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The Nemeth Braille Code for Mathematics

Introduction

The Nemeth code is based primarily on the presentation, appearance, or notation of the mathematics, in contrast to semantics. As Dr. Nemeth has written,

A Nemeth Code transcriber need not be proficient in mathematics; all that is required is to look up the symbols and follow the rules. That is what has attracted so many transcribers and what accounts for such a large collection of braille books in math and other natural sciences. Braille Into the Next Millennium

One of many advantages of the Nemeth code's being based on the appearance of printed mathematics is the simplification of communication between print-reading and braille-reading colleagues.

There is also a spoken form of Nemeth. Dr. Nemeth told me that he could train a sighted reader with no knowledge of mathematics to use this spoken form in about 15 minutes. He developed this approach so that he could enter braille while the reader was speaking. For example, for 1/2, the reader would say, "Start fraction, one, slash, two, end fraction."

Nemeth is a comprehensive six-dot braille code that is designed to cover all levels of mathematics from arithmetic to the most advanced graduate level. This includes, in addition to arithmetic, algebra, geometry, trigonometry, functional analysis, ordinary and vector calculus, differential equations, linear algebra (matrices, determinants, and systems of equations), group theory, set theory, number theory, symbolic logic, topology and much more.

In addition to linear representations, the Nemeth code addresses intrinsically planar (spatial) representations for adding columns of numbers, long division, matrices, tables, etc. Planar material should be paper braille because of its two-dimensional nature. It is crucial to appreciate that no one can understand mathematics, including graphs, without the ability to understand planar concepts. Structural formulas in chemistry are also planar. Dr. Nemeth likes to demonstrate to first graders how to set up addition problems on a Perkins. It is supposedly a bit easier with a Mountbatten.

Nemeth braille is an add-on to contracted braille (English Braille American Edition) so the text portions of technical books are written as in Grade 2 with a few modifications for consistency with the needs of the Nemeth code for technical material. Some modifications might be required for the text in languages other than English.

The Nemeth Rules

This is an overview of Nemeth to give you the basic idea of what it covers. There is is not enough detail to start transcribing. The standard Nemeth reference The Nemeth Braille Code for Mathematics and Science Notation, 1972 Revision comprises 250 large pages in print. There are 25 Rules illustrated with numerous examples.

Rule 1. Indicators

The Nemeth code, like other braille transcribing codes, is a markup language. Rule 1 lists the markup symbols, which are referred to as indicators in braille terminolgy. Braille markup, unlike print markup, is intended for the end reader rather than as a means to control rendering and is thus much terser than print markup. The markup symbols are designed to be easy to remember.

One type of markup is comprised of compostion indicators similar to print mark-up tags for special typeforms and other variants. A second type of markup provides templates for constructs like fractions; this is similar to print mathematical markup languages like LATeX.

Rule 2. Numeric Signs and Symbols

Nemeth uses lower numbers to represent the digits. The first digit of a number is prefaced by a Numeric Indicator (number sign) where the alignment of the lower cells might be difficult for the tactile reader to identify quickly, such as after a space, or where there would otherwise be ambiguity. The Numeric Indicator is thus used when necessary to avoid confusion with certain cells that are used as numbers and as literary punctuation marks. (Punctuation marks are occasionally prefaced by the dots-456 punctuation indicator for similar reasons.)

This rule also addresses the brailling of non-decimal numbers.

Rule 3. Capitalization

Capital letters are preceded with the dot-6 capitalization indicator. In eight-dot braille one could replace this rule with an equivalent rule for adding dot-7 to cells.

Rule 4. Alphabets

The Nemeth braille code provides for four alphabets in addition to the default Latin alphabet: German, Greek, Hebrew, and Russian. Letters in special alphabets are represented by using an alphabetic indicator before the letter or letter sign to indicate a transliteration from the Latin alphabet. For example, the ordinary lower-case Greek gamma is two cells, the Greek alphabetic indicator (dots-46) followed by the letter g (dots-1245). Letters in the Latin alphabet do not require the English Letter Indicator unless they are set in special type.

Rule 5. Type Forms

Plain type does not require an indicator but indicators are used for four special Type-Forms: Boldface, Italic, Sans Serif, and Script. The Boldface indicator is dots-456 and the Italics indicator is dots-46. The other two indicators use two-cell symbols: Sans Serif is dot-6 followed by dots-46 and Script is dot-3 followed by dots-56.

A boldface, Greek, capital letter thus requires four cells.

Where it is not obvious that a letter is intended as a mathematical symbol, it is preceded by the English Letter Indicator, dots-56, as in literary braille. (This convention is especially important for braille codes that use single letters as contractions.)

Typeform indicators are only used where the print typeform is mathematically significant such as Boldface for vectors or Script for sets.

Rule 6. Punctuation signs and symbols

The punctuation marks are mostly the same as in literary braille. Dot-6, rather than dot-2, is used as a comma separator between mathematical items. The punctuation indicator is used when necessary to avoid ambiguity.

Rule 7. Reference signs and symbols

There is a special notation for footnote reference marks and symbols for nine reference symbols including the paragraph and section marks.

Rule 8. Abbreviations

Discusses necessary modifications to contracted (Grade 2) braille for transcribing abbreviations.

