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What is the meaning of $\mathbb{N_0}$?

To put it into context, I have in my notes, $f^{(k)}$, $k \in \mathbb{N_0}$ is a continuous function on $[-\pi, \pi]$.

How is it different to saying $k \in \mathbb{N}$?

Zev Chonoles
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Al jabra
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    Perhaps you're allowing $0\in\mathbb{N}$. Sometimes $\mathbb{N}$ is taken to start at $1$. – Michael Burr Dec 26 '15 at 18:40
  • Usually you see mathematicians start $\mathbb{N}$ at $1$, while computer scientists and physicists start at $0$, but it all depends on which is more convenient at the time. – Mike Pierce Dec 26 '15 at 18:42
  • That would make the most sense, since the notation $f^{(k)}$ often is seen with taylor series, which begin with the "zeroth" derivative, or the function itself, which is sometimes written $f^{(0)}$ for the convenience of sigma notation. – pancini Dec 26 '15 at 18:43

2 Answers2

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There is no general consensus as to whether $0$ is a natural number. So, some authors adopt different conventions to describe the set of naturals with zero or without zero. Without seeing your notes, my guess is that your professor usually does not consider $0$ to be a natural number, and $\mathbb{N}_0$ is shorthand for $\mathbb{N}\cup\{0\}$.

Noah Schweber
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To add to the above: I prefer personally to use either $\mathbb{N}_0$ or $\mathbb{Z}_{\geq 0}$ if I want to be absolutely clear that $0$ is included. Similarly, one could use $\mathbb{N}^+$ or $\mathbb{N}_{> 0}$ to refer to the case that $0$ is not included. While they are a bit more cumbersome, they are more clear. Since there is some lack of agreement across all mathematicians, it is better to be clear.

Another way to do this is to include a note at the beginning of a textbook or paper to introduce your notation.

Anyhow, clarity is always better.

Simon Rose
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