Or can I expect the reader to know that that denotes the set of complex numbers and natural numbers, respectively?
Or must I have "...where $\mathbb{C}$ denotes the field of complex numbers..."?
2026-03-25 17:34:19.1774460059
When writing a paper, do I have to describe what $\mathbb{C}$ and $\mathbb{N}$ are?
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You don't have to, but it doesn't hurt. Just keep the description short, making sure to include a term that enables someone who doesn't know what complex numbers are or natural numbers to look it up. "...where $\mathbb{C}$ is the field of complex numbers..." that's perfectly nice, short and sweet. Or $\textbf{C}$, for those who don't like blackboard bold.
As pointed out in the comments, there is some disagreement over whether $\mathbb{N}$ includes 0 or not. It's best to preempt that distraction by being perfectly clear on whether you intend to include it or not.