Is 0 a Natural Number?
Subash, a user of my math site (Interactive Mathematics) asked recently whether 0 is a Natural Number or not. My reply:
Normally I have always taken the Natural Numbers to start at 1 and not to include zero. I used to get my students to remember the difference between Natural Numbers and Whole Numbers by saying the natural numbers can be counted using your fingers and the first finger looks like a 1, while the word “whOle” has a zero in the middle, thus the Whole Numbers include 0.
Thoughts by others:
According to Dr Math:
Natural Numbers are 1,2,3,4,5,… [...] and Whole numbers are 0,1,2,3,…
According to Wikipedia:
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, …) or a non-negative integer (0, 1, 2, 3, 4, …). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science.
The safest thing is to state whether you are including 0 or not when talking about Natural Numbers. You could write it something like:
“The Natural Numbers (taken as 1,2,3,4,…) are blah blah blah.”
So who cares? This situation is strange because mathematics is normally a very precise science and there is normally broad agreement about such definitions. Anwyay, it matters if students lose marks in assessments because there is disagreement about the definition. So the set theorists and the computer scientists should just conform… 
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11 Oct 2007 at 9:10 pm Link to this comment
i call the positive integers “the positive integers”
(and use the BlackBoardBold symbol ${\Bbb Z}^+$).
the “natural numbers” are (of course!) N = {0, 1, 2 …}.
and “whole numbers” is just slang for “integers”
(i.e., is left with no formal definition at all).
textbooks have done a pretty good job of muddying
the waters here … but i think these definitions come close
to industry standard (note that there’s no Bbb symbol–
like N, Z, Q, R, C, and even H– called “W”; this is because
this language was created by people who like to feel
smarter than their students [compare "ln" versus "log":
the big kids get to call the natural log by its right name
but *you* little boys and girls aren't ready for that yet...]).
12 Oct 2007 at 11:09 pm Link to this comment
Thanks for your input, Vlorbik.
I’m with you on the “ln-log” issue, something I have found silly for a long time, especially considering many students here in Singapore write it with an ‘eye’ rather than ‘ell’ as in
not
Grrr…
Don’t miss Towards more meaningful math notation where related issues are discussed.
20 Aug 2008 at 6:39 pm Link to this comment
Now that we have accepted 0-9 and 11-19 as counting tables,it is high time to give up various nomenclature and accept ‘0′ as Natural Number and include in set N
16 Sep 2008 at 6:53 am Link to this comment
so is 0 a natural number or not?
16 Sep 2008 at 3:29 pm Link to this comment
Hi Amanda
It depends (and I believe it shouldn’t “depend” – it should be agreed).
Here’s another 5 (incompatible) definitions: http://dictionary.reference.com/browse/natural%20number
19 Sep 2008 at 10:46 am Link to this comment
How about this convention?
There are the the positive integers (or natural numbers), {1,2,3…}, the negative integers, {-1,-2,-3…} and there ’s the zero, as a member of a unitary set or singleton {0}. According to this convention, zero is not a natural number.
20 Sep 2008 at 1:47 pm Link to this comment
This question came up today with one of the students I tutor. Checking various textbooks, British books tend to count 0 as a natural number while American books don’t. This is not the first time I have found differences between math taught in both countries. Check out the meanings of trapezoid and trapezium they are reversed between each country!
20 Sep 2008 at 3:22 pm Link to this comment
Al: Thanks for the comment about “trapezium” and “trapezoid”. It is interesting that in the US, “trapezium” is a quadrilateral with no parallel sides and as you said, the opposite in England (where “trapezoid” has no parallel sides).
As I said in the post, mathematics is not always as precise as it claims to be…
13 Mar 2009 at 8:44 am Link to this comment
I think that you are all in a language discussion and not in a math discussion, First you must define the “natural number” definition and then reach an agreement about whether 0 is a “natural number” or not.
13 Mar 2009 at 9:05 am Link to this comment
Yes, Julio.
But whose definition are we going to take? This particular textbook’s? Or that one? A computer science definition or a set theorist one?
Unfortunately, I still think the best thing to do is to make it clear which convention you are following each time you use it.
This includes when students are answering an examination question, especially if it is going to have an impact on the answer. It’s a bit tough if it is a multiple choice question, however.
