li][/ul] Yeah, I wish there was a convention, but there doesn’t seem to be. I hope I’ll be corrected… ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 20, 2004, 1:19am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/13 "2004-02-20T01:19:43Z")
[QUOTE=Shade] [li]I think ‘positive integer’ including 0 and ‘non-negative integer’ not including zero are universal, but I’ve seen people use them wrong.[/li][/QUOTE] Did you mean this the other way round? ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [February 20, 2004, 1:27am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/14 "2004-02-20T01:27:26Z")
> [@ultrafilter](#): > > Did you mean this the other way round? :smack: :smack: :smack: Yes. Damn. Thanks. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [February 20, 2004, 2:06am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/15 "2004-02-20T02:06:12Z")
“0 and 1 are more or less the same thing anyway.” -my Algebraic Topology prof ---
**Author:** ![C\_K\_Dexter\_Haven](https://avatars.discourse-cdn.com/v4/letter/c/b2d939/32.png) [@C\_K\_Dexter\_Haven](https://boards.straightdope.com/u/C_K_Dexter_Haven)\ **Post date:** [February 20, 2004, 3:27am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/16 "2004-02-20T03:27:38Z")
\<\< “0 and 1 are more or less the same thing anyway.” \>\> Quite true, of course, for an algebraist, since each is the unit under a group operation (addition and multiplication, respectively.) A topologist however, would probably say that 0 has a homotopy group equal to the integers, while 1 has a homotopy group of 0 (that is, non-technically, you can shrink the symbol 1 to a point, but you can’t shrink the symbol 0 to a point because it has a hole in it.) A donut is different from a pancake, for a topologist. Having re-read the Staff Report, it clearly defines what _I_ meant by the Natural Numbers, since it shows the sequence to be {1, 2, 3 …} so there is no ambiguity. Since mathematicians are divided (some are multiplied), and there is no consensus, I shall leave the Staff Report as it stands, with thanks for an interesting thread. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 20, 2004, 3:33am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/17 "2004-02-20T03:33:16Z")
> [@C K Dexter Haven](#): > > A donut is different from a pancake, for a topologist. And the donut is not different from a coffee cup. Do we really want these people telling us what’s the same? (for all you topologists who might be offended: ;)) > [@](#): > > Having re-read the Staff Report, it clearly defines what _I_ meant by the Natural Numbers, since it shows the sequence to be {1, 2, 3 …} so there is no ambiguity. Since mathematicians are divided (some are multiplied), and there is no consensus, I shall leave the Staff Report as it stands, with thanks for an interesting thread. After reading this thread, I’m starting to think that different specialists might have different definitions. The von Neumann ordinals (the first construction I gave) probably are the standard for set theorists and logicians, but analysts and arithmeticists might have a different opinion. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [February 20, 2004, 5:47am UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/18 "2004-02-20T05:47:38Z")
> [@C K Dexter Haven](#): > > Having re-read the Staff Report, it clearly defines what _I_ meant by the Natural Numbers, since it shows the sequence to be {1, 2, 3 …} so there is no ambiguity. Since mathematicians are divided (some are multiplied), and there is no consensus, I shall leave the Staff Report as it stands, with thanks for an interesting thread. Maybe you could just put in a parenthetical statement that some mathematicians start it with 0? I have to admit, though, since the whole point is whether to include zero, equating “Positive integers” with “Non-negative integers” is more confusing to me than this distinction over the Naturals. ---
**Author:** ![C\_K\_Dexter\_Haven](https://avatars.discourse-cdn.com/v4/letter/c/b2d939/32.png) [@C\_K\_Dexter\_Haven](https://boards.straightdope.com/u/C_K_Dexter_Haven)\ **Post date:** [February 20, 2004, 1:34pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/19 "2004-02-20T13:34:14Z")
Since 0 is neither positive (it’s not greater than zero) nor negative (it’s not less than zero), one normally says the Positive Integers are {1, 2, 3 …} and the Negative Integers are { -1, -2, -3…} and thus the non-Negative Integers are {0, 1, 2, 3…}. I didn’t think that was disputed, nor did I see any confusion? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 20, 2004, 3:36pm UTC](https://boards.straightdope.com/t/zero-is-a-natural-number/229762/20 "2004-02-20T15:36:27Z")
> [@C K Dexter Haven](#): > > Since 0 is neither positive (it’s not greater than zero) nor negative (it’s not less than zero), one normally says the Positive Integers are {1, 2, 3 …} and the Negative Integers are { -1, -2, -3…} and thus the non-Negative Integers are {0, 1, 2, 3…}. I didn’t think that was disputed, nor did I see any confusion? **Achernar** is probably talking about this paragraph in the staff report: > [@](#): > > Let’s ignore history and get back to mathematical development. After you have developed the Counting Numbers, you get the Positive Integers, and that’s when zero steps onto the stage. The Positive Integers (more technically correct, the Non-Negative Integers) are the set of Natural Numbers and zero, usually designated P = {0, 1, 2, 3…} At that stage in the development of your number system, zero becomes a number. That doesn’t seem right to me. [Next page](https://boards.straightdope.com/t/zero-is-a-natural-number/229762.md?page=2)