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Is 0 a Natural Number?

By Murray Bourne, 24 Aug 2006

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.

What's the answer?

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. Anyway, 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... 🙂


As you can see below in the comments, this topic has created quite a bit of discussion.

TL;DR The community is divided. As I said, just state whether you assume zero is included or not.

See the 162 Comments below.

162 Comments on “Is 0 a Natural Number?”

  1. vlorbik says:

    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...]).

  2. Murray says:

    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




    Don't miss Towards more meaningful math notation where related issues are discussed.

  3. Arvind says:

    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

  4. amanda says:

    so is 0 a natural number or not?

  5. Murray says:

    Hi Amanda

    It depends (and I believe it shouldn't "depend" - it should be agreed).

    Here's another 5 (incompatible) definitions:

  6. Jose says:

    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.

  7. Al says:

    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!

  8. Murray says:

    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...

  9. julio says:

    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.

  10. Murray says:

    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.

  11. Julio says:

    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.

  12. Murray says:

    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...

  13. jose says:

    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. Murray says:

    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.

  15. : / says:

    0 is not a natural number. it's a whole number.


  16. jose says:

    But whole numbers include natural numbers, so any number can be natural and whole... They are not mutually exclusive ... DUH!!!

  17. Barrington Burton,sr says:

    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?

  18. Bropink says:

    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.

  19. Murray says:

    Thanks for the fundamental viewpoint, Bropink.

  20. travis says:

    i belive that zero is not a natural nor a whole.

    from my belifes it is both.

  21. Shawn says:

    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"

  22. Murray says:

    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"...

  23. John Foster says:

    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!).

  24. Steve says:

    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.

  25. John Foster says:

    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.

  26. wonderlesswizardofmath says:

    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.

    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!

  27. asiya says:

    hi i want to know the natral numbers between 10&11 would u like to give me

  28. John Foster says:

    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.

  29. jose says:

    That's a very illuminating answer, John Foster. Many thanks.

  30. wonderlesswizardofmath says:

    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.

  31. Daniel says:

    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.

  32. Daniel says:

    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.

  33. Esquio Hernandez says:

    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.

  34. Murray says:

    @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!

  35. peter kitheka says:

    "0" most likely looks whole not natural!

  36. Euler says:

    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;

  37. Murray says:

    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.

  38. Euler says:

    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:, 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.

  39. Philip says:

    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.

  40. Nighat amin says:

    i am having confusion about the natural numbers and whole numbers, some author writes both are counting numbers. then where is the difference?

  41. Daniel says:

    i have always believed that in this world there can't be any number without zero but at the sametime in some things we dont need the zero but when counting in units we need it

  42. Lucas Mentch says:

    These "conventions" mentioned above are just that: conventions. There are times when it is easier to consider 0 a natural number and times when it is not.

    However, the asking whether 0 is a natural number is equivalent to asking whether you want to call the empty set "0" or "1". When the natural numbers are constructed rigorously in set theory, we have

    {} - the empty set
    {} , { {} } - the empty set and the set containing the empty set
    {}, { {} }, { {} { {} } } - the empty set, the set containing the empty set, and the set containing both (the empty set and the set containing the empty set)


    Thus, these sets either correspond to 0, 1, 2, 3, ... or 1, 2, 3, ... so as I said above, it comes down to whether you want to call the empty set "0" or "1". It seems more natural to me to call the empty set 0, since it contains nothing.

    Besides, 0 is the most "natural" number of all. Simply go into a room and think of all the things in the world. Then determine how many of each thing are in that room. For example, there might be 1 desk, 1 chair, 1 bed ... but there will likely be 0 tigers, 0 elephants, etc. Thus, of all the things in the world, there will be more things, 0 of which are in the room, than things 1 or 2 or 3 ... of which are in the room.

  43. Philip says:

    Lucas Mentch comment, above, describes O very well in his last paragraph.
    There is reality, what is there, that which can be enumerated as 1 2 3 etc., and there is what we think, what isn't there, and that which represents no-thing or 0 for abstract thinking.

    Whether 0 is a natural number or not seems to be answered by this. For me, what is real, nature, represents what is natural, what is unreal, or produced only to suit the demands of abstract thinking, is non-natural.

    Thinking, as a mathematical exercise is abstracted from reality and its connection to the natural world is representative of nature or of a fantasy that thought has dreamed up to suit it's own devices.
    0, Zero, nothing, are unreal and non-natural because they only exist in the mirror of abstraction's comparison with its own abstracted thought patterns.

    The pattern of decimal numbering demand sets, what I was taught to call columns of ten as a child, units, hundreds, thousands, etc. These columns form an abstract non-natural pattern in human brains and people become confused between what is natural, real, and non-natural, abstract, when the patterns set by these columns seem more real to their way of thinking than what they actually observe.

    0 is an abstract non-natural device, created in an abstract world to suit the rules of decimation, it has nothing to do with nature or with what is natural for human beings.

  44. a student says:

    omgh i am soo confused and i just need to know if it is or not for my homework. i think that it could be but i would say no because you cant multiply something like 1x1 to get to 0 but saying 0 isnt a natural number is kinda like saying that 0 isnt a counting number and that it is just a place holder. like 10ths place 100ths place 1000ths place etc... so i have no idea! i am so confused though!!!

  45. Philip says:

    In reply to a student, you don't really need to know if 0 is a natural number or not.
    You only need to know what the one who marks your homework thinks it is.
    If you read the blogs you can see that the naming of numbers is more or less down to confusion and personal preference.
    Your marks are given by personal preference so I would say what you were taught to say, collect the sweetie, and then remember that many rules of maths are preferential.
    2+3x3 = ?
    Maths is not an exact science, it is a preferential one.

  46. Anton Pech III says:

    It's really confusing everyone has a good point. But as I analize it based on your notions 0 is not a natural number if it represent on its own. It can only be a natural number when added to other numbers just like in 10,20, 30 . . .

  47. Philip says:

    As Anton Pech 111 points out, the problem is two different concepts, one symbol. Nothing and Unity.

    1-1 = 0
    1+9 = 10

    Maths requires a second symbol, but it seems to prefer the confusion that comes from one symbol doing two jobs.

    Hitler did much the same when he made staff appointments. Two people, one unclear job, and both vying with each other, trying to win his preferential favors.

    Maths is an authoritarian system, unclear preferences are unwittingly passed on by teachers who decide what 0 is, and confusion reigns until you see rules of preference for what they are. Until then, confusion can over-rule peace of mind and conflict's mindset becomes superior to commonsense.

  48. Tony Pechayco says:

    Yes Philip I agree with what you said that, 'unclear preferences are unwttingly passed on by teachers who decide what zero is'. So, do you know who was the first one to say that '0' is a natural number? Maybe it was just also over-heard as natural instead of neutral.

  49. Philip says:

    Hi Tony, I suggest a book called 'The Universal History of Numbers, by Georges Ifrah, if you want to explore the origin of the zero. However, he doesn't make the distinction between its use as a unifying symbol and a symbolic nothing.
    Like most mathematicians, he recognises its usefullness and its growth in several cultures, but doesn't draw the distinctions of use and application that may have existed in other mathematical systems.
    Given the linguistic variations of the several inventors of zero, I doubt that a simple misunderstandng of a concept took place as you suggest, and as I see it, neutral is as far from unity as it is from nothing. ( Nuetrality springs from neuter, and sexlessness, as neither masculine nor feminine, often appears in nature, even if human beings mostly judge this condition to be unnatural.)
    For me, authoritarianism rules by confusion, and by the acceptance of a social appearance that is similar to that described in the children's story of the Kings New Clothes.
    Maths spreads confusion through its preferences, and yet it is easy to see its naked reality once you stop trying to fit into an illogical system. Guess you could say that unclear preferences neutralise people's brains and stop imagination and inventive thinking from working together as equal partners.

  50. Richard and Luisa says:

    dr math
    hi! we are two chilean students in pedagogy in mathematics and computation, our teacher shows us that the 0 belongs to the naturals, but we want to tell you that your blog is very good because we can learn from the opinion of the people some topics about mathematics and the few diferences in this science. This topic is very important for some teachers because they don't know if the 0 is or isn't in the natural numbers; in our country in the primare school, the teachers say that 0 isn't in the naturals, but not because they think it but they follow the government programs.

  51. Murray says:

    Hi Richard and Luisa. Thanks for giving your background. All the best with your studies!

  52. Philip says:

    Hi Luisa and Richard,
    The Mayan concept of 0 may be a little closer to home for you than the confusion spread by the western system, but you'll have to rediscover the beauty of the Mayan mindset to understand their maths.
    Difficult after the dichotomies of rationality have invaded everything and everyone by pretending to be logical.

  53. Dan Bach says:

    I have always seen in books (and written in my own, Prealgebra) that natural numbers start with 1, and whole numbers start at 0.
    But when I was in grad school at Berkeley, the great Julia Robinson seemed to include 0 in the natural numbers in a theorem. I asked, "are you considering 0 as a natural number?" The confident reply came from Prof. Robinson: "Zero _is_ a natural number."
    (I didn't argue; she was the chair of my oral exam committee later that term!)

  54. Murray says:

    Hi Dan. I love that - answers to the great questions of math by force of personality (and power of assessment)!

  55. Philip says:

    Whole numbers are unified ones, unity exists, and some one has noticed it, so where does no-thing come from?
    Perhaps it's a financial abstraction of maths, and only relevant in as much as the monetary or possessive state of having and not having can dominate westernized minds with calculated conceptualizations of some thing.

    Preference started out as the order of courtiers and dignitaries, all trying to please a monarch, and being pleased by a ruler as they were shown favor. They were chosen ones, but the force of personality that chose them could vary from the benign to the insane.
    So, it was the position of monarch, the first one, that chose its favorite ones, and this assessment could be based on anything from promise to the needs of the state.

    Monarchs preferred those who kept them in power. Preference arranged the system, position mattered, power might, and generally did whatever it wanted to do, and poor common folk had nothing to compare with it, so they strived to be some one, too.

    I guess, when maths notices that natural numbers come out of unity, just as they form it, then the system could find balance again. But of course, it will mean letting go of the first position, the prime mover, the original one, and noticing that this was actually a unified everything before it diversified.
    The king will be dead, there will be no authority, but someone will shout long live the king and raise a new pretender to the throne because they prefer things that way. Somehow, assessing your own capabilities isn’t quite the same as being told that you’ve passed the test, or that you’ve failed it of course. This self-assessment seems to be pretence and it just becomes easier to accept the qualification of the system that promotes a preferential state of mind than to acknowledge and trust your unified self.