Rule 9. Contractions and Short-Form Words

Discusses necessary modifications to contracted (Grade 2) braille for contractions and short-form words.

Rule 10. Omissions

Explains how to transcribe problems like 3 + ? = 5. Nemeth uses the general omission symbol, dots-123456, to replace the question mark used as a sign of omission. The long dash, four repeated dot-36 cells, is used for underscores for fill-in-the-blank problems. Use of ellipses and of small shapes to represent omissions is also addressed.

Rule 11. Cancellation

Describes the braille format for planar displays showing where numbers have been cancelled out as in reducing fractions or to show carrying in subtraction.

Rule 12. Fractions

There are start and end indicators for three kinds of fractions: simple, complex and hypercomplex. There is also a special notation for the fractional part of a mixed number.

A complex fraction is a fraction where the numerator, denominator, or both, is a simple fraction. A hypercomplex fraction is one where the numerator, denominator, or both, is a complex fraction.

The horizontal fraction (division) line is dots-34 while the diagonal slash is dots-34 prefixed by dots-45..

Rule 13. Superscripts and Subscripts

Superscripts are indicated with the superscript indicator (dots-45) and subscripts with the subscript indicator (dots-56). Parentheses are not used with superscripted and subscripted expressions since the return-to-baseline indicator (dot-5) shows the extent of the expression. The subscript indicator is not used with chemical formulas. (There is a new add-on to Nemeth for structural formulas in chemistry and other special items.) There are simple rules for superscripts to superscripts, etc.

Superscript and subscript notation is not just for exponents and indices but for any situation where adjacent expressions are raised above or below the line such as one method for typesetting the limits of an integral.

Rule 14. Modifiers

A common mathematical notation is the modifier, which is a symbol that appears directly over or directly under another symbol, such as the small arrow sometimes used above a letter that represents a vector or the limits of a sum shown below and above a capital Greek sigma. There is a seven-step rule for brailling modified expressions.

1. The dot-5 multipurpose indicator starts the modified expression
2. Then comes the symbol or expression being modified
3. The dots-146 directly-under indicator is next if there is a modifier is directly under the main symbol. (If not, this and the next step are omitted.)
4. Then the expression for the subscribed modifier.
5. The dots-126 directly-over indicator is next if there is a modifier is directly over the main symbol. (If not, this and the next step are omitted.)
6. Then the expression for the superscribed modifier. For example a simple right-pointing arrow is dots-1246 followed by dots-135 (note that dot pattern resembles a print arrow).
7. The dots-12456 terminates the modified expression.

Rule 15. Radicals

The radical notation includes the radical symbol, dots-345, and the termination symbol indicating the end of the expression under the radical, dots-12456. There is special notation for radicals with indices, such as the cube root, and indicators for nested radicals.

Rule 16. Shapes

Rule 16 describes the simple markup language used for small shapes, including circles, triangles, diamonds, squares, hexagons, etc. The language is designed to be extensible in case there is a need to braille a new shape. Shapes can be indicated as either open, filled-in, or shaded. Shapes are identified by the letters, e.g. c for a circle and d for a diamond and, in the case of regular polygons, by numbers, e.g. 6 fora regular hexagon.

Shapes can also have interior modification such as a a letter inside of a circle or other shape.

The shape notation is extended for geometry and trigonometry to include symbols for angles including right angle, obtuse angle, etc. and special forms to show adjacent angles, complementary angles, etc.

Rule 17. Function names and their abbreviations

Shows how to express trigonometric and other standard functions—log, max, min, etc.—and the format for explicit function arguments.

Rule 18. Signs and symbols of grouping

Symbols for parentheses, brackets, braces (accolades), bars, angle brackets, and many more. These symbols come in symmetric pairs as in print.

Rule 19. Signs and symbols of operation

Ordinary binary operations—plus, minus, multiply, and divide—as well as operations used in set theory (union, logical product, etc.) and other special areas of mathematics.

The plus sign is dots-346 and the minus sign is dots-36. The multiplication dot is dots-16 and the multiplication cross used for a Cartesian product is dot-4 followed by dots-16. Just as in print, the multiplication sign is often omitted in linear notation.

Rule 20. Signs and symbols of comparison and relation

Includes equals, greater than, less than, parallel to, etc. as well as modified forms such as "approximately equal." Also includes set theory relations such as logical sum, logical product, inclusion, etc. There are several hundred of these symbols!

Rule 21. Arrows

A unqiue method for arrow shapes that shows the arrow direction, arrow shaft type (ordinary, double, wavy, dotted, etc.) and arrow head type (barbed full, barbed right lower, blunted full, curved,plus about 25 others).

Rule 22. Miscellaneous Signs and Symbols

More than 25 symbols including caret, check mark, dollar sign, factorial, infinity, integral, and per cent plus a method for defining new symbols.

Rule 23. Multipurpose Indicator

Rules for using dot-5 as a visible to avoid ambiguity in certain situations.

Rule 24. Spatial (planar) Arrangements

Long division, addition and subtraction, multiplication, taking the square root, determinants and matrices, systems of equations, etc.

Rule 25. Format

Special formats for tables, spatial arrangements, labeled examples and theorems, layout for proofs, etc.

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This page was first posted November 30, 2002.