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$\mathbb {N,P,Q,R,Z}$ all represent commonly used sets of numbers, but I can't find any good references on what they're called. I've tried double bar, double stroke, and googling what the ____ number symbol was called; inserting for integers, primes, etc.

What is it called?

Jacob Claassen
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My answer isn't authoritative, but I've always heard them called "blackboard" or "blackboard bold" letters. Note that the TeX command is \mathbb, where the "bb" stands for "blackboard."