# Mathematical Dopers: Does the set of the naturals include zero? **URL:** \ **Category:** In My Humble Opinion\ **Created:** [February 1, 2011, 7:46am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764 "2011-02-01T07:46:22Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [February 1, 2011, 7:46am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/1 "2011-02-01T07:46:22Z")
I realize that this is hardly on the level of, say, a poll about what you’d do with the money you’d get from winning a Fields Medal, but I would like to see if there’s a strong (or even weak) consensus on the issue of whether **N** includes zero. As a lagniappe for opening the thread, you now know how to spell ‘[lagniappe](http://en.wikipedia.org/wiki/Lagniappe)’. As if that wasn’t enough, as a little something extra, [here’s a link to Tex the World](http://thewe.net/tex/), a Firefox (and [Chrome](https://chrome.google.com/extensions/detail/mbfninnbhfepghkkcgdnmfmhhbjmhggn) and, possibly, [Opera](http://groups.google.com/group/tex-the-world/browse_thread/thread/8fe4f1617bb78455)) extension that allows LaTeX equations such as [; \mathbb{N}_{0} = \aleph_{0} = \omega ;] to render correctly in web pages written by people who know it exists. ---
**Author:** ![Nava](https://avatars.discourse-cdn.com/v4/letter/n/da6949/32.png) [@Nava](https://boards.straightdope.com/u/Nava)\ **Post date:** [February 1, 2011, 7:51am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/2 "2011-02-01T07:51:41Z")
IANAMathematician, but I distinctly recall that when we were bashing our heads against set theory we had to demonstrate the properties of both “natural numbers including zero” and “natural numbers excluding zero”. The properties were different (the exclusive set does not have the neutral element for addition, the inclusive set does) - but there was no assumption made about whether zero was or was not included, this condition was always listed explicitly. So, “natural numbers with zero” and “natural numbers without zero” are two different sets and there is, or at least there wasn’t in those courses, such a thing as “natural numbers”. First answer and you already got a “ **Derleth** , d00d, you left out…” ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [February 1, 2011, 7:58am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/3 "2011-02-01T07:58:02Z")
> [@Nava](#): > > So, “natural numbers with zero” and “natural numbers without zero” are two different sets and there is, or at least there wasn’t in those courses, such a thing as “natural numbers”. Bah! Now I wonder how common this actually is. It certainly can’t be universal, though you can’t deny it would end this debate once ad for all if it were. ---
**Author:** ![Indian](https://avatars.discourse-cdn.com/v4/letter/i/c0e974/32.png) [@Indian](https://boards.straightdope.com/u/Indian)\ **Post date:** [February 1, 2011, 8:38am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/4 "2011-02-01T08:38:20Z")
I learnt in school that, what comes to you naturally, when counting is “natural number”. When I start counting 1 comes to me first not zero. ETA: I understand this is a hotly debated topic.So don’t take my word for it. ---
**Author:** ![Manwich](https://avatars.discourse-cdn.com/v4/letter/m/e480ec/32.png) [@Manwich](https://boards.straightdope.com/u/Manwich)\ **Post date:** [February 1, 2011, 8:52am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/5 "2011-02-01T08:52:37Z")
At my university there was a fairly strong consensus that there is no consensus, so I didn’t vote. As **Nava** said, if it mattered it would be specified whether or not zero was included. Mathematicians don’t generally like to argue about definitions, they just care about the interesting consequences of definitions. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [February 1, 2011, 10:16am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/6 "2011-02-01T10:16:02Z")
I voted for what I was taught, which was that natural numbers were all positive. But, now that I think about it, I think it’s silly. There’s already a term for the natural numbers excluding zero: positive integers. I guess one could argue there is also a term for those including zero, nonnegative integers Crap, so I’m back to not having a clue. Such a term seems wholly unnecessary. ---
**Author:** ![Digital\_is\_the\_new\_Analog](https://avatars.discourse-cdn.com/v4/letter/d/a88e4f/32.png) [@Digital\_is\_the\_new\_Analog](https://boards.straightdope.com/u/Digital_is_the_new_Analog)\ **Post date:** [February 1, 2011, 1:36pm UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/7 "2011-02-01T13:36:49Z")
I went through an accelerated math program in middle and high school (Any other MEGSSS people out there?). It started with basics, and built up to things like set theory and abstract algebra. Fun stuff to take into college. Anyway…we were taught that **N** , the naturual numbers, is positive integers. Including 0 gave you **W** , the whole numbers. And if you wanted the negative integers, 0, and the positive integers…then **Z** is your set. It never occurred to me until reading this thread that this wasn’t universally accepter. Ignorance chipped away at. -D/a ---
