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

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 $k \in \mathbb{N}$?

• Perhaps you're allowing $0\in\mathbb{N}$. Sometimes $\mathbb{N}$ is taken to start at $1$. Dec 26, 2015 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. Dec 26, 2015 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. Dec 26, 2015 at 18:43

## 2 Answers

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 $\mathbb{N}\cup\{0\}$.

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.