  56. UKPhil says:

    I always explain to my students that the 'natural numbers' are the numbers that arose naturally when 'somebody' began to count. Not having something (sheep, rock, cave ...) is not a concept that requires counting to begin. But as soon as one has ONE object it is NATURAL to take another to gain 2 objects and counting begins. Look at the stars (from any point in the universe) .. 3 stars are seen naturally (in nature) and threeness is universal, as it twoness and unity, but if there are no stars then one looks elsewhere; I cannot believe that anyone would 'naturally' include 0 stars in the same category list as 1, 2, .. stars when looking up for the first time (or pebbles on a beach if you prefer) - it just seems completely unnatural to me.
    As has been stated above, 0 was invented as a place holder long after counting came about NATURALLY so I remain firmly in the "0 is not" camp and I will continue to tell this to my students (let's hope that this does not upset their first exposure to set theory!).

  57. Philip says:

    Looking at the stars also shows us the heavens, what we have learned to call space today, and we are looking at a unified concept.
    When counting the stars we often ignore this heavenly space, or see it as an infinite nothing when abstraction takes over.

    As Einstein pointed out, trying to measure space doesn't work because measured frames of reference are always surrounded by space as well as containing it, so a way of thinking that focuses only on the stars has to limit its own machinations and it has to ignore what might be called the unity that includes the heavens.

    Infinity is a relatively recent concept in the west, and yet various groups of people have used it, under other names and for thousands of years, to demonstrate the limited capability of the abstract thinking that focuses on its own things by ignoring reality.

    Math’s is an abstract language because its numbers can ignore natural things, and one star, apple or cave can be represented as 1, no matter what this 1 is supposed to be.
    However, as with all languages, trying to turn their symbolism back into significance is fraught with problems.

    Abstraction disappears into its own orifices, and yet its symbols still seem to represent something. The paradoxes of set theory show this anal search at work, and as fascinating as it might be to arrange meaningless symbols into sets that cannot belong to their own set, all that this actually demonstrates is the limitation of the rules applied.

    A number of square shapes can become a square and, linguistically, it is of its own set, but I can also accept that a number of circles cannot become a circle and this fact isn't paradoxical if I recall that the heavens always surround the stars, just as they do squares.
    But to think this way, I have to accept that unity exists before I turn it into an abstraction, and that the sophistication of infinity, like nothing, or even set theory, is an unnatural offspring of the misunderstood and long ignored, unifying zero.

    Nothing is the abstraction of something, it has been taken away from my view of a previous or even memorised reality, but if I look around, then I’ll find that beyond my loss, unity remains constant.

    Teaching children that 0, as nothing, isn’t natural, is admirable, but it doesn’t help to tell them only one part of the whole story.
    They might grow up to think that the unity of the heavens doesn’t exist.

  58. UKPhil says:

    wow Phil!

    I'm not sure that I really follow this deep philosophical take; hey I'll bet Einstein would vote "not 0" (but Hawking would probably go the other way). I'm way up there voting for the unity of the universe on the other hand. Also let's not forget the real masters of counting - the primes .. 0 has no place in their universe. Yup, I'm sure the Primes would vote "not 0".

  59. Philip says:

    Hi Phil,
    Sorry if my take is deep, I must be talking out of one of the orifices I mentioned, any language will do that to you, not just math’s, and I guess it comes from trying to discuss something that dwells in the dark, in symbols that can be read in the light.
    The problem with your masters of the counting universe, the primes, (Optimus and his friends,) is that they follow a fundamentalist theory, where one is excluded and where like-minded adherents are included. And like all fundamentalist ideas, its easier to follow rules that have been laid down by prime movers than to realize that one is essential, (even to a prime,) and that one can think for itself.
    However, if all ones are unique, and essentially primary, then the set of unity is also a unique one, as a paradox that doesn't exclude anything, and where nothing cannot exist.
    What value does infinity's endless search for an unreachable goal bring to a unified equation, apart from the denial of its existence?
    Guess the difficulty comes from fundamentalist theoreticians not accepting that one can represent anything if it follows the rules of unity, they seek like-minded followers, and we are back to authoritarian theories and the masters of the universe once more.

  60. Rich says:

    I visited this page about a year ago... And today, while revisiting, I feel like sharing some simple thoughts as a humble, ordinary guy.

    This comment of mine takes the total number of comments to 60. Comments numbered 1 to 60. No comment 'number 0'... ever!

    So I find UKPhil's explanations very logical.

    And now if I may digress slightly (and only because 'nothing' has been mentioned quite often in the posts here!): Is there truly such a thing as "nothing?" "No thing"?

    And why is there something rather than nothing?

    In my humble opinion (and with all due respect to Hawking ) no number of big bang, multiverse, string, or superstring theories can really answer that one!

  61. Murray says:

    @Rich: I've come across books that have a "Chapter 0", which explain basics before getting into the heavy content...

  62. H. James Hoover says:

    The debate about what particular set to call the natural numbers is silly. The point is that the set {0, 1, 2, ... } is the most "natural" for the world of computation.

    Robinson is correct - 0 is a natural number. Modern logical formulations of computation start by generating the Naturals with the Peano axioms, with 0 being the first element. Note that you can call the first element anything you want, say Fred.

    The "numbers" in the set inductively generated from the Peano axioms correspond to the number of times you apply the successor function to the initial element. That is why 0 is a Natural - it is what you get by applying the successor function no times.

    Much of the confusion comes from how to "index" the set of Naturals. Do you start at 0 or start at 1? That's not so clear. Sometimes starting at 0 is convenient, sometimes at 1. Languages like APL gave you a choice of 0-origin or 1-origin indexing for vectors. Star magnitudes start at -1.

    The observations that children don't know about 0 can only be made by someone that is not a parent. Ask any child how many candies are left in the bowl after Dad eats all of them, and they will say 0. In Canada, all children also know about negative numbers from an early age, since we have them for temperatures in the winter.

  63. Philip says:

    0 candies in the bowl after Dad eats them all!
    Guess that answers the question.
    If Dad leaves nothing for the kids then 0 must be natural.
    Teaching kids what's natural is what being a parent is all about, isn't it?

    Or is it that what's become normal in an individual, family, society or nation can seem to be natural until you start to think about normality and discover another way of doing things?

    Teachers used to say, "Nature abhors a vacuum," but perhaps nature wasn't natural and maybe what they, Dads and other parent figures did, by leaving nothing for the kids, was.

    For a selfish way of thinking, something or nothing is normal.
    For a natural way of thinking, something or something else is normal.
    Guess it comes down to Dad in the end.
    Is eating all the candies normal or is there a natural way to teach the kids?

  64. Fernando Montenegro says:

    Wow! What a thread!

    Really, now... Different definitions of 'natural numbers' are used in math, all equally valid (see post 30 and several others). If I choose to use the expression, I should make it clear that zero is or is not included (if it matters), and if I do not make that clear and someone realizes that I am including zero or not, while he/she does not, he/she should accept my convention for the nonce.

    As children came into play (re poster 62 in frigid Canada), I have a five-year-old who can add nonzero, one-digit positive integers (has notion of counting and adding). He has the concept that digits 1~9 represent quantities and are used to count. He also knows that the '0' on the elevator button means 'ground floor' and the '-1' means 'garage', but has yet to grok that '0' can also denote 'nothing', 'none', or 'empty' and be used "to count".

    Personally, I think of 'natural numbers' as the same as 'counting numbers' or 'positive integers'. But I have nothing but admiration for Peano's theorems, which start with "there is a zero" (a non-negative integer, though I don't think he called it 'natural').

    The rest is semantics, philosophy, psychology, personal preference, ...

    Is zero included in the natural numbers? You tell me what you mean, I'll go along.

    One reservation: Preparers of university entrance exams should be careful to avoid 'natural numbers', and takers of such exams should be prepared to challenge any disqualified answers!

    What a thread!

  65. Philip says:

    Good point about the 0 representing the ground floor on an elevator, although in Germany it is generally an E for Erdegeschoss,(ground floor,) and from what you say, your 5 year old sees 0 as a symbol for the ground floor, which for him is a tangible something or he's hanging in a void.

    I recall having a detailed discussion about 0 with a teacher not long after being a 5 year-old, well, I was actually on the losing side of an argument because teachers didn't discuss things with children in the early 1950s, or maybe it was only the teachers I came across that seemed to prefer parrots.
    My side of the discussion was asking whether 0 came at the start of the 1-9 sequence or at the end of it and why, but what this memory indicates is I couldn't make sense of what I was being told and that the teacher couldn't explain it in a way that my young, and for me, very natural mind could understand.

    0 as a symbol for the ground floor makes sense, it is a symbolic sign of life, but I'm not sure that where it goes from there is only semantics.
    As I see it, 0 becomes a thing of preference, like the parrots, and in this case, a confused thing of preference where two or more of them exist as a conflict in the same symbol, each of which are taught with authority and neither of which make sense to a child until someone has taken all the candies for themselves.

    As for going along with it, ‘When in Rome do as the Romans do,’ sounds like good advice until you find yourself in the arena, then what the Romans do isn't quite so attractive and going along with their meanings can be difficult.
    “Father forgive them for they know not what they do,” sounds alright if you are enlightened and hung on a cross, but as a child, or even as an adult who is being crucified by the authority of a preferred opinion, or if Dad has nicked all the candies, it can be a difficult call to make.

    For me, math’s limits the ability to think clearly because it isn't logical in a natural way that children can easily understand, and it is what happens in adult minds as we ignore the child in us and try to excuse our ill-considered preferences that leads to the semantics.
    I guess if Peno had given a clearer explanation of what he meant by zero, nothing or unity, or even the ground floor, then the value of the symbolic 0 could have been sorted out long ago, as could the symbols used in elevators.
    So, back to looking for the candies.
    Father, why hast thou forsaken me!

  66. april says:

    when there is no 0 in numbers, what would be the possible effect on learning math??

  67. Philip says:

    Without a 0, the decimal system wouldn’t exist, and you’d be back to finger and body part maths, what you might call counting on yourself, and maybe others would join in to extend the series.
    But, realistically, the existence of 0 isn’t the problem, its definition as nothing is because this denies that unity, such as an extended version of physical counting provides, is.
    I guess a world in which we could all count on each other is a bit much to imagine, but it’s an improvement on the nightmare in which dads continue to eat all the candies.

