Is 0 a Natural Number?
Subash, a user of my math site (Interactive Mathematics) asked recently whether 0 is a Natural Number or not. My reply:
Normally I have always taken the Natural Numbers to start at 1 and not to include zero. I used to get my students to remember the difference between Natural Numbers and Whole Numbers by saying the natural numbers can be counted using your fingers and the first finger looks like a 1, while the word “whOle” has a zero in the middle, thus the Whole Numbers include 0.
Thoughts by others:
According to Dr Math:
Natural Numbers are 1,2,3,4,5,… [...] and Whole numbers are 0,1,2,3,…
According to Wikipedia:
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, …) or a non-negative integer (0, 1, 2, 3, 4, …). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science.
The safest thing is to state whether you are including 0 or not when talking about Natural Numbers. You could write it something like:
“The Natural Numbers (taken as 1,2,3,4,…) are blah blah blah.”
So who cares? This situation is strange because mathematics is normally a very precise science and there is normally broad agreement about such definitions. Anwyay, it matters if students lose marks in assessments because there is disagreement about the definition. So the set theorists and the computer scientists should just conform… 
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14 Responses to “Is 0 a Natural Number?”
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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...]).
Thanks for your input, Vlorbik.
I’m with you on the “ln-log” issue, something I have found silly for a long time, especially as many students here in Singapore write it with an ‘eye’ rather than ‘ell’ as in
not
Grrr…
Don’t miss Towards more meaningful math notation where related issues are discussed.
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
so is 0 a natural number or not?
Hi Amanda
It depends (and I believe it shouldn’t “depend” – it should be agreed).
Here’s another 5 (incompatible) definitions: http://dictionary.reference.com/browse/natural%20number
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.
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!
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…
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.
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.
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.
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…
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.
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.