1. My Credentials: At this point in time, the core of the Unified Braille Code (UBC) is established. It encompasses both the literary and the mathematical aspects of the Code. The Code must still be enlarged to include chemistry, physics, and engineering on the scientific side, and foreign language passages, contractions, diacritics, phonetics, and related issues on the literary side. These are not trivial matters.

I have been a member of the Objective II Committee since the inception of the UBC project in 1991. I have participated in all the discussions of that Committee, and I am intimately acquainted with the provisions of the UBC. I am also familiar with the Nemeth Code and with the BAUK math code and its near clones, which are currently the only authorized math codes in the English-speaking countries around the world. I have taught math and computer science at the University of Detroit for 30 years both at the undergraduate and graduate level, and I have used braille exclusively in the preparation and the delivery of my lectures. I have taught in every area of mainstream mathematics, and I am therefore familiar with the possibilities and the limitations of braille in all these areas. With this background of credentials, I will attempt to assess the impact of the UBC in both the literary and the mathematical environment.

2. The Literary Aspects of the UBC: The UBC is very much skewed in favor of the literary code as it is currently used in the English-speaking countries around world. Here are some of the differences that a reader would probably notice if he were reading typical prose in the UBC.

(2a) A few of the contractions to which he has been accustomed would be missing. These contractions, and the reasons for their deletion, are: "com," "dd," and "ble." The "com" contraction is represented by dots 36, and this braille character is already overburdened by other meanings, such as minus sign, hyphen (both a hard hyphen that is present in print, and a soft hyphen that is supplied by the transcriber to divide words between syllables at the ends of lines), and as one of the components of a short or a long dash. A period frequently occurs in the interior of an abbreviation, where it could be misinterpreted as the contraction for "dd." The "ble" contraction conflicts directly with the number sign which plays a prominent role in the UBC, and about which I will have more to say later. However, there is a Contractions Committee as part of the UBC project which has not yet begun to function. When it does, the final result may be that additional contractions will be deleted, that new contractions will be added, or that the rules governing contractions will be altered. Any combination of these three possibilities or no additional changes at all could be the final result.

(2b) He would notice that the practice of sequencing has been abolished. This means that the words "and," "for," "of," "the," "with," and "a" can no longer be written together without a space between them. Similarly, the words "to," "into," and "by" can no longer be written without a space before the following word. As a result, the word "into" would always be written using the "in" contraction followed by the letters "t" and "o"; the word "by" would never be contracted; and the word "to" would be represented by dots 235 and would become a lower whole-word sign subject to the same rules as the words "be," "enough," "were," "his," and "was." This change would remove all the problems that ensue from slavishly using sequences wherever possible regardless of the occurrence of a "natural pause," and would make the erstwhile "natural pause" rule itself moot. In my view, this is a positive step. Braille is a written form of the English language, just as print is, and I do not think that those who develop braille codes should take upon themselves the prerogative of altering the orthography of the English language by omitting the space between words when a space between those words is present in print. For the same reason, I do not believe that the decision as to whether to omit or to retain capitalization, which is another form of tampering with the orthography of the English language, should come within the jurisdiction of those who develop braille codes. In commenting on the sorry plight of the English language, Prof. Higgins, in "My Fair Lady," observed that "the Americans haven't spoken it for years." How ironic, then, that the Americans should be so zealous in preserving capitalization, while the British, who no doubt feel that since they "invented" the language and therefore own it, have a right to dispose of it in any way they please! This "capital" issue has been on the table now for more than 60 years with no perceptible prospect that it will soon be resolved. How likely is it, then, that issues that are more substantial will be resolved within a reasonable period of time?