  68. redblue says:

    I believe 0 should be a natural number in number theory, as it is common to write an odd number x (with x as an element of the natural numbers) in the form of 2n+1 (with n being a natural number). you have to include 0 because if you wouldn't, 1 wouldn't be a natural number aswell because you couldn't represent it as 2*0 + 1

  69. euler says:

    Nice idea, however...

    An odd number given in the form 2n+1, where n is any integer, will generate every odd number (positive and negative). Whereas, 2n-1, where n is a "natural" number, will generate all the positive odd numbers.

    Of course, I could use a similar (flawed) argument to show that n=1 is the first natural number... an important part of number theory involves the primes, and it is well known that all primes, p>3, are of the form p=6n-1 or p=6n+1. (Note that the converse is not true: all numbers of this form are prime.) Hence it should be clear that n>=1, otherwise 6n-1 would generate a negative quantity.

    On another note... I think that any arguments which appeal to essential building blocks, and zero being a necessary component, need to consider the importance of -1. In any system of counting which begins with the notion of incremental components, we would be unable to incorporate one of the fundamental rules of arithmetic: subtraction, without -1. So is -1 among the natural numbers? Or do we simply see subtraction as an extension of arithmetic by including the "reflections" of the natural numbers? In which case, what purpose does zero serve in this domain? Arguably, 0 appears from the clash of equal and opposite quantities, but as we continue extending the system we soon have all rational quantities. The line needs to be drawn somewhere.

    Personally I can see no benefit in including zero as the "first" natural number. However, I do concede that with the advent of computing, it is often helpful to talk about the "zeroth" term, u(0). But equally, when working with iterative sequences it sometimes helps to begin with u(-1) or u(-2). Those familiar with solving Pell equations will often make use of these terms.

    The bottom line, and it has been said here several times already, there now exists sufficient uncertainty in the proper meaning that is is sensible to either use the phrases: "positive integer", "non-negative integer", depending on your intention. Alternatively include some proviso in your definition. For example, "N is a natural number: {1,2,3,4...}..."

  70. Philip says:

    Today is 31 December 2010, the last day of the year. The cycle of the seasons has been completed once again, and a new cycle will begin as the earth follows its path around the sun. That path can be described as 0, a circle, and however the year is divided, each division is part of the whole 0 cycle.
    Similarly, the sun rises as the earth turns, and its turning also describes a 0, a circular motion that defines a day as the passage of the risen sun to sunset and back to sunrise.

    The number of days can be described as 1 each, or as sets of days, as weeks or months, and in a tangible and quite natural way, the circle, cycle of 0, can be said to be the natural phenomenon that allows each 1 day to exist, just as the earth’s cycle around the sun allows a year to exist.

    Without the 0 cyclical movement 1 doesn’t exist and 1 cycle isn’t complete until the circle 0 has been drawn. In both these natural instances, 0 can be said to be a unified 1 or the completion of a cyclical movement that is now designated as 1 symbolic and naturally significant 0.

    1 earth completes each of the circles that describe a visual 0 path, and each 0 has a tangible form that is described by 1 earth and 1 sun.
    So, does 1 + 1 = 0 or does 0 = 1 + 1.

    Does the cycle 0 exist before that which is describing it, or does the cycle 0 only exist because 1 + 1 brings it to the attention of human minds.

    0 is a symbol, a tangible cyclical passage that describes unity, but it also describes the completion of 1 day or 1 year, (what might be called the end of days,) and then it starts to describe itself all over again.

    The end of a year does not result in nothing any more than the end of a day does, they both bring a new cycle, a new 0, that will be unified until the current natural course of events says that this cycle is no longer sustainable.
    What will happen then will also be described by 0s, circles and cycles, and new 0s, cycles will be formed as the movements demanded by gravity take over once more.

    The minds that first described a 0, what we now call zero, didn’t imagine it, 0 was real as a unifying circle or cycle, and the concept of someone taking all the candies to leave nothing described something else.

    The complexity of mathematical thinking ignores its own conception and relies on the rules that it prefers to justify its existence, but the 0 of maths had a natural birth and perhaps many of its current preferences are only the result of an unnatural system of education.

  71. Dalcde says:

    I once wondered whether 0 is a natural number and went through the whole textbook to find the answer, but it is not stated.

  72. Jaff Lawrence says:

    Concerning whether 0 is a natural number; I feel that there is a difference between natural numbers and counting numbers. Counting numbers are numbers used in counting real objects. If students are asked to count their fingers, none of them will start from 0. However, 0 is used to express the absence of a real object. Natural numbers are defined as the non-zero negative integers. In each case the elements should be listed for students to observe the inclusion of 0 in the set of natural numbers.

  73. Philip says:

    So, we now have counting numbers for natural, material fingers and natural numbers for imaginative, abstract things. And by this definition, we can start to feel that the fantasies of abstraction are more natural than the countable fingers of the material world.
    Additionally, 0 is the absence of both of them, and yet it is also the unified cycle of life that is naturally imaginative and material as a human body and mind.
    Oh! And of course we must list the elements for students to make sure that they include 0 in the abstract set of natural numbers, to ensure that they understand as we do.

    Is anything missing from this, because I’m still confused?
    The fact is that 0 unity is material and imaginary and that 0 nothing can be imagined and manifest by the actions of selfish people.
    So, why do we use a single symbol for opposing concepts?

  74. Raghuthaman says:

    What is a number?
    Modern mathematicians have not explained it accurately.
    Ancient Indian Vedic mathematicians had precise answer to this question!
    Each number is a condition. This is Ancient Indian logic!
    How and why a number is imagined as condition?
    1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 is a one to one count!
    0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 is ancient Indian count, which is "one less than each number of one to one count".
    Both counts are technically differrent!
    I mean, modern 'one to one count' and ancient Indian
    "one less states of one to one count" differs!

    While Ancient Indians used equal step increases(by one)they also used 'number' as 'condition'.
    first one has 'no one' before it, which condition "no" is zero!
    Second one has 'one' before it, which condition is one!
    Third one has 'two ones' before it, which condition is two. And so on number/condition increases by 'equal ones'as if in a 'one to one count'!

    What is differrence?
    Zero is not there in one to one count "1, 2, 3, 4, 5, 6, 7, 8, 9 and 10", whereas,'0' is a condition like any other number/condition! It is also reason to why number-increases are digital(equal step by step).

    It is high time to review number basics(modern mathematics and ancient Indian Vedic mathematics)

  75. Philip says:

    Nice one Raghuthaman, on a one to one basis or a one less one basis.

    The problem in transferring Vedic logic into English is the use of the word condition, which basically tries to impart something that is said in a united way, ( Com-together + dicere-say,) and if logic comes from logos (word ) then in a very real sense, whatever people say,( ancient Indians or modern schoolmasters,) the proof of the pudding is in the saying of it together.

    Unity comes from a humanly united front, not a nationalized one, and mathematics as a world language needs a humanly united way of saying things together.

    Many old numerical systems were linked to words, the Vedic system included, and these words described powers that were often linked to the transcendental states of focus that imaged them, but originally they were said and remembered and they were not written down until much later.
    Word mathematics of this order required vast amounts of memory and the ever expanding use of the imaginative intellect to raise the next powers, but each power was only valid once it could be imagined and said together.

    Personally, I like the ancient Vedic system, no one as a zero start makes sense if you are arranging ones in a logical way, and O (the circle or cycles of nature,) can be said to come before the ones that fill out the circle of the mind for our later recognition.
    You find the O circle used in this way in many ancient spiritual teachings, and spirituality in this sense was an imaginative exploration of the mind, what it held, could hold, and of what could be said about what was found there.

    For later spiritualists, Jesus, Logos, the word, came after God, (O or the no one circle in which Logos set about trying to imagine things to say.) But even Jesus had a few problems when others didn't speak his language or agree with his words, and as his story shows, he didn't find a united front, a together said way of life, on earth.

    The whole ‘is zero a natural number’ thing is an extension of the same problem, a disunited front, and people saying different things and either forcing others to agree with them, or pretending that they are in agreement when they are not.

    What is said is just that.
    And the problem is imagining a way of saying things that every one can agree on.
    Sticks and stones can break our bones and words can be poisonous and very painful.

    In decimal math’s, basically a column system until 9 units transcend into zero, (no one in the column, look to the next column,) what is said makes sense, but it uses a unifying zero because what has gone before is agreed on and the next column starts with the transcended one as one ten.

    The old Vedic system got left behind because it was easier to explain things in a decimal way, but in moving on, mathematicians lost their way and forgot where their ideas started from, the O circle or cycle of the imaginary and imagining mind that brought what they said to life in the first place.

    Out of unity came one, but the first unity was unexplainable, just as the name of God has been for thousands of years, and all that could be said about it, him or her, was that he, she or it was no one.

    The same is true today, but modern mathematicians use images of God (or his powers,) to reinforce their take on one, and we end up with an authoritarian Father God, (as the Great Architect or any other megalomanic imaginative construction,) that can say what he wants as he takes all the candies and leaves the rest of us late coming ones with nothing.

    No one comes first, but then again, he, she or it is just that, the unimaginable unifying mind out of which the Vedic ones were formed and in which they are still being formed,(with the help of paper, pencils and pens, slide rules, tables, calculators and computers.)
    Now does that sound natural to you?
    For me it is, but I'm willing to change my mind if it achieves a unity in which all the ones have a naturally equal and agreeable say.
    I guess that would leave us in a situation where we would have to agree to disagree on many things, but we will all fall back into the O circle in the end and be united there.

  76. Daniel says:

    Why is there someone posting walls of text that are impertinent to the discussion? Is this about math or philosophy? Geez. Get over yourself.

    I've always considered 0 is a natural number, and so did all my math teachers. Until I came to went to the US - the land where people still use Imperial system, where people say Darwin is a fraud, where pi is exactly 3.2, and so on.

  77. Philip says:

    Geez! Sorry to have been impertinent Daniel.
    Guess you've explained it all.
    Maths doesn't have a philosophy, American people say Darwin is a fraud, imperial systems are American and an American pi is a no nonsense 3.2 and so on.
    Bully for you and your non American teachers.
    Thank's for your help. It's hard work trying to get over your self, isn't it, but where exactly are we now?

  78. Pradip Baksi says:

    "It is high time for putting an end to the anachronism of beginning the series of natural numbers with one. In a pencil- box there are always some natural number of pencils - perhaps zero.A natural number is the cardinality (of the number of elements) of a finite set, in particular - that of an empty set." Vladimir Andreivich Uspensky,"Reflections on seven themes of philosophy of mathematics", in:Part Three of the Special Supplement: Marx and Mathematics to: MARX,K. 1994, MATHEMATICAL MANUSCRIPTS,p.505. Calcutta:Viswakos.ISBN 81-86210-00-8.

