Blackboard bold

Blackboard bold is a type of font 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 describe sets of numbers and are also referred to as double struck, although attempting to produce them by double striking on a typewriter is unlikely to give satisfactory results. The symbols were first introduced by the group of mathematicans known as Bourbaki.

In some texts, these symbols are simply shown in bold, and 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. Wikipedia too uses ordinary bold in place of blackboard bold, as browser support for the latter is far from universal.

TeX, the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the add-on AMS Fonts package by the American Mathematical Society provides this facility; a blackboard bold R is written as \\Bbb{R} in regular text and as \\mathbb{R} in math mode.

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). 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 very new and not widely supported.

The following table shows some of the more common uses of blackboard bold. The first column shows the letter as rendered by Wikipedia's TeX markup system. In those cases where Unicode encodes the symbol in the BMP, the second column shows this encoding, together with the symbol itself (which will only display correctly if your browser supports Unicode and has access to a suitable font). The third column mentions typical uses in mathematical texts.

		Sometimes represents the algebraic numbers, the algebraic closure of Q (although a Q with an overline is often used instead).
	U+2102 ℂ	Represents the complex numbers.
		Often used for finite fields, with a subscript to indicate the order.
	U+210D ℍ	Represents the quaternions. (The H stands for Hamilton.)
		Sometimes represents the irrational numbers, R\\Q.
		Sometimes represents either the real numbers or the complex numbers in contexts where either could be used and which doesn't really matter.
	U+2115 ℕ	Represents the natural numbers. Wikipedia includes 0 as a natural number, other texts may or may not include 0.
		Represents the octonions.
	U+2119 ℙ	Represents the prime numbers.
	U+211A ℚ	Represents the rational numbers. (The Q stands for quotient.)
	U+211D ℝ	Represents the real numbers.
		Represents the sedenions.
	U+2124 ℤ	Represents the integers. (The Z is for Zahlen, which is German for "number".)

Note that P ⊆ N ⊆ Z ⊆ Q ⊆ A ∩ R ⊆ R ⊆ C ⊆ H ⊆ O ⊆ S.

External link:

• http://www.w3.org/TR/MathML2/double-struck.html shows blackboard bold symbols together with their Unicode encodings. Encodings in the BMP are highlighted in yellow.