(2c) The reader would notice some changes in the punctuation system. The rules for handling single vs. double quotes, inner vs. outer quotes, specific vs. non-specific quotes, and oriented vs. non-oriented quotes would require the establishment of a whole new mindset concerning quotation marks. Furthermore, the UBC proposes an additional kind of quotes called "Italian quotes." The slash would be represented by dots 456, 34. The ellipsis would become a sequence of three periods (dots 256). There would be new braille symbols for parentheses (round brackets) and brackets (square brackets). In particular, the left parenthesis would become a two-cell symbol formed by dots 5, 126, and the right parenthesis would become a two-cell symbol formed by dots 5, 345. Thus, if this paper were being written in the UBC, we would indicate that we are now in subsection (2c) by writing

"<#b;c">

which is an 8-cell construct. Legal citations which refer to sections and subsections in which letters and numbers are enclosed within parentheses would become excessively and annoyingly long. Many legal citations contain several such enclosed letters or numbers.

(2d) The reader would notice a change in how numbers are handled by the UBC. The numbers themselves would still be in the form to which he is accustomed, that is, they would be announced by the number sign and the digits would be the letters from "a" to "j". But he would notice some changes in the rules for their use. For example, the decimal point would be replaced by the period, dots 256. American readers would find that mixed numbers would no longer contain a hyphen between the whole number and the fractional part of the mixed number. A hyphen after a number in the UBC would terminate the effect of a preceding number sign rather than prolonging its effect. The ordinal endings "st" and "th" could no longer be contracted after a number as they now can be in Standard English Braille. In an expression such as "50-ish in appearance," the "sh" contraction could not be used. He would have to learn a new dollar sign, a new pound sign, a new percent sign, and a new degree sign.

(2e) In addition to italic type, there is provision for other type forms, such as boldface type, sanserif type, and underlining. There is even provision for one additional type form whose nature is specified by the transcriber in response to a particular local need.

(2f) There would also be a few additional miscellaneous changes. The ampersand could no longer be replaced by the word "and," and the at sign could no longer be replaced by the word "at." There would be a new asterisk. The transcriber would no longer be at liberty to alter the punctuation within a date. The transcriber could no longer convert Roman numbers to Arabic numbers when writing a Biblical citation, and the print form of the citation would have to be maintained in braille. The letters "p" for "page," "v" for "volume," and the "ch" contraction for "chapter" could no longer be used as they now are in bibliographic citations.

(2g) Americans would notice that mathematical symbols like "plus," "minus," "equals," and other mathematial or chemical constructs would no longer be brailled as words; they would instead be represented by the symbols provided for them in UBC.

(2h) As a result of the preceding discussion, some important questions come to mind. The first one is: Do these changes collectively constitute an improvement to the braille code? One of the strong points of the UBC is that the notation it proposes is completely unambiguous. But this is only a necessary condition for a good code; it is by no means a sufficient condition. There are many factors other than that of ambiguity which operate together in a complex manner in determining the extent to which a braille code is viable and usable. As I pointed out in (2c) above, parentheses which require 2 cells for their representation add considerable length to legal citations. Most people who read braille would come across such citations only infrequently and, although they may be momentarily annoying, could probably tolerate them well. But could a lawyer, who deals with such citations all the time? Would that lawyer consider the proposed UBC to constitute an improvement?

The next question is: How well will these changes be accepted by the braille-using community? This can only be determined by conducting a well-planned and a well-run research project. A user may selectively accept some of the changes and reject the others. But for the UBC to be a coherent code, all the changes must be accepted as a package.

For those who never use braille for mathematics or the sciences, the literary aspect of the UBC is, for them, the whole universe. But to deserve the name "Unified," UBC must employ the same symbols and the same principles in mathematics and the sciences as it does in a literary environment. The next question, then, is: Do the changes that have been made to the literary portion of the code pave the way for a viable and usable code that can handle mathematics and the sciences? To answer this question, I therefore turn my attention to the scienticic aspect of the UBC, where I feel much more at home.