  79. Philip says:

    Pradip Baksi's comment is probably the most valid that has been made in this discussion so far.
    Natural numbers are a set, just as ordinal numbers are, and yet the latter relies on the former for its validity, and in effect that which is ordained, (agreed to by many minds,) is secondary to the initial and individual (Natural?) concept of enumeration.
    All sets have a point that they are connected to, and in the case of natural numbers, (similar to the pencil case,) the point is that of there being numbers in the first place.
    As Pradip points out, the described element of the set is the point and once this has been defined, the set exists no matter what it contains.

    Logically, an empty set is as valid as a full or part full set, and nothing is as natural as something once the set has been conceived.

    The use of zero as nothing is valid then, however, when a set contains an infinite number of related sets, as in the tens, hundreds, thousands and so on of decimal maths, then each related set has a finite expression that says this part of an infinite set is complete.
    In decimal maths, the zero fulfills this function and that of a set being empty, and its ambiguous use describes the fact of nothing in the set and it unifies the completed set.

    Where a human mind is the set that describes the point of any other set, it becomes a set that can contain any set, including that of natural numbers, and as such it is the original zero that becomes full of the symbolic forms of life that give it cause to think in sets.
    In effect, its associative ability becomes the original unifying concept, and this associate ability is described by the O that a human body sees and senses around it.

    O describes the mind to itself, and it unifies what it senses by association, including the nothing that empty sets can seem to contain when a mind ignores everything else, but a human mind is still full of its ability to define and associate things.

    This doesn’t make a human mind unique, other creatures do the same, but perhaps they don’t use ordained definitions to confuse themselves and others, as we do.

  80. LarryKing says:

    I disagree on the end part Philip humans are unique cause of love it make our mind really unique from the other creatures.In saying that we are not looks like you forgot that part.

  81. Philip says:

    Hi Larry,
    Your disagreement is noted, and respected, however for me, I guess it depends on what you call love.
    Defining love requires that it is lifted beyond lust, the romantic or even the philosophical, and the literal has no real idea about it, except that its black and while opposite is hate, and yet we all seem to know love when we feel it or see it in action.
    Compassion? Together or with feeling, being that feeling, considering it and intimately understanding what it means to feel life flowing through you and to be that life, can human's claim to be the cause and soul authors of this experience?
    Watch creatures that care long term for their young, and those that honor their elders, see elephants revisit the skeletons of their dead associates, swans and other creatures pair up for life, the bond between penguins, the family systems of African Wild Dogs, wolves, and on and on and on.
    Sure I don't see many valentine cards, bunches of flowers or even boxes of chocolates, but I do see a lot of feelings being portrayed, not discussed or debated upon, but being acted out, and as a common man, I can only say that these actions often appear to be very loving to me.

    A previous email told me to get over myself, and the author may have been right, but it is this getting over ourselves that represents the human quest as I see it.
    Then we can see that we are part of a whole that exists beyond human cause.
    Maybe this whole that I am part of is love, don't know, only that it shares life with me and that it is up to me whether I notice and share my life with it, or not.
    How do I share, that’s the question that we each have to answer for ourselves, but then that brings us to consciences and souls and O and unity and something that can seem to be nothing if we ignore the whole that we live within.

    Kind regards, almost love, or sort of affectionate in an almost stilted and polite way anyway.


  82. Murray says:

    While philosophical considerations do come into the discussion of whether or not zero is a natural number, the discussion has strayed too far from the fundamentals.

    Thank you for your contributions, but only mathematical comments will be approved from here on.

  83. Rod says:

    This is a long blog thread and I may have missed an already made point that I would like to put to this fine discussion. The concept that zero is not a "counting" number is as equally a convention as what is a natural number. The count of one is one added to zero, the basis from which all counting begins. That there are no apples left for me to eat, if I am starving to death, is very real count!

  84. Benazir Junejo says:

    please help me to clear this problem,
    If 0 is not a member of set of natural numbers then how the set will have numbers like 10, 20, 30,.........

  85. Murray says:

    Ah - interesting question, Benazir. "0" is just a representation of the number, just as 10, 20, 30 are. In some languages, there is no zero as such. In Chinese and Japanese, "ten" is written "+" and the number 20 is written (in effect) as 2x10.

    So your question would not even arise in those languages!

    The symbol for a number and the concept of that number are separate things, so even if "0" is not regarded as a natural number, it doesn't preclude 10 or 20 from being natural numbers.

    Hope that makes sense.

  86. Dan Bach says:

    Benazir - Please allow me to read your mind ;-} I think you meant if 0 is not used as a 'digit' then you can't make 10, 20, etc. But in the definition of natural numbers we just need to decide whether to include the quantity 'zero' in the set of natural values. Obviously there's some controversy about that. I didn't realize it might just be a West vs East kinda thing.

  87. Bonzomi says:

    I feel ...
    "Murray says:
    14 Mar 2009 at 10:34 am

    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." I need to answer that question even my vocabulary fails me. *struggles to find definiton of 'nominal'*

    Back to the question, I'd say that before '5' was invented, the person/community/walrus/anything that was responsible in inventing the number '5' had to choose something. It could be anything, '@','7','^','÷','?','*more random symbols*'. Just anything, anything at all. That person chose one of those possible things in the set of 'anything' (I hope I'm explaining this right lol @-@) and he/she/it got the symbol/number/drawing of '5'. Factors that influenced him/her/it may also be responsible to the decision of the existence of the number '5', but those factors can still be understood given that you can travel through time and search who made it or something. I'd say only God knows how it was done... for now... maybe... Possibilites'R'endlessz

    Right now if you actually want to go through all the pain creating a time machine just to find the reason of the form of '5', I'd say go for it and good luck doing that. Though you might end up with a problem-loop (e.g. To solve problem x, you need to solve problem y first. To solve problem y, you need problem x solved first.) I want to borrow your time machine if you ever succeed too.

  88. Fernando Montenegro says:

    WHY such resistance to acknowledge, accept, adopt the notion that the definition of "natural number" is NOT a transcendental truth but a human convention? The connotations of the word "natural" as used in everyday language seem to contaminate its comprehension when used in specialized terminology and discourse.
    Fulano says: When I say "natural numbers", I mean the positive integers, and I don't include zero, which is not positive.
    Beltrano says: When I say "natural numbers", I mean the non-negative integers, and I do include zero. which is not negative.
    Sicrano replies to both: No problem, as long as we undertand each other in this discussion. After all [see my post no. 64 above], even Peano proposed at first that the natural numbers were {1, 2, 3, ...} before revising his axioms to define them as {0, 1, 2, ...}.
    (Poor Murray, entrapped in this fascinating yet frustrating thread that rambles on and on and refuses to die!)

  89. mike toye says:

    In the beginning, GOD.....
    Before God created anything there was no thing! We have passed the era of NOTHING. There are things to talk, discuss and ponder over rather than no thing! Zero is not natural FULL STOP.

  90. Pradip Baksi says:

    Dear Philip,
    Thanks for your comment dated 22 May 2011. Please note that, as already indicated in my intervention, the said comment was not from me but, from Vladimir Andreivich Uspensky.I have merely translated and included his text in the Special Supplement to my translation of Karl Marx's Mathematical Manuscripts [CONTENTS:

  91. Taylor says:

    I did not read the 90 other comments but here is my theory. Think of what it means to be a "natural" number. Why "natural" of all words? To answer this let's first look at raw data, or numbers. These numbers have inverses, negatives, like everything else. For example, tangent and cotangent. In the natural world data is composed of objects rather than raw data. The objects essentially are complexaties of raw data.

    Even "natural" language has much object like data. For example, if one was giving directions they could say travel 5 miles NE. That in essence is a vector. Both parts of the vector are necessary and since the vector is composed of more than one type of raw data type (namely direction and magnitude), the vector is an object.

    The vector always has to have a positive value or else it would not exist or would be pointing in another direction contradicting its definition. However, a vector can have a value of zero. Other objects can also have instance values of zero but not negative values in nature. For example, a certain flower can have 5 petals but it could also have zero. However, this flower could not have a negative amount as mentioned earlier. Therefore, since the instance value 0 can exist in natural objects, a single natural number or an instance (raw data) of the natural object can be 0 or any other positive integer.

    This is my theory. This may help clarify why computer scientists and set theorists believe that 0 is in fact a natural number. If you are asked on a test about it... just copy this paragraph. Hope this helps!

  92. Rina Sengupta says:

    Natural numbers came naturally to mankind.To keep an account of their possessions they(I mean our ancestors) not only used their fingers & toes but also the parts of their arms and so on.0 came much later, while trying to solve certain equations, we know of. So 0 can not be a natural number.

  93. Euler says:

    I realise that this discussion goes in circles, because despite intuition the bottom line is how the phrase "natural number" is being used and understood.

    However, I personally think you are at the heart of the matter, Rina. To extend your idea. If I ask someone to show me one calculator or two books then I would be somewhat surprised if they showed me one pineapple and two space shuttles.

    Yet if I asked someone to show me zero calculators or zero books then I would not be surprised if they showed me zero pineapples or zero space shuttles instead. I simply would not be able to make a distinction. Surely we cannot say that is natural! (c;

  94. Rina Sengupta says:

    May be I was not able to put forward my idea clearly. I wanted to emphasize that as 'counting', to keep an account of any possession,came into 'existence' so did the natural numbers. Relation with the body parts was like 'one -to one' correspondence between elements of any two non empty sets. Ignorant may not distinguish between a calculator and a pineapple but not the quantity of the articles brought. It's very difficult to answer 'what is 2?' or any such number to a child.We have to bring in pairs of many articles to show them the relationship existing between them.Here, we try to develop the 'natural' ability of the child to understand the 'essence' of any number.
    Once that is achieved, one can gradually understand the other sections of the number system.

  95. Shamlu says:

    Hi ... May be this thought is "antique" ... It started with nature. Natural numbers... 1 sun, 2 eyes, 3 leaves of clover or 3-toed sloth, 4 from a quadruped, 5 fingers or toes in man, 6 corners of a hexagon in a beehive, 7 colours in a rainbow, 8 legs of a spider, 9 oceans(or till recently, planets), 10 digits of a man's hands.... Numbers which appear in nature, Beyond 10, just look at the flowers on a plant .. Thousands of blades of grass in a square patch, millions of stars in the sky or grains of sand on the beach. Numbers are natural because they appear in nature. I have defined natural numbers in this manner in my Primary MATHS Text Books for India, Pakistan, Sri Lanka and Nepal.