14 Mar 2009 at 5:42 am Link to this comment
Zac, we are never going to agreed in this discusion if one insist that “natural numbers” include 0 and other insist on the contrary. We must ask each other what is a “natural number” and not take one definition because they all contradict each other.
14 Mar 2009 at 8:30 am Link to this comment
Hi again, Julio
OK – how shall we decide on that definition? Majority view wins? (And how do we determine the majority?)
And then, how do we convince the minority to drop their pet definition and conform?
Actually, I still find it interesting that this is not a fixed and agreed thing – math is supposed to be incredibly consistent, after all…
14 Mar 2009 at 9:33 am Link to this comment
This is a very interesting discussion in mathematics. Perhaps the important logical distinction between nominal definitions (conventional abbreviating notations) and real definitions (which specify essences) could help. This discussion would seem to take definitions in mathematics to be real, when in fact definitions in mathematics tend to be nominal.
14 Mar 2009 at 10:34 am Link to this comment
Actually, Jose, math is akin to a game. There is a set of arbitrary rules that someone sets up. They ensure that the rules are basically consistent and then everyone plays by those rules.
For example, why is 5 the 5th counting number? Why name it “5″ at all? Why couldn’t it be “green” or perhaps “loud” (which is the way Daniel Tammet sees it)?
So you’re right – definitions in math can be nominal.
12 Aug 2009 at 6:30 am Link to this comment
0 is not a natural number. it’s a whole number.
duh.
12 Aug 2009 at 9:33 am Link to this comment
But whole numbers include natural numbers, so any number can be natural and whole… They are not mutually exclusive … DUH!!!
26 Aug 2009 at 2:17 pm Link to this comment
It seems that we as people have a way of thinking something that is simple and making it confusing, Well every one is born in a month, on a day and in a year! for one to say I was born in zero month, on zero day and in the zero zero zero zero year, Well would he/she be giving there birthday?
10 Sep 2009 at 5:18 am Link to this comment
In my eyes it is not natural. Definition of natural is keep on adding one to get another number, so you would start at 1. If your allowed to start at 0, then thats the same as starting at -11111111112424321515 because if you keep adding 1, you will still end up at 1.
10 Sep 2009 at 9:08 am Link to this comment
Thanks for the fundamental viewpoint, Bropink.
10 Sep 2009 at 11:40 pm Link to this comment
i belive that zero is not a natural nor a whole.
from my belifes it is both.
16 Sep 2009 at 2:45 am Link to this comment
I prefer to not use the term “natural numbers” at all. At my school, it is taught (in my math class) that the set of natural numbers starts at 1. But after finding out that some people use a different convention starting with 0, I just stick with “non-negative integers” and “positive integers” because these have much more absolute definitions. Unfortunately it still doesn’t solve the problem of other people using “natural numbers”
16 Sep 2009 at 8:51 am Link to this comment
Thanks, Shawn. Perhaps you’re right. Maybe we should drop “natural numbers” (and “whole numbers”, because it’s not even a technical term) and just use “positive integers”.
But there’s so many text books that use “natural numbers”…
18 Sep 2009 at 1:46 pm Link to this comment
Natural numbers are the ones we use to count things that are there. So I can count the number of students that are in my classroom. If there are no students in the classroom, I cannot count the ones that are there.
Cardinal numbers are the ones that are used to count the elements of a set. The empty set has no elements, and the name for this set’s cardinality is zero.
Computer programmers like to count from zero because zero based arrays can be easily manipulated by pointer arithmetic. These people find counting from zero more “natural” for the problems they encounter. But what they are counting is an abstract representation of objects (elements) in a container (a set). The word “natural” is being used in a different context from that used when discussing the Natural Numbers.
As an experiment, ask a 3 or 4 year old who can count concrete objects to count the number of elephants in the room (I am assuming there are no elephants in the room). After a quick search they will respond with “don’t be silly” or they will count the number of imaginary elephants in the room (never underestimate a young child’s capacity to find novel solutions). This is what “Natural Numbers” are about (not the imaginary elephants!).
26 Sep 2009 at 2:31 pm Link to this comment
Surely, John Foster, you’re not saying we should decide this question based on a 3-year-old’s answer concerning numbers of elephants?
We need precision and a convention we can all agree on.