3. The Interactive Nature of Mathematics: Besides reading and writing, there is an additional activity when doing mathematics, namely, that of manipulating expressions and equations in order to follow a mathematical argument or to solve a mathematical problem. The need to perform such mathematical manipulations is fundamental and vital for both blind and sighted users. Manipulation of mathematical expressions is akin to word processing. Subexpressions are identified, blocked off, and deleted. They may be inserted in another part of the main expression or into another expression. They may be replaced by another subexpression created by the user. They may be rearranged, etc. From the earliest grades on, homework assigned by math teachers has as one of its principal objectives the continual honing and improvement of this manipulative skill. Whereas word processing of literary text can be largely automated by using computer software for that purpose, mathematical expressions must be manipulated manually. This generally entails the rewriting of a major portion of the latest expression, inserting the revised subexpression at the appropriate places as the rewriting proceeds, and thereafter, it involves the reading of the revised expression to determine how the next revision should proceed. This alternation between the reading and the writing of mathematical expressions characterizes the interactive nature of working with mathematical notation. One of the major problems with UBC is its sheer mass. By the examples below, I will show that UBC makes the activities of reading, writing, and manipulating mathematical notation so burdensome for the braille user that it is impractical as a tool for serious work in mathematics.

4. "The Group of Six": In what follows, I will refer to the following group of 6 symbols as "the Group of Six." These symbols are: the plus sign, the minus sign, the equals sign, the general fraction line, and the left and the right parenthesis. In mainstream mathematics, these six symbols collectively constitute about 90 percent of mathematical notation, apart from letters and numbers. Yet in UBC, each of these symbols is assigned a 2-cell representation.

Example 1: (57 cells)
"7 ((f"./g")"-(f"./g")./,.dx)

In words: "equals, begin complex fraction, begin simple fraction, f of x plus delta x over g of x plus delta x, end simple fraction, minus, begin simple fraction, f of x over g of x, end simple fraction, complex fraction line, delta x, end complex fraction."

This example is encountered early in the first semester of a college calculus course. It is used in arriving at the formula for the derivative of a quotient, and is found in every elementary calculus textbook. It contains 15 symbols belonging to the Group of Six, and every symbol in that group is represented in this example at least once. Thus, there are 15 prefixes which would not be present if 1-cell assignments had been made for the symbols in the Group of Six. These 15 prefixes make no contribution to the notational information in the expression, and thus must be regarded as nothing more than "white noise." They constitute 26 percent of the braille characters in the example, and the example itself is 36 percent larger than it would be without the 15 prefixes. If you inquire as to the reasons for making 2-cell assignments to the symbols in the Group of Six, the response will be based on philosophical and conjectural irrelevancies which, in my view, must be dismissed in order to meet the practical need for a manageable code. And now that the mathematical aspect of the UBC is complete, there still remain a few 1-cell braille characters that are as yet unassigned.

5. The Tyranny of Upper Numbers: UBC uses upper numbers, that is, the letters from "a" to "j," for representing the 10 digits of the Arabic system of numeration. In this scheme, a number sign is required to establish a numeric mode for a sequence of one or more digits and some other related symbols. Furthermore, a letter sign is required whenever a letter in the range "a" to "j" which means a letter comes into juxtaposition with a symbol affected by the number sign.

Example 2: (7 lines, 40 cells per line)
"-#b;a9#c;b9#b"6#d;a9#b;b9#c a"6b

"-#b;a9#d;b9#b"6#d;a9#c;b9#c
"-#b;a9#c;b9#c"6#d;a9#b;b9

"-#b;a9#d;b9#b"6#b;a9#c;b9#c"6#d;a9#b;b9

In words: "Line 1, cell 1; minus 2 a cube b square, plus 4 a square b cube; line 2, cell 1; a plus b; line 3, blank; line 4, cell 1; minus 2 a to the fourth power b square, plus 4 a cube b cube; line, 5 cell 15; minus 2 a cube b cube, plus 4 a square b to the fourth power; line 6, blank; line 7, cell 1; minus 2 a to the fourth power b square, plus 2 a cube b cube, plus 4 a square b to the fourth power."