    ZERO or negative numbers do not appear in nature. ZERO is a concept which man discovered (remember, it already existed in man's fingers or toes) and it does not exist in nature ... So, I think, ZERO is not a natural phenomenon, neither is a negative number (a debt or one less than ZERO are man's concepts) a natural concept. Hence, are this should be quite an acceptable definition.

  96. Shamlu says:

    Correction ...
    Remember, 10 already existed in man's fingers or toes

  97. yeah says:

    may i ask?
    i have been reading all your comments, making me confused!
    what is really the truth?
    is 0 considered a counting number?! Is there any evidences?

  98. Rina Sengupta says:

    Shamlu, I liked your explanation. I feel number system is one of the most important & interesting topic of Mathematics.
    yeah,0 is not a counting number!

  99. Joe says:

    It's far simpler, it is logical and natural to follow a consistent sequence:
    00, 01, 02, 03, 04, 05, 06, 07, 08, 09,
    010, 011, 012, 013, 014, 015, 016, 017, 018, 019,
    020, 021, etc.

    When we run out of numbers after number '9' we repeat the sequence with a '1', '2', '3', etc. in front of the elements of the preceding sequence. To use 10 as the end of a base 10 sequence is inconsistent, and therefor illogical and un-natural.

    Just because man was too limited in his perception at the time to come up with the concept 'zero' does not justify excluding the number zero from the natural numbers.

    The argument that zero is not a natural number because man could not count zero fingers does not make sense. Cut of his fingers and thumbs and ask him to count his digits, then he will have to come up with a concept to describe this state of being: "I count zero digits on my hands."

    I guess to solve the problem once and for all is to genetically manipulate man to have 4.5 digits on each hand and foot. We can than count in sequences from 0 to 9, with the toes as indicators for the next 10 elements in the sequence. Then again, there will be hordes of people disagreeing, saying genetic manipulation is not natural!! Well, who is to say that man's lack of previous intellectual evolvement in this science does preclude genetic manipulation from being natural... 🙂

  100. Michael says:

    This is used earlier for natural numbers "using your fingers and the first finger looks like a 1" as far as I know when you take that finger down it is a 0. Would that not be considered a natural number?

  101. Vincent_P says:

    Clearly 0 should be a natural number, since if you want to define addition on the natural numbers such that (N,+) is an abelian group you are going to need the identity element (i.e. 0) in that set.

  102. Shepherd Moon says:

    Disclosure: I'm a total math layman so I'm just commenting recreationally, not in any way trying to settle this discussion for good.

    I did look up Natural Number in Wikipedia and it mentions the concept of a successor.
    * There is a natural number 0.
    * Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a + 1.
    * There is no natural number whose successor is 0.

    I wonder if it is possible to add an axiom (if that is the accurate term) that states:
    a + a = S(a)

    In other words, for every natural number, if you add it to itself, you get a successor number. Except with 0 if you add it to itself, you get 0, not a successor.

    I don't know what to make of this, but it does seem to make way to exclude 0 from the positive and negative integers. Not that the negative integers matter in this discussion of natural numbers, but if a < 0, then a + a ≠ S(a), isn't that true?

    Thanks for this seemingly simple but fascinating question.

  103. Shepherd Moon says:

    One more sorry. 🙂

    I was thinking of successor number in the general sense, that is, a number greater than a - not just a + 1.

    So where a >= 1, a + a > a.
    But where a = 0, a + a = a.
    And where a < 0 a + a < a.

    I'm not sure if the above properties mean 0 is or is not a natural number, but it certainly seems to possibly to distinguish it from the positive integers in this way - as opposed to something like a + 1, for which 0 behaves like all the other positive integers.

  104. Euler says:

    @Shepherd Moon: I like your idea of a successor because, like every argument which excludes zero, it appeals to the sense in which natural numbers are... natural?

    I mentioned in an earlier post that I teach Mathematics in England. What this gives me is a privileged insight into developing minds and as part of your teaching you need to be both aware of and prepare for any concepts which *seem* counter-intuitive. A classic example is arithmetic with negative numbers. In the most minds of most people we believe that adding makes bigger and subtracting makes smaller. The first time a student sees something like, 5 + ? = 2, it doesn't make sense. In other words, it is natural that adding two numbers gets an answer which is bigger than each of the numbers you start with.

    To formalise this idea: if x, y belong to the set of natural numbers then x + y > x, y. "Naturally" this excludes zero! (c;

    It seems clear that learning about directed numbers require a re-boot on our concept of number and it becomes necessary to add a family of less-natural numbers which satisfy peculiar results, like x + y <= x, y.

  105. Geoff Pointer says:

    I think people are getting too hung up about the meaning of the word Natural. We are after all talking about doing maths and modern maths is not necessarily something whose axioms have to arise from elements that would be intuitive for young children, for instance. The human intuition evolved in the wild and it's quite flawed from a modern scientific perspective.

    Forget about imaginary elephants, young children are generally pretty honest. If I asked a young child at random to go to an empty field and tell me how many sheep there are, I'm pretty sure I'd be told there aren't any. The guy above whose rebuttal of the point about one finger being held up being the number one as a starting point showed quite clearly that holding no finger up is just as easily seen to be zero.

    There is just as much disagreement in the world about the definition of Whole Numbers as there is for Natural Numbers, so no, the former is not a handy default name for the set of positive integers in union with the set containing only zero. And writing "Doh!!" after this claim is just a mirror for whoever said that. Very mathematical indeed. And besides, what's whole about zero? (Rhetorical, please don't try to educate me.)

    However, the point, I think, is about, the sets that we need in the maths that is being done right now. In a purely utilitarian fashion the symbol of \mathbb N is most "useful" if defined as including the number zero, then the asterisk can be used consistently throughout maths. That is, \mathbb R*, the reals without zero, \mathbb Z* the integers without zero and \mathbb N*, the natural numbers without zero.

    As someone has already mentioned, I don't see \mathbb W catching on.

  106. kk raghuthaman says:


    Our confusion arises from a basic understanding ‘number’. If ancient Indians had invented zero how they could have imagined it? This could confuse us. I think that ancient Indian ’0? is a very simple concept.
    Modern mathematician imagined a one-to one number application. (Say one to relate one). It is simpler and straightforward number imagining. No wonder modern count starts with one (1).
    Ancient Indians could not have imagined a complicated ‘zero’ for everyday number application. Is zero truly simple?
    Adjacent 2 digital numbers have ‘same gap’(1) in between them. Ancient Indians noticed this “exact one” in each gap.
    A related ‘digital count’ is explained below.

    0…..1…...2…..3…..4…..5….6…. 7…..8…. 9
    This is a count relating start of each 1
    Modern digital count(only whole numbers)is shown below.

    …..1…...2…..3…… 4….5…..6…..7…. 8…. 9…10
    This is a count relating ‘end of each 1’

    Vedic Mathematics by Shri Jagadguru Sankaracharya 1884-1960 is evidence to ancient Indians used “zero-start count”/ “before one/ones count”. It is equally simple as ‘modern one-to one count.
    Both these opportunities to count did exist (always).

    It is pointless to conclude a ‘simpler count’ by excluding related ‘facts’.
    I disagree with “zero is unnatural”. ’0? hunter animal ‘chases me’ is an ‘unnatural sense’ to each prey-animal?

    ‘Natural numbers’ and ‘whole numbers’ come after “what is a number”?
    Ancient Indians regarded each number is a ‘condition’, which is count ‘before one/ones’ by looking at a Vedic matrix 0…9. Vedic matrix technically eliminates several number imagining problems of today.

    ..0… 1… 2….3….4… 5….6….7….8… 9
    x00 —- —- —- —- —- —- —- —- —-
    1—- —- —- —- —- —- —- —- —- —-
    2—- —- —- —- —- —- —- —- —- —-
    3—- —- —- —- —- —- —- —- —- —-
    4—- —- —- —- —- —- —- —- —- —-
    5—- —- —- —- —- —- —- —- —- —-
    6—- —- —- —- —- —- —- —- —- —-
    7—- —- —- —- —- —- —- —- —- —-
    8—- —- —- —- —- —- —- —- —- —-
    9—- —- —- —- —- —- —- —- —- x99

  107. Joe Razavi says:

    Apparently lots of people in the past didn't consider 1 to be a number, because "a measure is not the thing measured", or for example because they expect multiplication to be strictly increasing -- similar to some arguments above.

    The Pythagoreans didn't include 2 either, because they thought that a number was something built up out of 1 and 2 by addition. My source here is an MIT OCW course document.

    Personally, I think there are 3 good choices: {1}, {0,1,2,...}, or {2,3,...} ! In the "real world" there is surely only one of everything -- the only thing which is the same as something is itself, so there are no multitudes of "same things". Any counting after that already relies on an abstraction of the objects: either we compare the world to an imaginary object (in which case, include 0), or else we pay attention to those differences between things which we think leave them "essentially the same". In that case, leave 1 out as well, since there are no differences to think about if there is only 1 of something.

  108. Murray says:

    @Joe: Thanks for that - and for the OCW resource. Very interesting!

  109. marikar says:

    interesting discussion.

    is zero an even number?

  110. Murray says:

    When 0 is divided by 2 there is no remainder. So yes.

  111. Ben McCann says:

    Riddle me this. A family from Mexico moves to the US. The US doesn't have a Cinco de Mayo celebration because May 5th holds no particular importance to their national history. Does that mean the immigrant family should stop celebrating Cinco de Mayo and start celebrating the Fourth of July instead, just so that any potential visitors don't feel confused?

    Such is the case with the set theoretic definition of the natural numbers. It carries deep meaning and importance to the field of set theory, a kind of significance that the word "whole" simply fails to connote in its entirety. I say, let the set theorists have their special N, and if students use the other definition, let the applicable problems be exempted from their tests and explained, at least the first time they make the mistake. It may be less practical, but it's more respectful.

  112. Murray says:

    Interesting take on it, Ben!

  113. Geoff Pointer says:

    @ Ben McCann Interesting, but in your metaphorical example there is no doubt which festival pertains to whom. With the "natural numbers" is there really a field of mathematics that can claim sole historical ownership of that name and hence demand what it's definition should be? I can see that opening a "whole" new can of worms.

