I'm a little curious how you define the set of natural numbers, \(\mathbb{N}\). That all the positive integers are members of \(\mathbb{N}\) everyone agrees on, but what about \(0\)? What I learned in school here in Sweden is that \(\mathbb{N}=\left\{0,1,2,...\right\}\), and then for the positive numbers \(\mathbb{Z}_{+}=\left\{1,2,3,...\right\}\). But what I have heard from many other people from other countries is that the define the natural numbers without \(0\), that is \(\mathbb{N}=\left\{1,2,3,...\right\}\). Is one more correct than the other? Not that it really matters if you define your notation first, but it can be annoying sometimes. Also, any ideas on why you would prefer one way over the other?

I guess my question is really: would you define \(0\) as a natural number or not, and why?

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TopNewestJust to add on:

This is a well known case of when mathematical notation can differ in various regions. As you mentioned, \( \mathbb{N} \) can have different meanings for different people, and there is no global standardization. As such, I tend to avoid using \( \mathbb{N}\) where possible, and instead say "non-negative integers" or "positive integers".

Even though natural numbers are supposed to represent the counting number system, the concept of 0 as a number (as opposed to a placeholder) only came about in 9th century AD in India. That's pretty late, considering the amazing amount of math that happened before. – Calvin Lin Staff · 4 years, 3 months ago

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