This is a ninth-grade elementary algebra example in multiplication. This simple multiplication example will not fit on a standard 40-cell braille line even when it begins in cell 1. Of the seven lines occupied by this example, two are blank. On the remaining 5 lines on which there is braille, there are 27 number signs, 18 letter signs, and 10 Group of Six prefixes, for a total of 55 braille characters that supply no notational information. The number signs and the letter signs only tell the reader how to interpret the braille; the Group of Six prefixes tell him nothing at all. The number signs and the letter signs, exacerbated by the Group of Six prefixes, have now enlarged the braille text to such an extent, that a standard braille line cannot contain the example. The example has been truncated on lines 5 and 7 to indicate the points of line overflow. When I was in London last January in connection with a UBC committee meeting, I showed this example to a gentleman who teaches blind children, and asked him how he would handle the alignment of the two like terms (the terms in a cube b cube in the two partial products) as sighted children are expected to do and still get the example across the braille line. He said that he would ignore the alignment. He did not offer an alternative format. I have asked my colleagues on the UBC Project II Committee how to handle this example in the UBC, but at this writing have received no reply.

The source of the problem is the use of upper numbers. No one minds an occasional number sign or letter sign, or even an occasional cluster of these in a literary environment. They will soon pass like a short stretch of dirt road, and we will be back to smooth reading. But upper numbers spawn number signs and letter signs. In a mathematical environment, numbers and letters are so closely packed together that the number signs and letter signs that they generate constitute an infestation of otherwise clean and smooth-flowing notation; we are traveling on a dirt road all the time. In the above example, there are 28 braille characters on line 1. 12 of these, or 43 percent, are either a number sign, a letter sign, or a Group of Six prefix. I will refer to these collectively as non-notational characters. Of the 16 remaining notational symbols, 4 have a neighboring non-notational character on the right, 5 have a neighboring non-notational character on the left, and 7 have neighboring non-notational characters on both sides. The longest sequence of notational symbols is 2 cells. The other lines of this example are similarly infested by non-notational characters. This phenomenon is not specific to this example, it pervades all of mainstream mathematics of which the examples in this paper are only a microcosm. No braille user can perform meaningful robust mathematical manipulations with all these interruptions and distractions. In the heat of following a mathematical argument or of solving a mathematical problem, he requires a smooth and uninterrupted flow of notational information. Upper numbers cause misalignment.

Example 3: (2 lines, 7 cells per line)
#a;e#hi bc#d;a

This example requires the addition of the two indicated hexadecimal numbers. As is evident from the layout of this example, it would be difficult to perform this addition from the raw placement of the symbols. To solve this problem, UBC contrives an "alignment mode." The alignment mode is entered and exited by a pair of symbols whose assignment is not yet completely decided. While in alignment mode, the digits are converted either to dropped numbers or to French (dot-6) numbers. Since, after this conversion, number signs and letter signs are no longer required, the alignment is automatic and the addition can proceed. "Alignment mode," however, is nothing more than a euphemism. It says: "Since the UBC Code cannot handle alignment by the application of its own rules, leave the UBC Code and achieve the required alignment by using a different number system wherein alignment is possible. When finished, reenter the UBC Code." So now we have two number systems--one for alignment and one for "business as usual." Somehow, "Unified" got evicted from UBC.

I can think of no relevant argument to support the use of upper numbers. The only irrelevant argument that I have heard is that upper numbers are customary and traditional. No custom is worth preserving or even defending that creates so many obstacles for the braille reader, and that wreaks so much harm and mischief. Does anyone think that upper numbers will survive when the chemistry code is developed?

6. The Emancipation of Dropped Numbers: Dropped numbers have none of the disadvantages of upper numbers. They do not clutter the notation with hordes of number signs and letter signs. They do not cause the text to expand to the point at which it overflows a standard braille line. They do not create misalignments. They do not require an elaborate set of rules to specify when numeric mode is in effect and when it is not. In fact, a code based on dropped numbers has no numeric mode. Dropped numbers are a familiar, de facto, replacement for upper numbers and are just as well known and easily used. Many codes are partially or entirely based on them, and it is inconceivable that any chemistry code can be developed without them. The Appendix at the end of this paper shows how the examples throughout this paper are written in the Nemeth Code, which is based on the exclusive use of dropped numbers.

