# 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.

-
voted to close. The question is subjective, as is clearly indicated by the first sentence of the wikipedia article on Natural Numbers – Tom Stephens Jul 23 '10 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. – Justin L. Jul 23 '10 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 '10 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! – Alex Kruckman Dec 2 '13 at 18:24
@Alex Kruckman: (belated joke) but 0! = 1. – Rob Arthan Apr 7 '15 at 20:59

Simple answer: sometimes yes, sometimes no, it's usually stated (or impled 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. – Charles Stewart Jul 21 '10 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. – Noldorin Jul 21 '10 at 9:12
Indeed, I can see why they were defined that way. There's a surprising lack of consistency in this area of naming. – Noldorin Jul 21 '10 at 13:50
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. – Martin Sleziak Sep 17 '11 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). – celtschk Aug 1 '13 at 17:12

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"? – Jonas Meyer Aug 1 '13 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 '13 at 13:38
This is wrong, -1 – Anixx Jan 8 '15 at 10:27
@Anixx : what is wrong? – mau Jan 8 '15 at 10:36
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 '15 at 11:34

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 '10 at 10:08
"take $0$ to be one" is confusing phrasing. – Jonas Meyer Aug 1 '13 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. – celtschk Aug 1 '13 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 $0$". – celtschk Aug 1 '13 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. – Jonas Meyer Aug 1 '13 at 20:30

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 at 0:56
@AJY, its worth pointing out that if we define $\mathbb{N} = \{0,1,2,\ldots\}$, then: $$\mathbb{N}_k = k+\mathbb{N}.$$ – goblin Apr 3 at 5:06
@goblin Correct, at least when we suppose $0 \in \mathbb{N}$. – AJY Apr 3 at 5:07

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 $0$;

• 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. – Asaf Karagila Jan 8 '15 at 10:30
@AsafKaragila Just saying, there has always been a huge non-Jewish population in Israel. There's a reason that three of the quarters of Jerusalem are the Muslim Quarter, the Christian Quarter, and the Armenian Quarter (also Christian). – Akiva Weinberger Jan 8 '15 at 11:21
@columbus8myhw: I'm not trying to bring up a discussion about what is natural and what is not. I'm trying to point out that to say that "$0$ is not natural at all" depends on your upbringing and your social norms which may or may not consider it natural. But, hey, way to give into the stigma about mathematicians unable to transcend the minor mistakes in an analogy, and nitpick it apart! Kudos! – Asaf Karagila Jan 8 '15 at 11:24
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. – Asaf Karagila Jan 9 '15 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? – N. S. Feb 6 at 1:05

As others have said, there's no consensus on this. However, if you need unambiguous notation, you can use: $\mathbb{Z}_{\geq 0}, \mathbb{Z}_{\geq 1}.$ This is a good option if you're writing something short and sweet, e.g. for posts to this website. In something longer, like an article or PhD, you may wish to spend a sentence or two establishing a convention that is more visually appealing. Personally, I use: $$\mathbb{N} = \{0,1,2,3,\cdots\}, \qquad \mathbb{W} = \{1,2,3,\cdots\}.$$ The motivation behind $\mathbb{W}$ is that its elements can be referred to as "whole" numbers (although as others have said, the term "whole number" is highly ambiguous unless and until you tell the reader precisely what you mean.)

Anyway, what I really wanted to say is that even if our only interest is number theory, nonetheless we still need both of these number systems, because prime factorization sets up an isomorphism between the monoid $\mathbb{W}$ (with multiplication) and the monoid of all finitely-supported functions $\mathrm{Prime} \rightarrow \mathbb{N}$ (with pointwise addition.)

-
Judging from other posts by you, I like your style of thinking about mathematics. I’m using $ℕ$ and $ℕ_0$ to denote natural numbers without and with zero respectively. Naturally, I’m now wondering why you do include zero in $ℕ$. I’m with Noldorin on this issue, thinking of natural numbers as the counting numbers with which we all grow up (that’s what makes them so natural). Is your use of $ℕ$ sheer pragmatics or do you think of zero as natural in some sense? – k.stm Feb 6 at 0:22
@k.stm, I really do think of $0$ as "natural"; for example, its the cardinality of the empty set, we use it to denote numbers (e.g. $408$ features the numeral $0$), and it appears in the identity matrix $$\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}.$$ The only exception to this feeling that $0$ is very natural and important is the special case in which I'm purely interested in the multiplicative structure of the natural numbers. In this rare instance, $0$ just ends up being a nuisance. Putting it another way, I like the monoid $(\mathbb{W},\times,1)$, but I dislike the semigroup $(\mathbb{W},+)$. – goblin Feb 6 at 3:20
The other comment I'd make is that its nice to start sequences at $0$, so that we can write powerseries like so: $$\sum_{n:\mathbb{N}} a_n x^n$$ This is one of many reasons why $0$-based numbering is a good idea. See also, Dijkstra's argument in favour of $0$-based numbering. – goblin Feb 6 at 3:28

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. – badp Jul 21 '10 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. – celtschk Aug 1 '13 at 17:17

I like to propose a notation that just crossed my mind.