  114. chithanh says:

    Actually, extending arithmetic on natural numbers with subtraction is a bit tricky, as natural numbers are not closed under "normal" subtraction.

    Those who take a rigorous approach to constructing arithmetic often use the predecessor function, which is defined recursively as

    That definition is ok, because every natural number is either 0 or successor of some other natural number according to Peano's Axioms.

    And it allows you to define a truncated subtraction ? as iterated predecessor (much like addition on natural numbers is the iteration of the successor function).

    So no -1 needed here. BTW, it will still work when you start natural numbers at 1 with p(1)=1 but things will get slightly more ugly then.

  115. William Wu says:

    Excellent explanation on whether 0 is a natural number.
    Sometimes it is actually a convention, for example the Peano axioms state that 0 is a natural number.

  116. Nasredeen says:

    I do not believe in Zero as a number!
    zero is not a number! It is rather a designated concept to
    serve the purpose of the transition from negative to positive numbers
    zero deceive us with its simplicity ( 1+0=1 , 1 x 0= 0)but this simplicity turns into hell when we try to manipulate operations such like (0/0) !/!
    But according to the limitations in our daily life - as humans - we do not have real life apps which encounter operations of the type (0/0) so no much talk is said about this dark side of the Zero!!!!

  117. 2piix says:

    Personally, I find the whole thing confusing. And in my experience, set theorists *don't* treat zero as a natural. In particular, the first infinite ordinal omega is taken to model the natural numbers. And if we recall, there's no such thing as the "zeroth" element of a list.

    And abstract algebra "tends" to treat zero as a natural number. If it didn't, we'd have special cases all over the place.

    I have followed a policy of inclusiveness since college. I assume 0 is a natural unless otherwise indicated. It never breaks anything. If anything, the whole numbers should start at 1. (And this is the convention I was taught.) What is "whole" about zero? Literally nothing! Sure, it's not fractional. But it is not a whole anything, and that's rather the point of 0.

  118. Joe Razavi says:

    2piix, I thought that in the Von Neumann (usual?) definition of the ordinals, 0 is an element of omega?
    This has something arbitrary about it; I think it is better to consider structural set theory, where the natural numbers are defined to be a set N with a special element 0 and a special function S: N -> N together with a condition saying that induction and recursion work. (See Of course, we could call 0, the special element, "1" if we wanted, but that would miss the point: the function S is a process which can be iterated (counting!) and 0 tells us that there is somewhere to start counting _from_ -- corresponding to _zero_ iterations of S.

    John Foster's early comments have also been praying on my mind. In my last comment, I was really trying to cast some doubt on the idea that "concrete referents" come before counting. Now I think I can say something about the reports of children. Imagine that every day a child reports the number of sheep in a field. If there are no sheep, they say "there aren't any". It would seem correct to record this as "0 sheep". As soon as we know what it is to count sheep, we know that writing "0 sheep" means "there aren't any sheep".
    But is this a slippery slope: perhaps in some society, it's obvious that once you know what a "leftover left slipper" is, you know that a "leftover right slipper" is -1 of them. If this seems contrived, because left and right slippers seem to have an independent existence, consider windings of an elastic band around a peg, where opposites can't live peacefully together. And perhaps for "homogenous materials", once I know how to count lumps of them, I should know the meaning of fractional parts -- though I find this last thought very mysterious, and would be grateful for an explanation!

    Sorry for the novel!

  119. GM says:

    I used to be a set theorist, and I certainly take 0 as a natural number.

    The situation is confusing for students. One way is to use "(strictly) positive integers" for 1, 2, 3, ... and "non-negative integers" for 0, 1, 2, 3.

    Zero is certainly a counting number. Ask a lecture theatre full of students something like this: let n be the number of people in this room born in Iceland. Surely n, whatever it is, is a counting number, and it is likely to be zero.

    The history is interesting (and I used to teach it), but it is not a good guide to current usage. For most purposes now, the more useful definition is to include 0 as a natural number, and my (blackboard bold) N certainly includes 0. (I'm talking about, say, upper secondary and above. I don't have any thoughts on how to teach primary school children about zero and negative numbers.)

    All this is ultimately a matter of agreed convention, as several posters have said. As far as things like the Peano axioms are concerned, we could make up two systems, one with zero and one without, and use either as a basis for building up the negative numbers, fractions, etc. This sort of building-up is of interest to mathematical logicians, who want to use the smallest number of assumptions necessary, but not really a guide to more general usage, even among mathematicians.

    Part of the problem is the word "natural". Even that isn't so natural. It is part of our intuition of (natural) numbers that they go on for ever; you can always add 1. But I have read that some traditional societies had a number word that meant something like "everything" or "the end", that was considered as the biggest number.

  120. Raghuthaman Kunnathully Kunhukuttan says:

    Actual Problem lay in language sense of natural.Language sense is never the same to users. Number sense has to be same.It is well known today that 0 and 1 are base numbers and all other n times'1' merged numbers are 2, 3, 4 and so on.
    0,1,2,3 is a count by before one/ones.
    1,1, 1, 1 first one has '0' before one, second 1 has 'a' before 1,third one has 2 before '1's and forth one has 3 before ones and so on!
    119.comment G M says gives a right direction.
    Count from '0'and onward(world wide)
    What is so un-natural about zero?
    To promote number sense and language sense have related problems.Nobody can impose a! rule that mathematics shall be unique all over galaxy.
    I find Vedic mathematics makes use of before one/ones count and a related Vedic matrix positions count as 0,1, 2, 3, 4, 5, 6, 7, 8 and 9. in y and x directions of matrix. A related yx merged count 00...99 or an n0s...n9s naturally work in an identical manner. Related Vedic mathematics is flawless mathematics. India alone can implement this but question of unique number application does exist! Please understand!
    Indians hardly recognize supreme kowledge today. Vedic mathematics is no exception to this! However my limited awareness of Vedic mathematics(dsiclosed by Shri Jagadguru Sankaracharya 1884-1960)Iam publishing as blogs on internet "" as blogs. It will give insight to basic issues of propagating mathematics and also basics of carrying a basic knowledge millinniums forward!
    Language sense of "natural" is definitely playing role in whole discussion. Be open for time being.
    Comment 119. GM says right thing. Use count 0, 1, 2, 3, 4..

  121. Quinn says:

    If you say 0 is not a natural number then it must be removed all together and we would have to learn math all over without multiples of 10. There would be no 10 because 0 is not a natural number. Mathematics is not something we created, it is still being discovered. Get over yourselves. Math is. You didn't add those numbers and have no right to remove one. Leave the 0 alone.?

  122. Jenson button says:

    I'm confused now.....
    My exam is coming up...
    So is 0 a natural no?

  123. Merely Useful says:

    How about this? The natural numbers are those which are used for counting whole things. Let's say there is no zero. For any subtraction a-b the domain of b is b<a otherwise you can have a result of zero which is an invalid result. Therefore if you have 5 apples.
    , you can only give away 4. It is invalid to give away 5. This is clearly an absurdity, ergo 0 is a natural number.

  124. Mahesh says:

    I think they (the first mathematicians) found 2 important sets, one starting from one and the other from zero, called the set starting from (0/1) as the natural set and the set starting from 0 the whole set, with the reason being anything in this long thread (even something shown false), as it will not matter now, or even that they had a outreach example that would now seem stupid). The main thing is to adopt a convention to avoid opinions in math. It will at least give some student from following their teacher's psychology...

  125. David R Smith says:

    In counting sheep, the method is,
    For every sheep that passes through the gate, you add one to the number you had before the sheep passed through the gate.
    For this to work, the start point has to be defined, and the number of sheep in the empty pen is ZERO.
    With the empty pen, the counter would say "I have no sheep.", or perhaps "I have not any sheep"
    The latter statement involves the Boolian negation operator and an indefinite plural quantity.
    Actually, in the first sentence, "no" could bee seen as a Boolian operator on sheep, but interpreting it so might imply that not(sheep) might be goats, so that the sentence with that interpretation is im precise, and should be, on a mathematical basis, be rejected.
    The then requires the word 'no' to be seen as a number representing 'zero'
    In this sense, then, zero is a natural number.

  126. Tshepang says:

    Guys,plz help me my head is cracking here is 9,16,81 natural numbers or not plz guys i need your help

  127. Roger Sheppard says:

    A simple number line places zero. If one limits one's number line to integers..ON EITHER SIDE OF gets negative integers and positive the Set of Z.

    This will include zero, a simple placement to indicate emptiness, OR importantly , that position where negative jumps the boundaries into positive and vice versa.

    As I said to my schoolboys "...I listened to the computers running last our Varsity.. The Prof. NEARLY found out how many times zero divides into three! [he had tried to divide zero into 6 he day before!]..he had zero in the denominator position, and says: "Why should it be that all numbers on a number line can divide into an element of Z...except zero? Doesn't zero it not a whole -number placement? Z = {...-3; -2; -1; 0; 1; 2; ...} Keep It Simple..Sing Country! That's where Brian May began!

  128. aaimah says:

    0 was discovered or invented by aryabhatta or you can say 0 was made by a man so its not a natural number....

  129. John C. Turner says:

    What an amazing, long-lasting blog, on the nature of 0! 'Much Ado about Nothing'.

    A few semi-serious, half-facetious, some-on-the-button, 'submitted with love', comments spring to mind.

    (o) Dare I say 0 is just a simple -1,0,1 problem?
    [hint-hint, nudge-nudge, wink-wink: Joke]. Actually, that
    joke is right on the money: since 0 is half-way between
    -1 and +1. All three are integers, and each takes some

    (i) In line 1, I used 0! That equals 1, a fine natural number.

    (ii) No blog has mentioned (I only skimmed them!) that
    it has often been said that 0 is the greatest invention
    that mankind has ever made. Bar none, THE GREATEST!

    (iii) Several books have been published in the last 16 years
    about the invention and uses of zero (i.e. 0). Try
    'The Book of Nothing', by J.D. Barrow, Vintage(2001). A
    fascinating and scholarly book. Much earlier,
    Shakespeare wrote many passages and paradoxes about
    'Nothing'; for example: Macbeth despairs that
    "Nothing is
    But what is not."
    A modern writer, Jean Paul Sartre, trying to convey
    information about the origin of Negation, wrote
    a 600-page tome on paradoxes about Nothingness and

    (iv) See Blog #65. The author asks if 0 is
    before or after 1-9. If he reads Fibonacci's famous book
    'Liber Abaci'(1202 A.D.)he will find on line 1
    his version of the Indo-Arabic numeration system as
    1,2,3,4,5,6,7,8,9,0 (if my memory serves me aright).
    Fibonacci goes on to demonstrate the
    arab methods of arithmetic, using these symbols, with 0,
    to Europe and hence to the Western World.

