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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.

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