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The Nemeth Braille Code for Mathematics and Science Notation is a plain text linear system used for representing math in braille. Nemeth braille, like all braille, is represented electronically by using the standard ASCII characters. This means that sighted persons only have to learn a few conventions in order to read Nemeth braille math visually.
This article is an introduction to the Nemeth braille math code used in the United States, Canada, and several Southeast Asian countries. It explains how sighted persons who don't know braille can learn to read Nemeth math visually.
The strategies in this article make it possible for a sighted teacher of mathematics, who has no knowledge of braille, to read a student's braille work simply by displaying the braille with an ordinary print font. (See also this information about automated back-translation of Nemeth math to print math.)
This article assumes no knowledge of braille and doesn't even show the braille cells as dots. (But click here for a version that does.) However, the article does assume a basic familiarity with mathematical notation. If anything isn't clear, please send questions to the address at the bottom of the page.
After studying this material, you should be able to do a fairly good job of reading Nemeth braille at least up through high school math. Reading is, of course, much easier than writing. Familiarity with the information on this page is not intended to substitute for the thorough knowledge required to convert print math to correct braille.
Nemeth braille mathematics is a plain text linear system with some similarities to related systems such as LaTeX and MathML. The main difference between Nemeth and the standard linear systems used for print math
is that Nemeth is much less verbose because it is designed for end use by tactile readers rather
than for machine processing. [Braille math systems have to be linear because tactile readers can't easily sense the relative vertical positions of the braille cells. Thus, for example, something like x squared is written as x^2 (using the corresponding braille cells) rather than just x with the
superscript 2 literally raised above the baseline.]
Braille is represented electronically using the ASCII characters. (The ASCII characters are the characters on a standard computer keyboard.) ASCII Braille is not something that has been invented just for this article; this method for representing computer-generated braille has been in world-wide use for almost 40 years. There are several different conventions for associating the braille cells to the ASCII characters but the only one used in the United States, and, luckily, the most useful one for working with Nemeth math, is North American ASCII Braille or, more informally, ASCII Braille. Braille notetakers use a very similar convention often called computer braille.
ASCII Braille or computer braille represents the braille cells for the letters and digits with the corresdponding ASCII characters for the letters and digits. The braille cell for the letter a is represented by the ASCII character a, the braille cell for the letter b is represented by the ASCII character b, the braille cell for the digit 1 by the ASCII character 1, etc.
ASCII Braille or computer braille uses arbitrary associations to represent most of the braille cells that are not used for either letters or digits. This is unavoidable since braille is not one-for-one with print:
From now on we will often refer to the braille cells by the common names of the ASCII characters used to represent or transliterate them. In other words, we will write "comma character" as shorthand for "the braille cell represented in ASCII Braille by (the same character code as) the ASCII comma character." This way of referring to the braille cells ensures that you will write the correct braille cell when entering braille from a standard keyboard.
Note that since braille doesn't have capital letters, ASCII Braille is rather random as far as sometimes using capital ASCII letters and sometimes using small ASCII letters for the braille letters. This article uses the small ASCII letters since this seems more natural.
Students who have braille notetakers generate electronic braille files in ASCII Braille or computer braille. These files are usually sent directly to a refreshable braille display or an embosser in order to convert the ASCII charcters to tactile braille. However, these same electronic braille files can also be printed on an ordinary printer or opened with a text editor for sighted viewing on a computer monitor. Select this link for information about obtaining braille files from braille-using students. Select this link for information about viewing braille files with WordPad.
It isn't necessary to have a braille keyboard to write braille. You can write braille, including braille math, using a standard keyboard and a standard text editor. To write braille math, simply enter the characters specified in this article. If you want to view the braille as simulated braille, you will need to install a compatible braille font. You can download Duxbury's SimBraille and Braille fonts for free.
ASCII Braille represents the braille letters by the corresponding print letters, the braille digits by the corresponding print digits, and 11 of the key braille mathematical symbols by the corresponding print symbols. These one-for-one representations make it easy for sighted persons to learn to read Nemeth Braille simply by using a print font to display ASCII Braille.
Let's start with some examples where Nemeth braille can be read off correctly using the ASCII transliteration to supply meaning. These examples includes two symbols that require some explanation.
First, numbers in Nemeth braille are sometimes written with a preceding hashmark character, #. The purpose of the hashmark character is to make the braille cells for numbers easier to read and can be safely ignored as far as interpreting.
The second unusual symbol is the equals mark which is period character, k character. (This looks funny in print but the braille dots actually resemble a print equals mark.) Note that the equals mark and other comparison symbols in Nemeth braille must be both preceded and followed by a space. However, the majority of symbols should be left unspaced from the adjacent symbols.
The following are all samples of Nemeth braille that you should be able to read directly. Hover your mouse over the braille to view the corresponding print.