27 Sep 2009 at 5:20 am Link to this comment
More formally, the Natural Numbers have a one-to-one correspondence to concrete objects. When we count zero things, we lack a concrete referent, so we do not use the natural numbers. Small children do not have a concept of abstract number distinct from the referents that are being counted. As a result they work within the Natural Numbers. It is when we extend the concept of number to cover the enumeration of things that do not exist that we find that zero has utility. At that point we move from the concrete to the abstract, as counting things that are not there requires numbers to represent ideas, not things.
For this reason, W ≠ N ∪ 0 , but there is an isomorphism between W and (N ∪ 0).
In terms of a definition, there is one – the set that contains the sequences that correspond to the process of determining the cardinality of non-empty sets (i.e. counting THINGS):
{ “1″, “1,2″, “1,2,3″, … }
Each sequence has a label, being the last element of each sequence. These labels are what we recognise as Natural Numbers.
This is however a definition of dubious utility when it comes to the problem of teaching children what zero is. There is a reason why so few cultures independently devised the concept of zero, and that’s that most ancient, pre-technological cultures without currency or finance had no need for an abstract concept where you had to count the things that were not there.
29 Sep 2009 at 2:49 am Link to this comment
The question is: Is 0 a Natural Number?
The answers are: Yes/No/maybe/kinda/sorta/sometimes/if you want it to be/go ask the 3 yr old kid over there
My answer: It is a matter of opinion.
John Foster says that Natural numbers “are the ones we use to count things that are there.” But that’s his definition…his opinion. That is actually the definithion for Counting Numbers (1, 2, 3…). Hence the word count in counting.
So…
FACT: Counting Numbers are the ones we use to count things that are there (1, 2, 3…). Hence the word “count.”
FACT: Whole Numbers are non-negative integers that are uncut, undivided, and not in pieces (0, 1, 2, 3…). Hence the word “whole.”
OPINION: 0 is a natural number.
OPINION: 0 is not a natural number.
STICK TO THE FACTS, JACK: Since we already have names for both, we can discontinue using the unnecessary phrase, “natural numbers!”
You were off to see the wizard and the wizard thanks you for visiting! Next!
29 Sep 2009 at 5:09 am Link to this comment
hi i want to know the natral numbers between 10&11 would u like to give me
30 Sep 2009 at 4:12 am Link to this comment
Yes, there are two names for the same thing, Natural Numbers, or Counting Numbers. This is common in mathematics, we often have more than one name, or more than one symbol to stand for the same concept. Usually which one we use is decided by the context. This is not tautological, unlike expressions like ‘Whole Numbers are non-negative integers that are uncut, undivided, and not in pieces (0, 1, 2, 3…). Hence the word “whole.”’.
It’s interesting who sat in the various camps regarding this question (is zero a natural number?).
Yes: Cantor, Peano, the Bourbaki
No: Euler, Kronecker, Sloane
It appears that the Formalists say yes, and the Intuitionists and Platonists say no. Perhaps the most interesting OPINION is that of Ribenboim (1996), who states “Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P.”
Convenience is driven by context. For myself (and this is indeed my opinion) I do not see the “Whole Numbers” as merely being an extension of the “Natural Numbers”, any more than the Integers, Z, is an extension of the Integers modulo 5, Z_5. They have different algebras, that make sense in their own contexts, although there are mappings between the sets, that also make sense in certain contexts.
I contend that the Natural Numbers are those that children use once they move beyond “one, two, many”. In their context there is no need for zero. It mystifies me why some people feel so strongly that this is not a valid position.
Ultimately the questions seem to be “Are Counting Numbers the same thing as Natural Numbers?”, “Are the Whole Numbers the same as the Cardinal Numbers?”, and “Are the Counting Numbers and Whole numbers merely proper subsets of the Integers?”. I contend the answers are Yes, Yes and No.
30 Sep 2009 at 10:12 am Link to this comment
That’s a very illuminating answer, John Foster. Many thanks.
2 Oct 2009 at 12:34 pm Link to this comment
It isn’t tautology if a writer’s or speaker’s objective is to make certain that he or she is very clear to the reader or listener. Me, myself & I; full & undivided attention; the truth, the whole truth & nothing but the truth are examples of this.
The mere FACT that this is post #30 should be enough for the readers of this post to conclude that zero being a natural number IS a matter of opinion.
I realize that certain people have very strong opinions one way or another. It seems that they want to influence others with those opinions. I have my own opinions but I will not try to influence anyone with them. There are books that include zero and books that do not include it. There are teachers that include zero and teachers that do not include it.