7. Reading by Inspection Vs. Reading by Deduction: Whether or not a braille user understands the mathematics conveyed by a notational expression, he must, a priori, be able to extract the information he needs from that notational expression. I will show by three examples that, whereas a sighted reader can extract the information he needs by direct inspection, the UBC reader must extract the same information by deduction. I suggest that you read the braille first, taking note of your mental processes, before you verify your comprehension of the notation by reading the print or consulting the verbalized form of the example. I also suggest that you consult the Appendix to see how the Nemeth Code handles these same examples.

(7a) Let us revisit Example 1. The purpose of my previous analysis of that example was to show what impact the Group of Six prefixes have on the size of the notation. My analysis of this example at this point is concerned with the fractions which it contains. The example contains an outer fraction, more frequently referred to as a complex fraction, which contains two subsidiary inner fractions, called simple fractions. For each of these fractions there is an indicator to tell the reader when the fraction begins, and another indicator to tell him when the fraction ends. In the UBC, there is no distinction between the indicators that initiate or terminate a complex fraction and those that initiate or terminate a simple fraction. Furthermore, the UBC does not distinguish between the fraction line that belongs to a complex fraction and the fraction line that belongs to a simple fraction. If you are reading the print version of this example, the merest glance will tell you that the main fraction line is abreast of the equals sign. It is the most prominent feature of the expression and extends across the full width of the complex fraction. In the braille version, the main fraction line is not distinguished from the subsidiary fraction lines and is buried deep within the structure of the fraction, close to its end and far away from the equals sign. The user is expected to determine which is the main fraction line by some unspecified process. In the case of fractions, no UBC rule can help him to distinguish between a simple and a complex fraction; he must self-generate a deep insight into the structure of the notation to arrive at such a distinction. Since the UBC is unambiguous, it will always be possible to identify the complex fraction line in any expression that is well-formed. But will the identification come about by merely inspecting the text, as a sighted person can do, or by analyzing the text and making deductions, which the UBC reader must do? I challenge anyone who uses the UBC to devise a simple set of instructions that a young student can follow to enable him to identify the main fraction line, not just in this example, but in the general case. And not just in complex fractions of the first order, as in this example, but in complex fractions of higher order as well. In this example, there are two consecutive fraction indicators at the very beginning of the fraction, so that the reader knows that there is a simple fraction within a complex fraction. In general, however, the simple fraction may begin much farther along in the notation, so that the reader is unaware that he is dealing with a complex fraction until much later, when he first encounters the simple fraction. My solution is to attach a prefix of some kind, such as dot 6, to the begin-fraction indicator, the end-fraction indicator and the fraction line in the case of a complex fraction, and omit this prefix from the corresponding components of a simple fraction. Under such a scheme, the reader would immediately know that he is dealing with a complex fraction; then he could simply scrub along until he found the fraction line with the attached prefix; that is the complex fraction line. It partitions the complex fraction into its numerator and denominator which could, in turn, be subjected to the same procedure. My colleagues rejected this approach.

(7b) Next, I offer a set of four short examples which deal with the relation between superscripts and fractions.

Example 4a (18 cells)
x9(#a./#bn"6#a)"6y

In words: "x sup, begin fraction, 1 over 2 n plus 1, end fraction, end sup, plus y."

Example 4b (13 cells)
x9<#a/bn"6#a>

In words: "x sup, one-half n, plus 1, end-sup."

Example 4c: (11 cells)
x9#a/bn"6#a

In words: "x sup one-half, end-sup, n plus 1."

Example 4d (17 cells)
x9<(#a./#bn)"6#a>

In words: "x sup, begin fraction, 1 over 2 n, end fraction, plus 1, end-sup."