**Author:** ![Student\_Driver](https://avatars.discourse-cdn.com/v4/letter/s/3da27b/32.png) [@Student\_Driver](https://boards.straightdope.com/u/Student_Driver)\ **Post date:** [February 1, 2011, 1:50pm UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/8 "2011-02-01T13:50:17Z")
I voted “yes” because I happen to think that way. Luckily, I’ve had good professors over the years who have reinforced the fact that “natural numbers” is not properly defined, and I’ve adopted their recommendations to start proofs and papers by giving a definition that explicitly states that 0 is a member of the set. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 1, 2011, 3:29pm UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/9 "2011-02-01T15:29:08Z")
People’s opinions on this seem to track their subfield pretty closely. Logicians, set theorists and theoretical computer scientists would generally say yes, but analysts and topologists would generally say no. I’m not sure about algebraists. ---
**Author:** ![Capt.Ridley\_s\_Shooting\_Party](https://avatars.discourse-cdn.com/v4/letter/c/cc9497/32.png) [@Capt.Ridley\_s\_Shooting\_Party](https://boards.straightdope.com/u/Capt.Ridley_s_Shooting_Party)\ **Post date:** [February 1, 2011, 3:38pm UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/10 "2011-02-01T15:38:18Z")
Yes (theoretical computer scientist). ---
**Author:** ![Blaster\_Master](https://avatars.discourse-cdn.com/v4/letter/b/cab0a1/32.png) [@Blaster\_Master](https://boards.straightdope.com/u/Blaster_Master)\ **Post date:** [February 1, 2011, 4:26pm UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/11 "2011-02-01T16:26:33Z")
> [@Digital\_is\_the\_new\_Analog](#): > > Anyway..we were taught that **N** , the naturual numbers, is positive integers. Including 0 gave you **W** , the whole numbers. And if you wanted the negative integers, 0, and the positive integers..then **Z** is your set. This is exactly what I was taught whenever it was covered in Middle School, and it was consistent all the way through my number and set theory classes in undergrad. I do think I had one professor who avoided it and said “Positive Integers” and “Non-negative Integers” for the two sets, but I don’t remember what class that was. FTR, I’m a Computer Scientist. ---
**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 1, 2011, 4:40pm UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/12 "2011-02-01T16:40:14Z")
We had a long thread about this very question back in 2004: [Zero is a natural number](http://boards.straightdope.com/sdmb/showthread.php?t=240366). My answer is the same as it was then: [QUOTE=Thudlow Boink] 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. [/QUOTE] ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [February 2, 2011, 4:22am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/13 "2011-02-02T04:22:44Z")
When I was in school (High school or middle school, probably) we were taught that the naturals started with 1, and the whole numbers included 0 **which you could remember because 0 has a hole in it**. How can you argue with something that easy to remember? (I didn’t vote.) ---
**Author:** ![Nava](https://avatars.discourse-cdn.com/v4/letter/n/da6949/32.png) [@Nava](https://boards.straightdope.com/u/Nava)\ **Post date:** [February 2, 2011, 7:23am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/14 "2011-02-02T07:23:53Z")
> [@Digital\_is\_the\_new\_Analog](#): > > Anyway..we were taught that **N** , the naturual numbers, is positive integers. Including 0 gave you **W** , the whole numbers. And if you wanted the negative integers, 0, and the positive integers..then **Z** is your set. Wheeee! Yet another example of how which language you use changes your definitions/labeling systems. In Spanish, “whole” and “integer” are the same word, entero. So we couldn’t use the same labels as D/a was taught: they would have made no sense, as we would have had the set of números enteros which would be a non-identical subset of the números enteros… say what? ---
**Author:** ![Jamaika\_a\_jamaikaiake](https://avatars.discourse-cdn.com/v4/letter/j/7993a0/32.png) [@Jamaika\_a\_jamaikaiake](https://boards.straightdope.com/u/Jamaika_a_jamaikaiake)\ **Post date:** [February 2, 2011, 2:34pm UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/15 "2011-02-02T14:34:38Z")
Yes. I’m a set theorist. If the natural numbers is the set of finite ordinals (which it is!), then it certainly includes 0. Here’s the natural numbers: 0=empty set 1={0} 2={0,1} 3={0,1,2} 4={0,1,2,3} 5={0,1,2,3,4} etc. See? 0 is in my list! ---