A hypothetical situation:
I happen to give a student my opinion. That student does his math work based on my opinion. The teacher teaches based on a different opinion. Where do you think that would leave the student?
You may notice in my previous post (#26) that I never stated that zero was or wasn’t included. Instead, my answer was that it is a matter of opinion. It will continue to be a matter of opinion until it is proven well enough (one way or the other) to become fact.
My suggestion: Refer to your own teacher/textbook or use the titles you’re sure about (counting numbers & whole numbers). If the question was, “Is 0 a whole number or counting number;” then I doubt there would be any disagreements and think this thread would be much shorter.
9 Dec 2009 at 7:48 am Link to this comment
I realize I’m sort of late to the party, but here is my take on it anyway:
We already have that ℤ+ = {x ∈ ℤ | x ≥ 1} = {1,2,3,…}
Thus if we want to represent the set {x ∈ ℤ | x ≥ 0} = {0,1,2,3,…} we’ll have to write ℤ+ ∪ {0}, which is cumbersome to write all the time. Especially if you are referring to that set often. Therefore, I find it more practical to define the set of the natural numbers to include 0, i.e. N = ℤ+ ∪ {0}.
If I’m writing something in a course I’m taking, I’ll use whatever convention the textbook uses, but otherwise I’ll say that 0 ∈ N.
9 Dec 2009 at 7:50 am Link to this comment
Seems like the blog doesn’t like unicode. I also see I made a typo, the positive integers were obviously supposed to have been the integers greater than or equal to 1, not strictly greater than 1.
10 Dec 2009 at 8:20 am Link to this comment
I consider a natural number as a value to something that you can see or is present. I can see 1 apple, 2 grapes, 10 trees.
I consider a whole number as a value attached to counting the number of the same things I can see. If I see no apples, then I see 0 apples.
I my explanation 0 is not a natural number.
10 Dec 2009 at 8:38 pm Link to this comment
@Daniel: This blog does accept unicode, but for some strange reason it chewed yours.
Anyway, I edited it and I think I have what you originally intended (including the corrected typo).
@Esquio: Thanks for your input. Often the simplest explanation is the best!
14 Dec 2009 at 5:54 pm Link to this comment
“0″ most likely looks whole not natural!
19 Dec 2009 at 3:42 am Link to this comment
What a fascinating discussion! And what a great website, zac; I’ve just stumbled on it and added it to my RSS feed reader. Thanks!
Back to the discussion…
Arguably it is a matter of opinion on one level, but I’m with John Foster on this. The term “natural” strongly suggests a sense of intuition. Hence Euler, et al., having such a bias.
I must say that I am unaware of England making such a distinction between 0 being included among the natural numbers. I teach mathematics in England and on the journey into the world of rational versus irrational numbers my older students take a brief tour into the world of “natural” numbers; we discuss the abstract and philosophical notions and implications of non-integers, negatives, and zero. For example, you cannot have half a piece of paper or half a chair. You can remove pieces, but it remains what it is until it is no longer what it was. (I hope that makes sense.) In other words, fractions exist to define relative comparisons or measures, whereas natural numbers define the actual quantity of (usable) items.
So in the same way we talk about how unnatural the concept of zero actually is. It is quite natural to talk about three books or one calculator, but it makes no sense to talk about zero anythings. If zero were natural then an infinite number of them would occupy some space. The room where I am typing this comment right now contains zero elephants with one written on its back, zero elephants with two written on its back, and so on. There are an infinite number of zero elphants with N written on its back and yet there is space for me to be here. There is nothing natural about zero! (c;
19 Dec 2009 at 8:03 am Link to this comment
Thanks for your reply, (the esteemed) “Euler”! You raise some great points. This bit gave me pause for thought – “you cannot have half a piece of paper or half a chair”, since functionally the situation is somewhat different. If I rip a piece of paper in half, I can still use the individual pieces of paper, but half a chair is as useless as no elephants with N written on them! An observer will say “that is a piece of paper” if I give her one of the halves, but will say “that’s 1/2 of a chair”. Your philosophy of numbers course sounds very stimulating.
Thanks for the input about conventions in your part of England. It’s interesting that these things are not even necessarily standard across one whole country, let alone universally.
I wrote about your Project Euler here:
http://www.squarecirclez.com/blog/project-euler/1558
19 Dec 2009 at 5:43 pm Link to this comment
Esteemed Euler?! To clarify. He is. I am not. He is simply my hero of mathematics.
zac, it’s not quite a philosophy of numbers course as much as high school mathematics with a bit more than the students bargained for. But they enjoy the opportunities to think outside of the basic curriculum diet.