Were you reading the notation by inspection, or were you deciphering the notation by applying UBC rules to determine when a superscript or a fraction begins and ends? In the UBC, the beginning and end of a superscript is sometimes implicit, as in Examples 4a and 4c, and sometimes they are explicit, as in Examples 4b and 4d. Sometimes the beginning and the end of a fraction are implicit as in Examples 4b and 4c, and sometimes they are explicit as in Examples 4a and 4d. Of course, there are well-formed rules for determining when either a superscript or a fraction is delimited implicitly or explicitly. However, when we are dealing with notation in which superscripts contain fractions, the number of possibilities multiplies. Thus, we could have a superscript and a fraction which are both delimited explicitly, both delimited implicitly, or the two situations in which one is delimited explicitly and the other implicitly. In the above four examples, each explicit/implicit superscript/fraction combination is represented exactly once. See if you can identify them. (7c) Here is an example taken from the UBC final report which was released in March, 1995.

Example 5: (36 cells)
e99"6y59>

In words: "e has a superscript consisting of an indexed x plus an indexed y. x is indexed by a subscript of i plus 1 and a superscript of p sub i. y is indexed by a subscript of j plus 1 and a superscript of q sub i."

The purpose of this example is to show that the UBC is completely unambiguous, even in an example of this complexity. The example is a convincing demonstration of this fact. To see how much inspection and how much deduction is required, let us find the plus sign that precedes y and ask the following questions: At what level is y? Is it a subscript or a superscript, and what symbol does it index? Are there other symbols at the same level as y?

Between the superscript indicator that affects y and the symbol y itself there are 18 cells of braille containing 3 indications of level changes, of which one is implicit. Could you remember all the level changes that occurred from the beginning of the example until you reached y? The enclosure mechanism which the UBC uses for explicitly delimiting superscripts or subscripts is a modification of a popular computer program used for controlling typesetting machinery that produces well-formatted mathematical printing. Typesetting equipment can easily keep track of an unlimited number of level changes, but the span of attention of a human being is limited and his memory is fallible. You cannot answer any of the questions above about y unless you start from the beginning of the notational expression and keep an accurate and complete mental history of all the level changes that have occurred up to the point of interest. In the Nemeth Code, all superscripts and subscripts are relative to the base line.

In summary, the purpose of this Section 7 is to point out that there are important areas of notation, namely fractions and indices (subscripts and superscripts), in which the UBC reader cannot grasp the notation by simple inspection, as a sighted reader can, but that he must use deductive reasoning, based partly on UBC rules and partly on intuition, to construct a mental image of the notation under his fingers. Only then can the reader extract the mathematical message from that mental image. With the need to read important notational passages deductively rather than by inspection, the reader is not operating on a level playing field. He cannot read, write, or manipulate mathematical expressions with the ease of his sighted colleagues. My long years of experience in dealing with mathematics have taught me that, to avoid reading by deduction, all delimiters must be explicit.

8. UBC and Literacy: As you may know, there is a sustained effort by the major blindness organizations of and for the blind in the United States to get braille literacy bills passed by the various state legislatures. At the present time, about 25 states have enacted such bills which are now law. In 1991, Gov. John Engler of Michigan appointed me to serve as chairman on the Michigan Commission for the Blind. In the two years that I served in that capacity, I had many contacts on many issues with both state senators and representatives. Many of those contacts were concerned with my advocating the passage of a braille literacy bill in Michigan. If I were an opponent of a braille literacy bill, I could bring no more potent a weapon than a few examples written in the UBC to convince a legislator of how clumsy, how unmanageable, how difficult, and how stressful that Code would be to blind children, and that other means of acquiring information, such as tape recorders, computers, live readers, and, where applicable, closed-circuit TV devices would be more appropriate. Two typical examples that I might use to make my point are the following:

Example 6a: (16 cells)
"<"-#a">9

In words: "minus 1 to the k plus 1 power."

