# Is $0$ a natural number?

Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?

It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers.

• It may be Italian education, but I've always been told, from 1st grade to 3rd year of my engineering degree course (present), that 0 ∈ ℕ, and never had any reason to believe the countrary. (We have ℕ₀=ℕ\{0} when the need does arise.)
Jul 21, 2010 at 12:06
• voted to close. The question is subjective, as is clearly indicated by the first sentence of the wikipedia article on Natural Numbers Jul 23, 2010 at 17:05
• While definitely subjective, it might be the case that the asker genuinely does not know about the controversy and is in need of an answer to say "There is no answer". Whatever the case, I still voted to close. Jul 23, 2010 at 20:49
• @Justin, I know that there are mixed views (as indicated in the second paragraph of my question). But for the case of 1 being classified as a prime number, it seems the consensus view of the Mathematical community is that it should not count as a prime number. My actual question is 'Is there a consensus on whether zero is a natural number?' (although the question's title is simpler), so a suitable answer would be 'No, there is no consensus' combined with a quick demonstration from a few Mathematical dictionaries or articles that there are conflicting definitions.
– bryn
Jul 24, 2010 at 2:37
• @Nick The responses to this question indicate that the definitions you propose are far from universally accepted. I agree that it would be great if everyone agreed on a standard, but I would argue strongly for the convention that 0 is a natural number. The convention $0\in\mathbb{N}$ doesn't mean you have to start counting at 0! Dec 2, 2013 at 18:24

Simple answer: sometimes yes, sometimes no, it's usually stated (or implied by notation). From the Wikipedia article:

In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers $\{1, 2, 3, \dots\}$ according to the traditional definition; or the set of non-negative integers $\{0, 1, 2,\dots\}$ according to a definition first appearing in the nineteenth century.

Saying that, more often than not I've seen the natural numbers only representing the 'counting numbers' (i.e. excluding zero). This was the traditional historical definition, and makes more sense to me. Zero is in many ways the 'odd one out' - indeed, historically it was not discovered (described?) until some time after the natural numbers.

• I see plenty of both these days, but when I was at school and at university, I almost only saw them defined to be {0, 1, ..}. The elements of {1, 2, ..} were called the whole numbers in my school days. Jul 21, 2010 at 9:06
• Maybe it's because I'm a student of physics that we do things slightly differently, but we seem to call only the counting numbers 'natural numbers'. 'Whole numbers' is just an informal way of describing all integers. Jul 21, 2010 at 9:12
• I would say that in number theory you will probably see $\mathbb N=\{1,2,\ldots\}$, but in set-theoretical textbooks $0$ will be included as a natural number. (It is a natural approach in that contexts, since they're defined as finite ordinals. Sep 17, 2011 at 12:09
• @Kaveh: Not including $0$ simplifies other things. For example, you can then define rational numbers as (equivalence classes of) pairs of an integer and a natural number; no explicit exception for $0$ needed. And also the equivalence relation can then be easily stated by $(a,b) \equiv (ac,bc)$ for any $c\in\mathbb N$ (again, no exception needed). Aug 1, 2013 at 17:12
• @celtschk: Those things that you mention still occur with more general fraction fields derived from integral domains, in which you still have to distinguish between $0$ and non-zero elements. Jun 3, 2014 at 7:37

There is no "official rule", it depends from what you want to do with natural numbers. Originally they started from $1$ because $0$ was not given the status of number.

Nowadays if you see $\mathbb{N}^+$ you may be assured we are talking about numbers from $1$ above; $\mathbb{N}$ is usually for numbers from $0$ above.

[EDIT: the original definitions of Peano axioms, as found in Arithmetices principia: nova methodo, may be found at https://archive.org/details/arithmeticespri00peangoog : look at it. ]

• "$\mathbb{N}$ is usually for numbers from $0$ above." Can you point to evidence supporting this "usually"? Aug 1, 2013 at 4:01
• For evidence that the issue exists, mathworld.wolfram.com/NaturalNumber.html is a source; as for the "usually", I should dig my old books, I think.
– mau
Aug 1, 2013 at 13:38
• That $\mathbb{N}$ usually includes zero. It is totally wrong. Jan 8, 2015 at 10:43
• Clapham-Nicholson, The Concise Oxford Dictionary of Mathematics (4th edition): «natural number - One of the numbers 1, 2, 3, ..., Some authors also include 0. The set of natural numbers is often denoted by N.» Eric Weisstein, Concise Encyclopedia of Mathematics (2nd ed) - «N: The SET of NATURAL NUMBERS (the POSITIVE INTEGERS Z : 1, 2, 3, ...; Sloane’s A000027), denoted N; also called the WHOLE NUMBERS . Like whole numbers, there is no general agreement on whether 0 should be included in the list of natural numbers
– mau
Jan 8, 2015 at 11:34
• Interesting enough, Peano constructed naturals starting with 1, but construction of naturals with set theory starting with empty set 0 also satisfies Peano axioms. Nov 20, 2019 at 22:13

I think that modern definitions include zero as a natural number. But sometimes, expecially in analysis courses, it could be more convenient to exclude it.

