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BackNem 2.0 Implementation: Part II
Backtranslation of Nemeth Math

This webpage is the second part of a two-part article which presents a detailed description of the approach that the BackNem 2.0 application uses to backtranslate Nemeth braille text and math and English Braille American Edition (EBAE) text to print. The first part covers the basic organization of the application, the methods used to distinguish text items from math items, and the methods used to backtranslate text items. (A significantly shortened technical description of the entire application is also available. )

This article is currently incomplete; please check back later for more information. Meanwhile, there is a very brief summary at the link.


The Nemeth code is a unified code for representing both text and math. A more technical term that means the same thing as text is narrative material and a more technical term that means the same thing as math is notational material. A good starting point to read more about the Nemeth code is this online copy of a speech titled Nemeth Code: A Historical Perspective given by Caryn Navy.

The first part of the present article dealt with the handling of narrative material. This second part deals with the handling of notational material. Nemeth braille includes rules for transcribing both inline and displayed notational material and for both linearand planar arrangments. Inline math is usually linear whereas displayed math can be either. The next section outlines how planar math is handled in BackNem 2.0.

Planar Math or Spatial Arrangements

Planar math or spatially arranged math is math that is laid out on a page in two dimensions. Planar math encompasses arithmetic and higher math.

The main difficulty with backtranslating Nemeth spatial arrangements is not the backtranslation to print per se but, rather, determining exactly what one is dealing with, especially given the possibility of Nemeth errors. BackNem necessarily assumes that spatial arrangements are separated from other material by the required blank lines both above and below. The following sections address the limitations of the current version of BackNem as far as backtranslating spatial arithmetic and higher-level planar mathematics.

Spatial Arithmetic: Introduction

Spatial arithmetic refers to planar arrangements of addition, subtraction, multiplication, and division problems. Learning The Nemeth Braille Code, the famous book by Ruth H. Craig, gives numerous examples of spatial arithmetic as does Sections 178-182 of Rule XXIV of the Nemeth Code Book.

Distinguishing Nemeth spatial arithmetic from narrative material requires considerable care because spatial arithmetic uses numbers without a preceding Numeric Indicator. This means that some numbers can be confused with certain of the whole-word lower-sign contractions.

BackNem 2.0 avoids this problem by limiting its handling of spatial arithmetic as described in the next section.

If the current handling of spatial arithmetic in BackNem doesn't meet your needs, I'm happy to accommodate specific requests for extensions. However, there are other alternatives.

If you have an electronic braille file that contains spatial arithmetic, you can just print it out using a monospaced print font. Sighted persons shouldn't have any problem learning to read simple Nemeth, such as spatial arithmetic, directly, without the need for backtranslation. This article gives more detail.

However, if you want to backtranslate scanned hard copy embossed braille or need other capabilities not available in BackNem, please read about WInsight.

Spatial Arithmetic: Limitations in BackNem 2.0

The current version of Nemeth backtranslates spatial addition, subtraction, and multiplication problems but not long division problems and not any type of problems involving cancelled number indicators. Use of numerical values consisting entirely of three or more 3's or three or more 7's will cause problems since BackNem isn't currently "smart" enough to distinguish the use of the corresponding braille cells as digits from their use as separation lines and as carried number indicator lines, respectively.

Spatial arithmetic may not include extraneous material with the exception that problems may optionally be numbered with main division numbers according to Rule XXV, Sec. 192, a. of the Nemeth Code Book. Subdivision numbers are not allowed.

The syntax of a main division number must consist of a Numeric Indicator followed by a one- or two-digit integer number followed by a Punctuation Indicator and a period punctuation mark. Other forms of the main division number are not allowed.

Addends, subtrahends, minuends, multiplicands, and multipliers can include optional commas and decimal points in addition to digits. All of these with the exception of multipliers can also include optional dollar signs. None of them can include units or any other items other than those specifically mentioned.

The plus sign of operation is optional for addition problems but the minus sign and multiplication cross are required for subtraction and multiplication problems, respectively.

Higher-level Planar Mathematics: Introduction

Sections 182-183 of Rule XXIV of the Nemeth Code Book specifies spatial arrangements for three constructs of higher mathematics: matrices, determinants, and unified expressions. All of these constructs employ special braille symbols, known as enlarged grouping symbols, as grouping symbols.

BackNem 2.0 doesn't have much trouble dealing with higher-level planar mathematics that is properly formatted and otherwise error-free. There are two reasons for this. First, the symbols that the Nemeth code uses for the enlarged grouping symbols of planar mathematics are distinct from those used for ordinary grouping symbols. Second, the same approach used to distinguish the semantics of spaces in linear math can be used to distinguish spaces that separate matrix elements from spaces with other uses.

Higher-level Planar Mathematics: Restrictions

BackNem 2.0 places the following restrictions on its handling of matrices and determinants:

  1. Any spaces within elements must be in accord with the following restrictions:
  2. Matrix elements may not contain other matrices or determinants, spatial fractions, comparison symbols, or spelled-out function names
  3. Each row, including its enlarged opening and closing grouping symbols, must fit on a single line of braille; in other words, matrix elements may not be run over to another line
  4. Each row must have the same number of elements
  5. Rows may not be empty (and thus represented by a line of dot-3 cells )
  6. The keying technique may not be used

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