To me, zero is no number I would call natural. It is finite, though. So I propose to use $$\mathbb N = \{1, 2, 3, …\}\quad\text{and}\quad\mathbb F = \{0, 1, 2, 3, …\},$$ where “$\mathbb F$” stands for finite numbers or rather finite ordinals or finite cardinals. This fits nicely with the usual practice of saying “a finite number of …” or “all but finitely many” with where always mean cardinalities of finite sets, including the empty set.

This may yield some confusion in cases where people use “$\mathbb F$” to denote a field or an arbitrary finite field. I’m unaware of other uses for “$\mathbb F$” which are not too specific to a certain area of mathematics.

It actually plays relatively neatly with finite fields of a specified prime cardinality, as for a prime $p$, one may view $\mathbb F_p$ as $\mathbb F / p \mathbb F$ (analogously to the notation $ℤ_n$ for $ℤ/nℤ$), which is a natural semiring that turns out be a field. This, of course, breaks down for finite fields $\mathbb F_q$ whose cardinality is a higher prime power $q$, in which case $\mathbb F_q \not\cong \mathbb F / q\mathbb F$.

-
This is fine; I don't see why its got those downvotes. Better to call these the "finite ordinals" though, as opposed to "finite numbers." So, under this system of conventions, $\mathbb{F}$ is the set of finite ordinals. – goblin Feb 6 at 3:22
@goblin Hm, what’s wrong with calling them “finite numbers”? – k.stm Feb 6 at 8:30
Ask yourself: which elements of the real line deserve to be called "finite"? – goblin Feb 6 at 8:51
@goblin None? I don’t follow you … Oh, actually, I get it: We use “of finite measure” a lot. So, convinced. Finite ordinals/cardinals it is. – k.stm Feb 6 at 8:52
By the way: Are people downvoting because they don’t like the proposal or because they genuinely think it’s a bad answer? – k.stm Feb 6 at 8:57

In consideration of symbols representing the counting (Natural) numbers, If $0$ is not included in the set $\mathbb{N}$, then there is no definition for the symbols $10, 100,\dots,100000$ etc. Since $0$ is undefined within the set $\mathbb{N}$.

For the system based on the number $10$ (Or any other base for that matter), the series $10$ to the $n$ is undefined for $n=0$ and we might just as well use the Roman symbol X or maybe even an Egyptian Hieroglyphic.

-
It seems very unclear what you are trying to say and how it relates to the question. Moreover, the question has been properly answered already. – Rasmus Jun 22 '12 at 14:04
@Rasmus: I think he is trying to say that digits are natural numbers so if zero was not a natural number we could not write $10$ as a natural number. I guess that in the naive way of thinking the representation of a number is closely related to the number. – Asaf Karagila Jun 22 '12 at 15:09
Not so naive, as the lack of zero was an impediment to the introduction of positional notation. – Jean-Claude Arbaut Mar 15 '13 at 19:14
Whether or not $0$ is in the set $\mathbb N$ has no bearing on whether or not the symbol '$0$' may be used in representations of the elements of $\mathbb N$. This answer is incorrect. – Jonas Meyer Aug 1 '13 at 4:13
While the answer isn't entirely clear, I think it makes a very good point. Excluding $0$ from the definition of $\mathbb{N}$ makes life difficult when you want to define base $b$ representation. For, you want to say that $104 = 1 \cdot 100 + 0 \cdot 10 + 4 \cdot 1$, but you can say no such thing if you don't have $0$ within your set of numbers. – 6005 Mar 12 '14 at 17:58

Unambiguous names for the set of nonnegative integers $(0, 1, 2, 3, \dots)$ are whole numbers Unambiguous names for the set of positive integers $(1, 2, 3, \dots)$ are counting numbers and $\mathbb{N}^+$.
As other answers (and your own question) indicate, natural numbers, or $\mathbb{N}$, can have different meanings, depending on context. If which usage you mean is not clear from context, you are better off using one of the other terms.

-
ℕ₀ is not an unambiguous name, we use it as ℕ₀ = ℕ \ {0}. – badp Jul 23 '10 at 9:55
@badp: I've never seen it used that way, but if there is someone else out there who does, I suppose that makes it ambiguous. Who's the we? – Larry Wang Jul 23 '10 at 14:53
"whole" and "counting" are ambiguous, too. Why is $-1$ not "whole"? Why is $0$ not a counting number? – Jonas Meyer Aug 1 '13 at 3:57
Indeed, Wikipedia confirms that "whole number" is even more ambiguous than "natural number" because it may not only take both meanings of "natural number", but in addition also the meaning of "integer". – celtschk Aug 1 '13 at 17:04
@Nick: We don't all agree on what these words mean, hence they are ambiguous. I think we agree that names can mean whatever we precisely define them to mean. We cannot stop people from choosing conflicting conventions; we can make clear what we mean by stating it when there is possible ambiguity. – Jonas Meyer Nov 26 '13 at 5:11