No this isn’t about some kind of science fiction, this is about five Unicode characters that are useful for mathematics, but are generally invisible or should be. The characters are the zero-width space (U+200B), function apply (U+2061), invisible times (U+2062), invisible comma (U+2063), and the new invisible plus (U+2064). This post discusses each one in the context of mathematical text.

The zero-width space is a handy character that has no glyph “ink” and hence no ascent (height above the base line), no descent (depth below the baseline) and no width. In Word 2007 math zones you can insert it (type 200B <Alt+x>) into an empty argument if you don’t want a dotted box character to appear. RichEdit uses it for optional empty arguments to suppress the dotted box except when the insertion point resides inside an empty argument.

The function-apply character (U+2061) is used in the linear format as a binary operator that builds into a math function object. For example in a math zone, if you type sin2061<Alt+x> x and click on “Professional”, you get the math function object sin *x*. Naturally it’s easier just to type sin<space>x and have formula autobuildup do this for you, but underneath it’s the function apply character that’s controlling the build up process.

The invisible times (U+2062) is a bona fide binary operator and you can break on it and align to it. Unfortunately we didn’t have enough time to develop the uses for invisible times, so it’s not currently very useful. Unlike in Word 2007, it shouldn’t display a glyph, except for a thin space if at the end of a math zone. With it you could then effectively break an equation before any character, not just on binary, relational and some other operators. It would be nice to be able to have it display a multiplication times symbol × if it ends up being the best point for an automatic break. Word 2007 displays the invisible times as a dotted box surrounding a times sign, which is the glyph for it in the Cambria Math font.

The invisible comma (or separator) is supposed to convey the semantic of separating two variables or indices. For example the indices *ij* on a matrix element *a _{ij}* could be separated by the invisible comma to emphasize that

*ij*isn’t the product of

*i*and

*j*. Word 2007 displays the invisible comma as a dotted box surrounding a comma, which is the glyph for it in the Cambria Math font.

The invisible plus (U+2064) is new with Unicode 5.1 and is supposed to carry the semantic of connecting a whole number like 3 with a fraction like ½ to give a quantity 3½ that has the value 3.5, not 1.5 (3/2). The invisible plus is well intended, but it’s also tricky to use. For one thing in ordinary arithmetic, addition is considered to have lower precedence than multiplication. So the value of the expression 4×3 + 1/2 is 12.5, not 14 (4×3.5). But 4×3<invisible plus>1/2 has the value 14. In this usage, the invisible plus has a higher precedence than multiplication.

Some more discussion of the invisible operators is given in Section 2.14 of Unicode Technical Report #25.