Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of producing blackboard bold is to double-strike a character with a small offset on a typewriter. Thus they are also referred to as double struck.
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Origin
In some texts these symbols are simply shown in bold type: blackboard bold in fact originated from the attempt to write bold letters on blackboards in a way that clearly differentiated them from non-bold letters i.e. by using the edge rather than point of the chalk. It then made its way back in print form as a separate style from ordinary bold, possibly starting with the original 1965 edition of Gunning and Rossi's textbook on complex analysis.
Rejection
Some mathematicians do not recognize blackboard bold as a separate style from bold. Jean-Pierre Serre uses double-struck letters when writing bold on the blackboard, whereas his published works consistently use ordinary bold for the same symbols. Donald Knuth also prefers boldface to blackboard bold, and consequently did not include blackboard bold in the Computer Modern fonts he created for the TeX mathematical typesetting system.
The Chicago Manual of Style in 1993 (14th edition) advises: "blackboard bold should be confined to the classroom" (13.14) whereas in 2003 (15th edition) it states that "open-faced (blackboard) symbols are reserved for familiar systems of numbers" (14.12).
Encoding
TeX, the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the add-on AMS Fonts package (amsfonts) by the American Mathematical Society provides this facility; a blackboard bold R is written as mathbb{R}. The amssymb package loads amsfonts.
In Unicode, a few of the more common blackboard bold characters (C, H, N, P, Q, R and Z) are encoded in the Basic Multilingual Plane (BMP) in the Letterlike Symbols (2100–214F) area, named DOUBLE-STRUCK CAPITAL C etc. The rest, however, are encoded outside the BMP, from U+1D538 to U+1D550 (uppercase, excluding those encoded in the BMP), U+1D552 to U+1D56B (lowercase) and U+1D7D8 to U+1D7E1 (digits). Being outside the BMP, these are relatively new and not widely supported.
Usage
The following table shows all available Unicode blackboard bold characters.
The symbols are nearly universal in their interpretation, unlike their normally-typeset counterparts, which are used for many different purposes.
The first column shows the letter as typically rendered by the ubiquitous LaTeX markup system. The second column shows the Unicode codepoint. The third column shows the symbol itself (which will only display correctly on browsers that support Unicode and have access to a suitable font). The fourth column describes known typical (but not universal) usage in mathematical texts.
In addition, a blackboard-bold Greek letter mu (not found in Unicode) is sometimes used by number theorists and algebraic geometers (with a subscript n) to designate the group (or more specifically group scheme) of n-th roots of unity.