Pros of considering $$0$$ not to be a natural number:

• generally speaking $$0$$ is not natural at all. It is special in so many respects;

• people naturally start counting from $$1$$;

• the harmonic sequence $$1/n$$ is defined for any natural number n;

• the $$1$$st number is $$1$$;

• in making limits, $$0$$ plays a role which is symmetric to $$\infty$$, and the latter is not a natural number.

Pros of considering $$0$$ a natural number:

• the starting point for set theory is the emptyset, which can be used to represent $$0$$ in the construction of natural numbers; the number $$n$$ can be identified as the set of the first $$n$$ natural numbers;

• computers start counting by $$0$$ (see the explanation of Dijkstra)

• the rests in the integer division by a $$n$$ are $$n$$ different numbers starting from $$0$$ to $$n-1$$;

• it is easier to exclude one defined element if we need naturals without zero; instead it is complicated to define a new element if we don't already have it;

• integer, real and complex numbers include zero which seems much more important than $$1$$ in those sets (those sets are symmetric with respect to $$0$$);

• there is a notion to define sets without $$0$$ (for example $$\mathbb R_0$$ or $$\mathbb R_*$$), or positive numbers ($$\mathbb R_+$$) but not a clear notion to define a set plus $$0$$;

• the degree of a polynomial can be zero, as can be the order of a derivative;

I have seen children measure things with a ruler by aligning the $$1$$ mark instead of the $$0$$ mark. It is difficult to explain them why you have to start from $$0$$ when they are used to start counting from $$1$$. The marks in the rule identify the end of the centimeters, not the start, since the first centimeter goes from 0 to 1.

An example where counting from $$1$$ leads to somewhat wrong names is in the names of intervals between musical notes: the interval between C and F is called a fourth, because there are four notes: C, D, E, F. However the distance between C and F is actually three tones. This has the ugly consequence that a fifth above a fourth (4+3) is an octave (7) not a nineth! On the other hand if you put your first finger on the C note of a piano your fourth finger goes to the F note.

I would say that in the natural language the correspondence between cardinal numbers and ordinal numbers is off by one, thus distinguishing two sets of natural numbers, one starting from 0 and one starting from 1st. The 1st of January was day number $$0$$ of the new year. And zeroth has no meaning in the natural language...

• You do know that "natural" (in the pedestrian sense of the word, not the mathematical one) is completely subjective and dependent on your upbringing and social norms. It is unnatural to eat a cheeseburger in Israel (at least it was, say, 20 years ago) and it is unnatural to go out and drink beers during Passover. But do you consider cheeseburgers unnatural? In a few years, many children brought up in a vegan households will consider it unnatural to eat a cheeseburger as well (for different reasons). Others might find drinking to be unnatural. $0$ can be natural if you were taught it should be. Jan 8, 2015 at 10:30
• @AsafKaragila I agree with you. But I think (please confirm) that in every country a child is taught to attach numbers to things (i.e. counting) starting from 1. I think that also mathematicians do that: Problem #1 is the first problem in the list. In my experience only computer scientists do count from zero and only when talking with computers or other computer scientists... Jan 9, 2015 at 12:02
• Emanuele, I'm not saying that in Israel, or somewhere else people start counting from $0$ as children. But my point is that what you might see as unnatural is only the result of your upbringing, so it is definitely not a valid mathematical or even a philosophical argument. The term "natural" loses its natural meaning when transferred to mathematics. There's nothing normal about $\Bbb R^{42}$ and there's nothing regular about $\omega_1$. Those are words, and if we insist on keeping their "natural language" meaning to them, then we're in trouble regarding most things in mathematics. Jan 9, 2015 at 12:05
• generally speaking 2 is unlike the other primes, it is special among primes in soo many different ways.... Should we exclude him fro the list of primes? Feb 6, 2016 at 1:05
• As a computer scientist, I think it's natural to count from zero. In fact, it's extremely strange if a programming language indexes arrays from 1, contrary to the norm. Feb 16, 2017 at 17:59

According to ISO 80000-2:2009: Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, page 6;

$$\mathbb{N}=\{0,1,2,3,\ldots\}$$ $$\mathbb{N^*}=\{1,2,3,\ldots\}$$ • i think Edsger W. Dijkstra has a useful comment on this. Oct 3, 2018 at 18:51

These lecture notes from a combinatorics course given for many years by N.G. de Bruijn suggest a helpful alternative:

Due to the confusion caused by N. Bourbaki about the natural numbers, we feel obliged to define: \begin{align}\Bbb N_0 & = \{0,1,2,\ldots\}\quad \text{ and } \\ \Bbb N_1 & = \{1,2,3,\ldots\}. \end{align}

(Page 4)