**Author:** ![Digital\_is\_the\_new\_Analog](https://avatars.discourse-cdn.com/v4/letter/d/a88e4f/32.png) [@Digital\_is\_the\_new\_Analog](https://boards.straightdope.com/u/Digital_is_the_new_Analog)\ **Post date:** [February 3, 2011, 3:44am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/16 "2011-02-03T03:44:32Z")
> [@Jamaika\_a\_jamaikaiake](#): > > Here’s the natural numbers: > … > > See? 0 is in my list! I’d make the same argument, but I’d claim those are the whole numbers. I’d even use the same definitions for the numbers. I’d then define the natural numbers as {x (- **W** | x != 0} Or if I wanted to be fancy, I’d define addition..then use that to define greater than..then define natural numbers as {x (- **W** | x \> 0} Hmm..I remember how to define Q (rational numbers)..and I seem to remember using dedican cuts to define real numbers..but I can’t remember how we defined negative integers on the way there. :smack: ETA: This was part of my all time favorite homework assignment in middle school..“Prove that 2+2 = 4”. With the right definitions, it’s simple and obvious. But as a middle school kid? Wow. -D/a ---
**Author:** ![lee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lee/32/7455_2.png) [@lee](https://boards.straightdope.com/u/lee)\ **Post date:** [February 3, 2011, 4:02am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/17 "2011-02-03T04:02:10Z")
I start counting with zero. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 3, 2011, 4:24am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/18 "2011-02-03T04:24:22Z")
> [@](#): > > People’s opinions on this seem to track their subfield pretty closely. Logicians, set theorists and theoretical computer scientists would generally say yes,… Except FORTRAN programmers, I imagine. ---
**Author:** ![Jamaika\_a\_jamaikaiake](https://avatars.discourse-cdn.com/v4/letter/j/7993a0/32.png) [@Jamaika\_a\_jamaikaiake](https://boards.straightdope.com/u/Jamaika_a_jamaikaiake)\ **Post date:** [February 3, 2011, 4:57am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/19 "2011-02-03T04:57:38Z")
> [@Digital\_is\_the\_new\_Analog](#): > > I’d make the same argument, but I’d claim those are the whole numbers. > > I’d even use the same definitions for the numbers. I’d then define the natural numbers as {x (- **W** | x != 0} Well, that’s not the same definition. My definition of the Natural Numbers is the set of finite ordinals. Since the ordinal numbers are precisely those sets which are transitive and well-ordered, they certainly contain the empty set, that is, zero. ---
**Author:** ![Jamaika\_a\_jamaikaiake](https://avatars.discourse-cdn.com/v4/letter/j/7993a0/32.png) [@Jamaika\_a\_jamaikaiake](https://boards.straightdope.com/u/Jamaika_a_jamaikaiake)\ **Post date:** [February 3, 2011, 5:29am UTC](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764/20 "2011-02-03T05:29:41Z")
> [@Digital\_is\_the\_new\_Analog](#): > > Hmm..I remember how to define Q (rational numbers)..and I seem to remember using dedican cuts to define real numbers..but I can’t remember how we defined negative integers on the way there. :smack: [Dedikind](http://en.wikipedia.org/wiki/Richard_Dedekind). One possible way to do this is to first consider the set NxN, the sets of ordered pairs of naturals. Then consider the equivalence relation (a,b)~(c,d) if a+d=b+c (IOW, a-b=c-d, but subtraction isn’t defined on the naturals!). Then we can think of the resulting equivalence classes as the set of integers. Then you can define addition and multiplication on the integers by [(a,b)]+[(c,d)]=[(a+c,b+d)] [(a,b)]_[(c,d)]=[(a_c+b_d,b_c+a\*d)] (NB: the brackets indicate the equivalence class containing the given ordered pair) EXERCISE 1. Show that these are well defined. That is if [(a,b)]=[(a’,b’)] and [(c,d)]=[(c’,d’)], then [(a,b)]+[(c,d)]=[(a’,b’)]+[(c’,d’)] and [(a,b)]_[(c,d)]=[(a’,b’)]_[(c’,d’)] EXERCISE 2. Show that these operations satisfy the axioms of a [ring](http://en.wikipedia.org/wiki/Ring_%28mathematics%29). Notice that you do a very similar thing when you define the rationals from the integers: DEFINITION: Consider the set Zx(Z-{0}), the set of ordered pairs of integers whose second coordinate is not zero. Then define the equivalence relation ~ on this set by (a,b)~(c,d) if a_d=c_b. The rational numbers is the set of equivalence classes of this relation. Furthermore, we define addition and multiplication as follows: [(a,b)]+[(c,d)]=[(a_c+b_d,b_d)] [(a,b)]_[(c,d)]=[(a_c,b_d)] EXERCISE 3. Show that these are well defined. That is if [(a,b)]=[(a’,b’)] and [(c,d)]=[(c’,d’)], then [(a,b)]+[(c,d)]=[(a’,b’)]+[(c’,d’)] and [(a,b)]_[(c,d)]=[(a’,b’)]_[(c’,d’)] EXERCISE 4. Show that these operations satisfy the axioms of a [field](http://en.wikipedia.org/wiki/Field_%28mathematics%29). [Next page](https://boards.straightdope.com/t/mathematical-dopers-does-the-set-of-the-naturals-include-zero/569764.md?page=2)