    (v) That's 'nuff from me. I'll get back to my efforts to
    solve the twin primes conjecture ... they start with
    (3,5), (5,7), ... Poor (2,3) doesn't get a look-in.
    Shall I call it number 0 in the sequence, to give it
    a break?

    Bye, j0hn.

  130. Murray says:

    Thanks, John. Liber Abaci starts:

    “The nine Indian figures are:
    9 8 7 6 5 4 3 2 1.
    With these nine figures and with sign 0 which the Arabs call zephir any number whatsoever
    is written...” Source]

    Thanks for alerting me to this!

  131. Adrien de Croy says:

    Natural numbers are those that occur in Nature.

    0 does not occur, it is the lack of occurance.

    You cannot give me 0 chickens. I will call it 0 cats.

  132. Champ says:

    Leave all other concepts and just learn natural numbers (N) are counting numbers,
    N = 1, 2, 3.......
    And whole numbers (W) are counting numbers including zero.
    W =0, 1, 2, 3......

    I'm preparing for JEE Mains and JEE Advanced for (IITs). And I was simply taught this. Don't get confuse in other stupid facts as it will lead to a great misconception and finally you will always be wondering "WHEATHER TO INCLUDE ZERO OR NOT" while smart students simply use the above simple concept and save the valuable time during exams.
    SAM from INDIA

  133. Barry says:

    Zero is a natural number.

    There is an empty plate and 5 beans right next to it.
    How many beans are there on the plate?
    You look at the plate and there is nothing on it as beans.
    so the answer is: 0 spelled as zero.

    now you put one bean on the plate and ask again:
    how many beans are there on the plate?
    You look at the plate and you see one bean sitting on it. so the answer is 1, spelled as one.

    Space is always part and parcel of forms of natural numbers as beans, carrots, fingers, etc. much less the stick figures which we call as 1, 2, 3, etc.

    I hope this practical truth helps to decide the issue.

  134. Holosiva says:

    I think we shouldn't argue about whether zero is a natural number or not; it depends on how we define things. For example, in an exam, if we want to ask questions concerning the natural numbers, the examiner should make clear of which natural number system is being used. Then it would not be a problem after all. I prefer to work with N={0, 1, 2, 3, ..} because it fits the Peano Axiom and the definition 0*n=0. I don't mind working with N={1, 2, 3...}, because the choice of convention would not alter much of my mathematical studies in the future anyway. This is just my humble opinion.

  135. Marcel says:

    I am a math teacher and also a student (College in progress). In my modest opinion this is a matter that shows 2 things: (1) specialists in math as a majority don't care about the subject enough to come up with a definitive answer or (2) don't have the formal structure to do it.
    Other sciences usually have a committee like the international system of measures and and other things that are a matter of opinion are discussed via formal discussion.
    I read most of the posts here (not all). Isn't there any formal discussion about this topic in scientific papers, not textbooks??

    I mean, textbooks are good and all, but you cant take them without questioning.

    I think one fundamental problem to this discussion os whether the person is a nominalist, or a platonist or whatever. You wont agree if you have different perspectives. For some will be just a matter pf deciding for the other a thing to be discovered, which changes a lot what is the point of the discussion.

    As a matter of fairness and practicality I think the said textbook, test or whatever, needs to explain either on a side note or what is using, because what we cam conclude as a fact of this discussion is only that we have different uses in different situations, textbooks and countries, which is problematic and far from the math ideal of precision.

    Personally I am a nominalist so I just want a better and universal definition so we don't keep talking about 2 different things without knowing. Depending on the situation when learning about corpus and rings usually things need to be well defined in order for us to understand how an unusual operation works and what type of group something is. If saying Natural numbers people don't understand what is being said, either we define it universally or just drop it already and just use integers, positive integers, etc, that people can understand and be precise. I think that is the whole point of mathematical language, don't you think?

  136. Vivi says:

    For those who consider 0 unnatural, unnatural, if a child has a number of candy and after a while you eat, if you ask how many does he have? Reply none! Number 0 corresponds to this,, nothing,, you have 0 lei bank account.
    Divisibility of natural numbers without 0 would be much poorer. Ex. property that any integer divide any integer. (0 divides a ∙ n as 0 = 0, regardless of the integer n). Thus the relation of divisibility on N is reflexive, antisymmetric and transitive.
       Someone above said that if a person wants to count the stars can say that numbered 0 stars. Yes, if the cloud.
    Excuse my English!

  137. Vivi says:

    Excuse my
    ,,0 divides 0, since 0 = 0 ∙ n, regardless of the integer n,,

  138. Richard Hill says:

    Just except the convention and get on with life. It's nothing after all and that is how much we should care about it 😎.

  139. Dick van Dijk says:

    Gullberg and its book "Mathematics from the birth of numbers" is showing too much different names.
    A new "natural start of the beginning with nothing" shows in Nature's unique & unambiguous way why there are no negative numbers in nature and why zero and its beta-symbol 0 is a "non-natural(counting) number".

    This new start not only enforces to accept Nature's logic, but also to respect its logistic order, disclosing -step by step-
    Nature's "All Unifying Theory" and its process of "creating something out of nothing".
    If you are interested to understand why Nature's dynamic math is far more simpler that static & immobile math of humans in all its beauty, see my website "".

  140. Valareos says:

    I think 11 years on this subject is proof enough that it is a matter of opinon, but I will put my 2 cents in from a person who is looking to teach mathematics.

    Historically, 0 was not a natural number, because it didn't exist in antiquity. And as math is built up on the shoulders of giants, the idea of Natural Numbers starting with 1 stayed. Nowadays, 0 is natural, so we feel that that should be a natural number. Instead of asserting my opinion though.. I want to look at this for the sake of conventions.

    First, using N starting at one, and Z (the set of Integers)..

    If I wanted the set of Positive integers, I could say N or {Z}^{+}
    If i wanted the set of Positive integers AND 0, I would have to do N U 0 or {Z}^{+}U 0

    ok, what if N started at 0?

    set of Positive numbers N/{0} or {Z}^{+}
    set of Positive Numbers AND 0, N or {Z}^{+}U 0

    Comparing the two, you can see that having N start with 0 is better convention wise, because set of positive integers can be written as {Z}^{+}, and N for set of positive integers

    Is 0 a Natural number? Historically no, but it should be, because I don't need two different notations for the set of Positive integers, but I do like having a simple notation for Positive integers and 0

  141. Valareos says:

    correcton above.. Comparing the two, you can see that having N start with 0 is better convention wise, because set of positive integers can be written as {Z}^{+}, and N for set of positive integers plus 0

  142. EetSumtinSpecial says:

    Big deal. There are too many people in "academia" always trying to make others believe they know a lot. Many authoritative set theory books adopt {0, 1, 2, 3, ...} to be set of natural numbers. It is mainly in beginning algebra books that we see idyots insisting that n = {1, 2, 3, ...}

    So who do you want to follow?

  143. John T says:

    Given, my first love (with respect to pure mathematics) will always be set theory, so I'm perhaps biased, but the set of all natural numbers includes 0. If it excludes 0, then the set of all natural numbers and the set of all positive integers are identical, and this set has been labeled twice at the expense of a commonplace label for the set of all non-negative integers. If the set of all natural numbers includes 0, then each of these sets has a unique name.

  144. Valareos says:

    John T, I believe you have reiterated my argument, but in a much more eloquent way.

  145. polygame says:

    What is it about this question that has captured the imagination of so many people with such varying sophistication for such a long time? It is really quite fascinating and I cannot resist the temptation to weigh in just for fun.

    In Mathematics it often happens that the same term is used in different contexts to mean entirely different things. For example the word 'normal' can refer to a type of subgroup that is the kernel of a group homomorphism, to a vector that is orthogonal to a manifold or to a set of productions in a context-free grammar that is in a standard form (think Griebach or Chomsky normal form). The same is true of the term 'natural', we have the natural numbers, natural logarithms, natural transformations between functors and we have natural isomorphisms. However, the latter two concepts likely do not feel natural to the average adult, talk less of a seven year old.

    So, whether or not something is considered natural depends very much on your background. To be sure, the concept of zero may not be something you are born with but it doesn't take long before you become quite comfortable with it. (If it weren't for the potential confusion with the complex numbers, we could call {1, 2, 3, ...} the set of comfortable numbers . . .). However, it requires a bit more sophistication before the notion of a natural isomorphism actually feels natural.

    It is also true that the language of mathematics evolves over time, just as natural (oh dear) language does. Isaac Newton used the term 'fluxions' which was a precursor to the modern term 'limit' and abstract algebra used to be called modern algebra. Other examples abound.

    The language even seems to evolve during one's mathematical education. When we were in primary school, prime numbers were those that could only be divided by themselves and 1. Of course this includes 1. By the time we got to college the 1 was dropped so that the smallest prime number was now 2. What happened? Well for one thing 1 has some rather special properties that the other primes don't (it is a unit and an idempotent), but the main thing is that the prime factorization theorem would no longer be true as it is currently stated since the factorization would not be unique. The theorem is still true but its statement would become quite a bit more awkward, as would many other theorems about primes that would have to say things like "If p is a prime, where p > 1, then ..." instead of "If p is prime, then ..."

    Nowadays in mathematics when a new concept is discovered or invented and we want to call it something, we often seek some term from the vernacular that somehow gives us a bit of intuition about the concept. For example the notion of a sheaf of rings on a topological space is reminiscent of a sheaf of wheat - the common meaning element being something like 'a bunch of similar-looking things emanating from points on a surface'. Once the term is chosen something wonderful happens, the meaning of the term is purified and made precise by stripping the term of all its previous connotations and attaching the unique special meaning to it by way of a mathematical definition. This way, as distinct from natural language, the term becomes unambiguous. Or at least it should.

    Which brings us to the ambiguity in the term 'natural numbers' and how it should be resolved. It has been pointed out that number theorists like to start their natural numbers at 1, while set theorists like to start at 0. There is a good reason for this and it and it comes down to ease of expression. Set theorists find the set {0, 1, 2, ...} appears more frequently in the statements of theorems and proofs than the set {1, 2, 3, ...}, while number theorists find it the other way around.

    Thus this term's definition is now context dependent. This may be troubling to some since all the number systems


    are the same in every context with {\mathbb{N}} being the only exception.