• Yeah. I had a professor who used $\mathbb{N}_{k} = \{ k, k + 1, k + 2, \ldots \}$ to talk about counting integers from $k$ on (he often used $\mathbb{N}_{2}$ to talk about base-$b$ expansions), a practice I adopted (though you'll wanna write somewhere in there what you mean).
– AJY
Feb 6, 2016 at 0:56
• @AJY, its worth pointing out that if we define $\mathbb{N} = \{0,1,2,\ldots\}$, then: $$\mathbb{N}_k = k+\mathbb{N}.$$ Apr 3, 2016 at 5:06
• @goblin Correct, at least when we suppose $0 \in \mathbb{N}$.
– AJY
Apr 3, 2016 at 5:07
• The dig at N. Bourbaki is very implicit and curious, as if there were inconsistency in their books as to whether $0\in\mathbf{N}$ (which would be conceivable, given that they are written by many different people). As far as I can tell, Bourbaki unanimously has $0\in\mathbf{N}$. There are just some differences in the ways in winch they (don't explicitly) say this, compare for instance Théorie des Ensembles Ch III, §4 1 ..."un cardinal fini s'appelle aussi entier naturel" with Algèbre Ch. I §2 5 I "Considérons le monoïde commutatif N des entiers naturels". Mar 2 at 13:22
• ... The only inconsistency I can find is that the (rather confusing) numbering of their books, chapters, sections, paragraphs never starts with $0$, but this can at least in part be defended by the fact that the numbering often uses Roman numerals. Mar 2 at 13:27

There are the two definitions, as you say. However the set of strictly positive numbers being the natural numbers is actually the older definition. Inclusion of $0$ in the natural numbers is a definition for them that first occurred in the 19th century.

The Peano Axioms for natural numbers take $0$ to be one though, so if you are working with these axioms (and a lot of natural number theory does) then you take $0$ to be a natural number.

• Bourbaki included zero in 1935. That's not all that recent...
– user126
Jul 21, 2010 at 10:08
• "take $0$ to be one" is confusing phrasing. Aug 1, 2013 at 4:00
• @JonasMeyer: Actually it makes sense in a non-trivial way: From the axioms, the natural numbers are basically a set with a first number, and then a successor for each number. At that point, you cannot actually distinguish between the natural numbers starting at $0$ and the natural numbers starting at $1$, except for the (completely arbitrary) name for the initial number. Basically, it is the different definition for addition and multiplication which distinguishes the two choices. Aug 1, 2013 at 16:57
• Now if you say "one" is the name of the initial natural number, then "take $0$ to be one" would be interpreted as "take $0$ to be the initial number", that is, "start the natural numbers with $0$". Aug 1, 2013 at 17:00
• @celtschk: Interesting point, thank you. It makes sense also in the way it was probably intended, namely "$0$ is one of the natural numbers," i.e., just as $15$ is one. In any case it is not a real problem, just potentially confusing. Aug 1, 2013 at 20:30

I remember all of my courses at University using only positive integers (not including $0$) for the Natural Numbers. It's possible that they had come to an agreement amongst the Maths Faculty, but during at least two courses we generated the set of natural numbers in ways that wouldn't make sense if $0$ was included.

One involved the cardinality of Sets of Sets, the other defined the natural numbers in terms of the number $1$ and addition only ($0$ and Negative Integers come into the picture later when you define an inverse to addition).

As a result when teaching the difference between Integers and Natural Numbers I always define $0$ as an integer that isn't a Natural Number.

• Obviously, defining ℕ from 0 and addition also works perfectly. I don't know what difference would 0 make to calculating the cardinality of P(ℕ) either.
Jul 21, 2010 at 12:08
• The cardinality of sets of sets can certainly be $0$: All members of the empty set are sets. Indeed, in ZF all sets are sets of sets. Aug 1, 2013 at 17:17
• @badp Ok, you have 0 and addition. 0+0=0. Where's the rest of N? Oct 17, 2016 at 2:54

The Peano-Dedekind axioms (as used in proving propositions by use of the Principle of Mathematical Induction) define the $$\mathbb{N}$$ as either $$\mathbb{N}$$ = $$\mathbb{Z^+} \cup \text{0} = \text{{0, 1, 2, ...}}$$ or $$\mathbb{N} = \mathbb{Z^+} = \text{{1, 2, 3, ...}}$$, that is, it depends on the context (usually this "context" may be seen from the given proposition to be proved, at least in the case of using PMI).

This may also depend on the country. I do not know about the prevalent convention in the United States, but generally speaking:

• French authors do include $$0$$ in the natural numbers,
• German authors do not include $$0$$ in the natural numbers.

I am not aware of any comprehensive survey on the topic, even though it tends to be contested.

Peano used 1 as the first natural number in his arithmetic theory. Geometric: If a line is breadthless with infinite points, then the first line you can create is a 1. Written 1 if it were a map you could scale it. This 1 can then be used as a metric to index a length of each successive natural number. On a Cartesian Graph 0 is used to define one end point of the 1, but without the second end point at 1 there is no length, and no number according to some. Also consider, if you were creating a set of lines each representing a natural number, what would 0 look like? There would be no length to 0, and hence it is not a member of the set of lines.