"^^A" does not work in TeX?

Question

On page 45 of The TeXbook,

TEX has a standard way to refer to the invisible characters of ASCII: Code 0 can be typed as the sequence of three characters ^^@, code 1 can be typed ^^A, and so on up to code 31, which is ^^_ (see Appendix C).

The ^^@, ^^B and ^^C and so on all works but ^^A doesn't.

^^@\par
^^B\par
^^C\par
^^D\par
\bye

produces

but ^^A causes an error

! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   ^^A
l.5 ^^A

Why can't use ^^A in TeX?

Answer 1

The error message you get, indicates that ^^A results in a math token. In plain.TeX you can find:

\catcode`\_=8 \catcode`\^^A=8 % underline and downarrow are for subscripts

(Note: This part of plain.tex can also be found in Appendix B of “The TeXbook”, page 343.)

From this point ^^A is a math subscript. This corresponds with the terminal output of

\showthe\catcode`\^^A
\bye

8.
l.1 \showthe\catcode`\^^A

Note: The later

\mathcode`\^^A="3223 % \downarrow

in plain.tex is irrelevant for the catcode of ^^A. And because the catcode of ^^A is neither 11 (letter) nor 12 (other) the mathcode is also not relevant for the output. But it can become relevant, e.g., after changing the catcode of ^^A:

$x^^A{y}$

\catcode`\^^A=12 $x^^A{y}$
\bye

For more information about \mathcode, how it works and the difference to \catcode see “The TeXbook” and “What is the differences between mathcode and catcode and how can I use mathcode?”

Answer 2

You can refer to invisible ASCII characters, but the TeXbook doesn't say you can, or should, use them just for fun.

For instance, ^^@ (the null byte) has category code 9, so

Hell^^@ w^^@rld

would result in “Hell wrld”. And indeed you obtain three characters in output and not four.

To the contrary, ^^A (the 1 byte), has category code 8. Here's the table of category codes as set up in plain TeX:

Why the choice for ^^@? Because some operating systems used the null byte in order to fill their fixed length records. There should be no surprise in seeing category code 5 assigned to ^^M. Two cases remain, namely ^^A and ^^K.

The keyboard which Knuth used to work with could produce ↓ and ↑ when pressing CtrlA and CtrlK, respectively, and he liked these visual hints more than ^ and _. A consequence of this is that

$x^^A0$

has the same effect as $x_0$. And, of course, $^^A$ raises an error, because it would be like $_$.

You find the symbols on Knuth's keyboard in the table on page 369. This table also explains why some of these bytes are assigned a mathcode.

For instance CtrlD would produce ∧ on Knuth's screen and indeed we find

\mathcode`\^^D="225E % \land

in plain.tex.

Just for completeness, you can use ^^00 instead of ^^@. This convention is also described in chapter 8 and was added in TeX version 3.

Answer 3

Because on an old computer system with old keyboards that Don Knuth used, the character set had a down-arrow as character 1 (^^A). That seemed like a useful synonym for the underscore as math subscript character, so it has catcode 8. Likewise for an up-arrow character at location 11 given catcode 7 like ^. The character set is shown in Appendix C.