    Within computer science, which is a branch of mathematics, something made explicit in many programming languages is still present implicitly in the rest of mathematics and this is the concept of a namespace. Essentially every piece of published mathematics has its own namespace and is free to use whatever terms it wants within that namespace as long as it defines its terms precisely and uses them consistently within the namespace. Another way of saying the above is that {\mathbb{Z}},{\mathbb{Q}},{\mathbb{R}},{\mathbb{C}} are in the global namespace, whereas {\mathbb{N}} is defined in a namespace local to the subject matter (set theory, number theory, algebra, etc.).

    Or we might look at this as letting {\mathbb{N}}_{1}={\left\lbrace{1},{2},\cdots\right\rbrace}{\quad\text{and}\quad}{\mathbb{N}}_{2}={\left\lbrace{0},{1},{2},\cdots\right\rbrace} as natural numbers versions 1 and 2 and that set theory has chosen not to upgrade!

    Finally, there is evidence that the definition of natural numbers actually is evolving. Certainly in abstract algebra and category theory, which are more recent in the development of mathematics than number theory, the almost universally accepted definition is {\mathbb{N}}_{2} and there is an algebraic reason for this: {\mathbb{N}}_{2} is a commutative monoid under both addition and multiplication, whereas {\mathbb{N}}_{1} is only a monoid under multiplication. Since multiplication (of positive integers) is defined using repeated addition, addition is seen to be more primary and so one could argue that in some sense {\mathbb{N}}_{2} precedes {\mathbb{N}}_{1} and should therefore be the definition in the global namespace.

  146. Student says:

    Hi I need quick help here, so is zero a natural number or not???

  147. polygame says:

    Your question suggests that you are expecting a context-independent answer to a context-sensitive question. This blog has not managed to achieve this yet. If you were a set theorist or an algebraist you would prefer that 0 would be considered a natural number, but if you were a number theorist you might not. If you are a student that is studying both, then you have to look at the context. How does your professor define it? What does your text book state? What makes more sense in your context?

    The only way to get an answer that will satisfy you is to first understand why you need it. Is there a specific problem whose solution depends on the answer (such as "do the natural numbers form a monoid under addition?") or is it to win a bet with a fellow student? And why the hurry? This blog has been debating this question for nigh on twelve years, it's unlikely to give a definitive answer on a tight deadline!

  148. ir D. van Dijk says:

    see comment 139, zero is a non-natural (counting) number and it has unique & unambiguous reasons for this...
    When human beings think they can deny Nature, they continue with disasters and ambiguities.

  149. Jenaro Tomaszewski says:

    I was always taught that natural numbers are positive integers. Whole numbers start at zero and integers include negative numbers

  150. Antonio Adrian says:

    Any natural number will follow this:

    A-B is smaller than A

    Except for 0:

    A-0 equals to A

  151. Dr. Cocktor says:

    Wonderlesswizardofmath is right. Its a matter of opinion. I think we should drop the term ‘natural numbers’ and just call the set of integers starting at 1 ‘positive integers’ and call the set of integers starting at 0 ‘whole numbers’.

  152. asd says:

    ISO standard ISO80000-2 defines {N}={\left\lbrace{0},{1},{2},{3},\ldots\right\rbrace} and {N}^{\star}={\left\lbrace{1},{2},{3},\ldots\right\rbrace} So even if you don't agree with {0} being natural number, you should use this notation with the corresponding sets of numbers.

  153. Murray says:

    Thanks for alerting me to ISO80000-2, "Mathematical signs and symbols to be used in the natural sciences and technology". That standard defines the Natural Numbers as "The set of positive integers and zero".

    I'm happy to go with that!

  154. Raj says:

    I'm not a mathematician, but I got some interests on this. (I'm a software engineer / developer)

    On some video from Numberphile, a guy said, in some conjuncture (may not be the exact word) it says, "all natural numbers can be written as a unique product of prime numbers".

    Which indicates, for every natural number, there exists exactly one unique product. e.g. 12 = 2 x 2 x 3. No other combination can make 12, and vice-versa, 12 cannot be expressed in any other way.

    However, for 0, there exists (option 1) absolutely no such combination, if you consider 0 not to be a prime, or (option 2) an infinite amount of such combination, if you, for some weird reason, count 0 as a prime.

    So by definition of that statement, 0 cannot be counted as a natural number. Also, is the same reason why 0 does not have any true factors, because factors are honestly a natural number thing. maybe, due to it's nature, it can be extended to all integers, but not really a all-integer thing.

  155. Florence says:

    Natural numbers are counting numbers. in agreement with that,we can't count nothing(zero).There must be a student,two students or three for us to count.Therefore zero is not a natural number.

  156. Vince says:

    Surely the logic here doesn't work past the first integer, "1". 1 x 1 = 1, 1 x 1 x 1 = 1, ad infinitum. Ergo there's an infinite amount of combinations for any prime number. What you could say however is that 0 is a special prime number.

    I ended up here, because I was wondering why we don't start counting from 0, it seems odd to shift our paradigm 1 integer into the positive numbers from the centre of all numbers. I guess we're just used to it like we are used to base 10 (base 10 actually has no meaning kek) instead of base 12.

    P.S. I think you probably meant conjecture.

    P.P.S. I have no basis in maths, I just have random thoughts sometimes which I have to search online to satisfy my curiosity XD

  157. Yosefa Febrina says:

    There are so many misconceptions about the number zero. And some people accept zero as a natural number. Zero is the integer that come before the positive 1 and follows by negative 1. In most numerical system, zero was identified before the idea of ‘negative integers’ was accepted. Zero is the lowest nonnegative integer. The real natural number following zero is 1 and no natural number come before zero. Zero may or may not be counted as a natural number, depending on the definition of natural numbers. In mathematics, the natural number are those used for counting or measure things. But some people measure things with a ruler by aligning the 1 mark instead of zero mark. It is difficult to explain to them why they have to start from zero, because the definition of the natural number itself. Until now, there’s no absolute answers about the issue. But I think zero is included in natural number. But sometimes, in mathematic courses, it could be more convenient to exclude it.

  158. Jonathan says:

    To the best of my knowledge, zero as a natural number originates from Bourbaki's Elements of Mathematics: Theory of Sets in which they define the notion of "natural number" as (essentially) "the numbers corresponding to the finite ordinals". Equivalently, we could consider this definition as "the numbers corresponding to the finite cardinal numbers". In other words, the natural numbers are the numbers used to count the number of elements in a finite set.

    The empty set is a finite set. It has zero elements.

    Thus, zero is a natural number.

  159. Tim says:

    “TL;DR The community is divided.“

    You might even say that the community is divided… by 0. 🥁

  160. idahoMathLover says:

    While we consider the existence of the natural numbers to be axiomatic, whether those numbers include zero or not is ultimately arbitrary as well. Many mathematicians (probably the majority today) view that zero should be regarded as a natural number They have arguments for this view; however there are also many great arguments not to include zero. To begin, we start with the fact that we are not defining the natural numbers. Instead the existence of the natural numbers as a set follows from the axiomatic existence of a set that has as elements one, the successor to one (two), the successor to two (three), and so on, i.e., numbers that inherently exist in the natural world. So the question becomes, in keeping with this axiomatic view of the natural numbers, does zero inherently exist in the natural world?

    Those who argue in the affirmative usually posit some hypothetical such as this: "If you had exactly one apple and you ate it, how many apples would you have?" The problem here is that this presupposes the existence of the number zero as the answer to the question. However, representatives from many cultures in the world's history would simply reply that the question does not make sense. (Consider that there is no Roman numeral for zero.) Instead of saying that a person has zero apples, they might say that the person does \emph{not} have apples. So the inherent natural world property at work here is the existence of a negation to a statement, not the existence of zero. Since we already have negation, we do not need zero to describe the lack of some item.

    What about mathematical reasoning? Since, though we call them natural numbers, we are actually dealing in the world of mathematics, any good argument not to include zero should also deal with the mathematics of it. Consider our simple, elegant definition for rational numbers: the set of all quotients with an integer dividend and a natural divisor. If zero is included as natural, we must then explicitly add that zero is not allowed as the divisor. (In fact, most texts define the rationals as the set of all quotients with an integer dividend and a nonzero integer divisor, but this is less concise and less elegant. We do not need to consider negative divisors, since (assuming y\neq0), \frac{-x}{-y}=\frac{x}{y} and \frac{x}{-y}=\frac{-x}{y}.) There are in fact many definitions and theorems requiring the explicit exclusion of zero if zero is included in the natural numbers. Of course, we might instead use the set Z+, the strictly positive integers (equal to {N} in this text), but this is not quite as elegant.

    A related argument is the fact that every natural number as we treat them has negative exponential powers, but zero does not (though admittedly this is also true of the integers, in which zero is obviously included). Further, all natural numbers n have the property that if x > y, then x^n > y^n. Zero does not share this property. The truth is that zero is a very complicated number and simply does not fit with several theorems we could prove about the natural numbers as we treat them in this text.

    One of the most common arguments to include zero is to provide addition on the natural numbers with an identity property. However, that property is inherent because it is true for the integers. Any thing that is true for all integers is certainly true for all natural numbers, so while we exclude zero from the natural numbers as a set, we do not deny that adding zero to any natural number n gives n. But adding zero is simply not a natural-world occurrence, thus this is purely an abstract phenomenon, and is better relegated to higher arithmetic than natural arithmetic, which is usually used only at the elementary/primary school level.

    Advocates of including zero argue that to do so would make the natural numbers a monoid under addition, but without the additive inverses (i.e., negative numbers), the only unit for this monoid is zero; hence it is already not a group, and the "group of units" of this monoid would thus be the singleton \{0\}, therefore no group of any interest may be defined around this monoid. There is simply no significant advantage to artificially forcing the monoid definition to apply, since closure and the associative properties may be handled separately.

    These are the reasons zero is excluded from the natural numbers in this text, and why we encourage other authors to re-adopt this standard. (Note that Peano in the formulation of his axioms originally excluded zero from the natural numbers but later revised that decision.)

  161. idahoMathLover says:

    Sorry, I mean to indicate that quote as coming from the appendix of a textbook I am currently writing.

  162. Jarom Luker says:

    The comment about ISO 8000-2 totally misrepresents the information found there. What it actually says is that N =\left\{1,2,3,\ldots\right\}, whileN_0=\left\{0,1,2,\ldots\right\}. There is no mention whatsoever of N^*, either in ISO 80000-2:2009 or the updated ISO 80000-2:2019.

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