You’re right about the degree to which you can remove pieces of paper and still describe it as a piece of paper compared to removing pieces or parts of a chair, but I still think that the idea has some merit, albeit tentative. If you asked for a piece of paper and I gave you a fragment of paper measuring 1 mm by 1 mm then you would think I was crazy. It might contain the same matierial as paper but it would not function as paper. The phrase “one piece of paper” refers to a usable and practical measure of paper. Admittedly the point at which it is no longer describable as a piece of paper is somewhat subjective, but you would never describe it as half a piece of paper unless you were comparing it with, say, a piece of A4 paper which had been torn in half. In which case you are comparing its size, not really describing it as half a piece of paper in terms of its function. Simialrly with the chair, if I continue to remove parts of it then at some point it ceases to be describable as a chair. Even if I took a chainsaw to through the centre of it then you might look at one “half” and say, “That’s half a chair.” But you would only be saying that in the sense that you recognise it as one half of a complete chair. Technically what you’re looking at is no longer a chair. It does not function as a chair any longer.
But I recognise that even here with all this philosophising I am skating on very thin ice, and I wouldn’t be foolish enough to defend my points with any authority. I simply don’t possess it.
The bottom line in the discussion of “Is 0 a natural number?” is that there is sufficient confusion to invalidate it as a universally acceptable phrase. You might have noticed at Project Euler, where our problems often venture into the realm of Number Theory, that we are careful to define the set of whole numbers not including zero, as the set of positive integers. However, at one of my other websites: http://mathschallenge.net, I do use the phrase “natural number” without stating if it including zero or not, but the nature of the problem would exclude it as a possibility. For example, “Given that n is a natural number, when is n^4 + 4 prime?” It doesn’t matter whether or not you include zero, it will not affect the solution.
28 Jan 2010 at 7:47 pm Link to this comment
The problem with O in mathematics is that it is used to symbolise nothing, no-thing, and yet, mostly, it refers to unity, a whole or united thing.
For human beings, no-thing is an abstract concept, meaning it has no-thing to do with our real experience of life, and in effect no-thing has to be imagined as a total blank, say the paper that something is written on, but of course this paper is a whole thing and it is only our focus on the writing that makes it a blank, no-thing, background.
No-thing is this background to focus, and once, space, the heavens, were seen as a black background of no-thingness against which the stars appeared as things. Nowadays this no-thingness is thought to be filled with fields, sequential influences and almost-things, and the no-thing is in doubt. At the other end of the scale we have atoms, quantum particles, strings and the something that they appear from, but if we focus on the strings and ignore the fields etc. that bring them into our imaginary view of the sub-ataomic world, we see the background something that describes them as no-thing again.
However, if I have an apple and someone steals it, takes it away, then I have no apple, no-thing.
If I recover my apple I have a unified thing and if I cut it into sections it is a divided thing, and maths is based on this unified principle even as it ignores its own reality. This is the division of unity into things, or the many things, like a lot of apples, that create a unified concept.
What is lost in maths today is the concept of a unified background, the unity that things appear from within or the unity that is being enumerated as things, and the sooner maths re-invents itself into a concept of unity and sees its no-thing for what it is, the better for everyone.
No-thing exists in my human experience when something is taken away, but what I experience before this event is a unified concept that can be divided into things or the things that can represent another unified concept.
O as part of the numerical symbol for ten, a hundred and so on only describes a decimal form of mathematical unity, and the modern decimal system is based on this unified concept of ten things.
O shows that the integers in a column have been unified as the 1 in the next column, it represents a unity of the ten in this previous column, and so on with 100 and 1000 etc., but unfortunately, mathematics ignores its own begginings and limits its focus to the abstract background that it prefers. This says that no-thing exists in the unified column and this ability to ignore reality, the paper that maths is written on or the human being that first divided things in a numbered or quantified way, is forgotten today.
Reintroduce the concept of unity as the background that maths is built on and which it uses all the time, and maths could make sense to everyone, but of course abstract thinkers will probably choose to rely on the unwritten rule of preference that created the zero. Their abstract way of thinking depends on it. They will take away our humanly unified reality and leave us with an abstracted no-thing again.
Can I have my unified life back please.