Representing an augmented matrix in Nemeth braille

[Donald Winiecki]

I am in production-mode on college-level maths materials for a math/electrical engineering dual major.

I am presently working content including augmented matrices. I find no guidance in provisional Nemeth training materials, or the 1972 and subsequent updates on Nemeth code for how to represent the vertical bar inside an augmented matrix.

Should this bar be represented as a vertical column of simple pipes (dots 456) down through the matrix, or should this be represented with the multi-row bar used for determinants (dots 6,1256), or something else entirely?

I am attaching a picture of a small augmented matrix, in case the Nemeth terminology varies from vocabulary being used in the class.

Best,

_don winiecki

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[Lindy Walton]

Hi Don. The vertical bar you are looking for is called an Enlarged Sign of Grouping in the Nemeth Code. You can find the enlarged single vertical bar listed in the table on page 123 of the 1972 codebook. Its use is illustrated in a determinant in example (2), Section 126, "Use of Enlarged Grouping Symbols".

Let me know if you have further questions. You have a big project ahead of you.

Lindy

[Donald Winiecki]

Thanks very much Lindy!

I thought that I should be using the enlarged vertical bar, but the example shown in section 126, example 2 is different from what appears in an augmented matrix.

In an augmented matrix the vertical bar appears inside the matrix itself as a dividing line between two matrices combined into one form.

Another NBA-er found the following: https://www.nationalbraille.org/forums/topic/vertical-line-within-a-matrix/

Is the above-noted forum advice still relevant?

Best,

_don

[Lindy Walton]

Thank you for finding that previous post from 2013! I was unfamiliar with the term "augmented matrix". I see now that the vertical line in an augmented matrix is not being used as a grouping sign, but rather as a separator. I think the best solution would be to draw the line as a tactile graphic, but if that option is not available to you, using the (456) "vertical line" makes sense. This is not the (1256) "vertical bar" (which has mathematical meaning). This "column separator" usage is similar to the vertical line used in a stem-and-leaf plot which became an option for us in the 2007 BANA Update. You'll note that there is a space before and after the vertical line when used in this manner. Let me know what you think.

Lindy

[Donald Winiecki]

Yes, the vertical line (dots 456) with preceding and following space is both tactually distinctive for the student (and instructor, who has no existing knowledge of braille) and convenient for me to produce. I will standardize on that symbol for augmented matrices, and include a transcriber's note on the T-note page to indicate this usage.

While it may be TMI, the process is for me to produce the materials in BRF and then make them available for the student in a GoogleDrive folder. She can then emboss them at her location on her own embosser. This would have been okay before our current non-face-to-face situation, but it is absolutely essential now.

Thanks muchly!

_don

[William Wheeler]

As a mathematician who is learning to author braille documents, may I please suggest that the 1256 symbol is more appropriate for augmented matrices than the 456 symbol, both because of the mathematical meaning of the vertical line in an augmented matrix (where it is a relation symbol with mathematical meaning, not merely a separation symbol) and because of mathematical typesetting practice, for instance in TeX and LaTeX.

The augmentation line in an augmented matrix denotes an = relation in the linear equations that correspond to the augmented matrix.  For example, the augmented matrix for the equation  2x + 3y = 4 is  [2  3 | 4].  The augmentation line indicates that there is an equality relation between the left side of the equation whose coefficients are 2 and 3 and the constant 4 on the right side of the equation.

In Don's png, there are two columns on the right hand side of the augmentation line because the augmentation matrix there is being used to find the multiplicative inverse of the matrix on the left side, so the top line  [1  2 | 1  0] indicates that one is really concerned with both the equation  x + 2y = 1  and, independently, the equation  x + 2y = 0.

On page 230 of The Nemeth Braille Code for Mathematics and Science Notation, 1972 Revision, Nemeth indicates that the 1256 symbol is used as (i) a grouping symbol, (ii) a relation/comparison symbol "(is a factor, divides)",  and (iii) for "such that" in set expressions, for instance, examples (13) and (14) on page 144 of the book.

In mathematical typesetting, e.g., TeX and LaTeX, the vertical bar as a grouping symbol is denoted by the macro \vert, and the vertical bar as a relation symbol (e.g., is a factor, divides) and as the "such that" symbol in set expressions is denoted by the macro \mid.  In the typesetting, both \vert and \mid produce the same print symbol, namely a vertical line, but no space is inserted on either side of \vert (because it is a grouping symbol) but extra space is inserted on both left and right sides of \mid (because it is a relation symbol like = and <).   (These spacing conventions are the same in both mathematical typesetting and braille.)

For these reasons, may I please suggest that 1256 symbol with a space on either side is the better transcription of the mathematical meaning of the vertical augmentation line in an augmented matrix.

Sincerely,

William Wheeler, Dept. of Mathematics, Indiana University, Bloomington, IN

• This reply was modified 3 years, 11 months ago by William Wheeler.

[Donald Winiecki]

Thank you for that information William!It is definitely helpful to have assistance from a mathematician on this.

If I understand your message properly, while the 2346 symbol is used in both augmented matrices and in set notation, the way to interpret a difference in meaning between 2346 in an augmented matrix, and 2346 in set notation is accomplished by surrounding spaces. To signify the proper meaning in an augmented matrix, spaces always separate the 2346 indicator and any members of the matrix. To signify the proper meaning in set notation there is no space between the 2346 indicators and members of the set themselves.

[William Wheeler]

Thank you for this response, Donald.

My effort to be succinct may have caused confusion.  In set notation, the 1256 indicator does not, in my experience, normally appear between the members of a set but rather as an abbreviation for the phrase "such that" that separates a variable expression from the defining condition for the set.  In English print, the expression  "{x | x is an integer between 1 and 10}" would be read as "the set of all x such that x is an integer between 1 and 10".   The vertical line | stands for the phrase "such that".  This instance of vertical line symbol | should have extra space on either side of it.   (Nemeth:  page 136; page 143, section 145; page 144, examples 13 and 14.)

 

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