Example 6b: (10 cells)
("-#a./#b)

In words: "the fraction minus 1 over 2."

The use of braille has been declining to dangerous levels of non-use in recent years, and there is a real concern as to the future of braille. The UBC Code can only elevate the intensity of that concern.

9. Conclusions: At the end of my evaluation of the impact that the UBC would have on the literary aspect of the Code, I posed the question as to whether the changes proposed by the UBC constituted an improvement over the current literary code. The answer was a definite "maybe." If, indeed, there is any improvement, it certainly is not incremental. But now I ask the same question regarding the mathematics aspect of the Code. This time the answer is a resounding "no." An expenditure of thousands of dollars to conduct an evaluation, and a delay of months while the evaluation is being conducted are not needed to conclude that 6 cells for enclosing a digit or a letter within parentheses, that 10 cells for writing "the fraction minus 1 over 2," and that 16 cells for writing "minus 1 to the k plus 1 power" are unreasonable by any criterion of reasonableness. Nor is an expensive and time-consuming evaluation required to conclude that when a simple elementary algebra multiplication problem overflows the braille line, or that when two 4-digit hexadecimal numbers cannot be added by using the Code's own rules, that something is seriously amiss with the UBC. I have submitted my reasons in painstaking detail in the preceding sections. The insistence on the use of upper numbers which infests the notation with hordes of number signs and letter signs is the principal reason. Not far behind is the mischief that the Group of Six creates. With a Code that employs these two mechanisms, we have invited "an elephant into the kitchen," and all normal activity must come to a halt. If this Code is adopted, the clock will have been turned back two or three generations when the predominant perception was that if you are blind, you cannot and need not concern yourself with robust and creative mathematics; the best you can hope for is to perform routine and trivial computations and manipulations. I have heard all the philosophical arguments, and all the rationalizations, and all the conjectures, and all the irrelevancies, and all the fictions, and all the statistics in support of the UBC in general and its various mechanisms in particular. The real world of notation, of which the examples in this paper are representative, and to which I have confined myself as the basis for establishing my position, belies them all. And now ICEB is proposing to subject this Code to an evaluation process. If a commercial enterprise had a product with known defects as serious as those in the UBC, and then proceeded to test market that product, it could rightfully be accused of being intellectually dishonest. After more than 60 years of papers, conferences, meetings, and all the other methods of communication known to modern man, it is time to come to an agreement as to how to write "two plus two equals four."

APPENDIX

Digits
All digits are dropped

The Group of Six
+ (dots 346) plus
- (dots 36) minus
.k (dots 46, 13) equals
/ (dots 34) fraction line
( (dots 12356) left parenthesis
) (dots 23456) right parenthesis

Fraction components
? (dots 1456) begin-fraction indicator
# (dots 3456) end-fraction indicator
/ (dots 34) fraction line
, (dot 6) complexity of fraction indicator

Level indicators
~ (dots 45) superscript indicator
; (dots 56) subscript indicator
" (dot 5) base-level indicator

Miscellaneous
.,d (dots 46, 6, 145) upper-case delta

EXAMPLES

Example 2: (31 cells)

-2a~3"b~2"+4a~2"b~3
a+b
3333333333333333333333333333333
-2a~4"b~2"+4a~3"b~3
-2a~3"b~3"+4a~2"b~4
3333333333333333333333333333333
-2a~4"b~2"+2a~3"b~3"+4a~2"b~4

Example 3: (4 cells)

1e89
bc4a

Example 4a: (13 cells)
x~?1/2n+1#"+y

Example 4b: (10 cells)
x~?1/2#n+1

Example 4c (11 cells)
x~?1/2#"n+1

Example 4d: (10 cells)
x~?1/2n#+1

Example 5: (30 cells)
e~x~;i+1~~p~~;i~+y~;j+1~~q~~;i

Example 6a (8 cells)
(-1)~k+1

Example 6b: (6 cells)
?-1/2#

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