#1+2 .k #3
x .k #0
#9.8-7.6 .k #2.2
#4*5 .k #20
#3(a+b) .k #3a+3b
f(x, y)
#10's place
As the examples in the previous section illustrate, the ASCII transliteration of Nemeth braille supplies the correct meaning wherever the 26 letter characters and the 10 digit characters are used in math expressions. The ASCII transliteration also supplies the correct meaning for the special characters in this table.
| ASCII Character | Meaning of Braille Cell in Nemeth Math |
|---|---|
| + | plus character represents cell for plus sign |
| - | hyphen-minus character represents cell for minus sign |
| * | asterisk character represents cell for times (dot, not cross) |
| / | slash character represents cell for horizontal fraction line of any length |
| . | period character represents cell for decimal point in a numerical item |
| ( | left parenthesis character represents cell for left parenthesis |
| ) | right parenthesis character represents cell for right parenthesis |
| # | hashmark character before numerical item represents numerical indicator |
| , | comma character after mathematical item, always followed by a space, represents cell for comma separator in math | ' | apostrophe character represents cell for apostrophe |
| ^ | superscript indicator |
The Nemeth code specifies braille representations for more than 400 mathematical symbols and other special characters as well as for the almost 200 letters of the Greek, Hebrew, Russian, and Fraktur alphabets. This means that the Nemeth code has to have some sort of system for dealing with all these different characters while limited to only 63 braille cells.
The two most common methods for representing a large number of different characters in other systems with a limited number of characters are numerical character codes, as used in MathML, and mnemonic abbreviations, as used in LaTeX.
\alpha. These can be easy to remember but they also have too many characters for convenient tactile reading. Another problem is that any special characters,
such as the slash, that are required to distinguish menomics from other items add unnecessary clutter. Since neither character codes nor print-based mnemonics are useful approaches for representing characters in braille math, braille math uses a special prefix-root system. The Nemeth representation of Greek letters is an example. Nemeth uses the braille cell dots-46 (represented by the ASCII period character) as a prefix before a letter root to indicate that the letter is to be understood as a small Greek letter according to a specified transliteration. The braille prefix thus acts in a similar manner to the use in a math entry system of a special keyboard shortcut, e.g. ALT-G, to signal that a Greek character is intended. With this system there a few important mathematical symbols, e.g. the integral sign, which are represented by simply a single-cell root. However, most symbols are represented have both a prefix and a root. The next section describes examples of some common Nemeth symbols.
| Character(s) in braille symbol | Interpretation | Standard print glyph |
|---|---|---|
| & | factorial | ! |
| ! | integral sign | ∫ |
| > | square root | √ |
| \ | vertical bar | | |
| .k | equals | = |
| @+ | vee | ∨ |
| .+ | union | ∪ |
| @( | left bracket | [ |
| @) | right bracket | ] |
| .( | left brace | { |
| .) | right brace | } |
| ,= | infinity | ∞ |
| @,p | paragraph symbol | ¶ |
| @,r | Crossed R, prescription | ℞ |
The table to the right shows the Nemeth representation of some common mathematical symbols. The first four symbols are represented with a one-cell root, the next eight with a one-cell prefix plus a one-cell root, and the final two with a two-cell prefix plus a one-cell root.
You might be wondering why the braille cell for the factorial sign is mapped to the ASCII ampersand character instead of the ASCII exclamation point character. There are two reasons. One reason is that, unlike print, Nemeth braille uses a different character for an exclamation point than for a factorial sign. The other reason is that the braille cell used for the factorial sign is the same braille cell used for the contraction for the word and in literary braille.* A helpful memory aid is factorial means multiply n and n-1 and so on.
The use of the ASCII exclamation point character for the braille cell that represents the integral sign is a purely arbitrary association. However, the cell itself resembles an integral sign, a useful item for the braille reader to know.
The integral sign is one example of the use by the Nemeth code of "visual mnemonics" where there is a relationship between the shape of the braille cells and the shape of the corresponding print glyph. This approach enhances communication between print users and braille users. The Nemeth code also uses symmetry in that pairs of symbols that are symmetric in print—e.g. a pair of parentheses or the the less-than and greater-than signs—have symmetric dot patterns in braille.
The two-cell representation of the equals sign was explained earlier.
In examining the remaining symbols we note a sort of ad hoc mnemonic system that is based partly on grouping symbols into families according to their meaning. Thus the logical sum and union characters both use the braille plus sign as a root. Also, the brackets and braces use the corresponding braille parenthesis as a root.
The at-sign character prefix indicates a variant form as does the period character prefix. The at-sign character represents a braille cell with just one dot whereas the period character represents a braille cell with two dots; perhaps the former prefix implies less variation than the latter. The period character prefix is also used before letters to indicate italic style and its use here is something like "fancy style vaguely related to italics."
The comma character prefix means capital when immediately before a letter and, more generally, "big" or "enlarged."
To be continued....
*The braille exclamation point punctuation mark is mapped to the ASCII digit six since the Nemeth code uses the same braille cell for both the exclamation point and the digit six.
This page was first posted on 4/21/2005.
Revised 11/02/2007.
Send questions to: info at dotlessbraille.org